PROBLEMS SOLVED BY GREEDY METHOD

Presented By-NEHA JAINIII-BTech CSE

The 2-terminal one to any special channel routing problem

The 2-terminal one to any special channel routing problem

DefGiven a set of terminals on the upper row and another set of terminals on the lower row we have to connect each marked upper terminal to the marked lower row in a one to one fashion This connection requires thatNo two lines can intersectAll lines are either vertical or horizontal (track)

2 feasible solutions

ACTUAL PROBLEM

TO minimize the of tracks

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

The 2-terminal one to any special channel routing problem

The 2-terminal one to any special channel routing problem

DefGiven a set of terminals on the upper row and another set of terminals on the lower row we have to connect each marked upper terminal to the marked lower row in a one to one fashion This connection requires thatNo two lines can intersectAll lines are either vertical or horizontal (track)

2 feasible solutions

ACTUAL PROBLEM

TO minimize the of tracks

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

The 2-terminal one to any special channel routing problem

DefGiven a set of terminals on the upper row and another set of terminals on the lower row we have to connect each marked upper terminal to the marked lower row in a one to one fashion This connection requires thatNo two lines can intersectAll lines are either vertical or horizontal (track)

2 feasible solutions

ACTUAL PROBLEM

TO minimize the of tracks

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

2 feasible solutions

ACTUAL PROBLEM

TO minimize the of tracks

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

ACTUAL PROBLEM

TO minimize the of tracks

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

SIMPLIFIED VERSION

At each point the local density of the solution is of lines the vertical line intersects

The problem to minimize the density The density is a lower bound of of tracks

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Redrawing solutions

Optimal solution

(density =1)

Another solution

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Reaching the solutionhellip

Upper row terminals P1 P2 hellip Pn from left to

right Lower row terminals Q1 Q2 hellip Qm from left to

right m gt n It would never have a crossing connection

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Assumption

Suppose that we have a method to determine minimum density d of a problem instance

Now we have to solve our problem greedilyhellip

If you were greedy you would always desire to have the minimum density (Agreed)

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

THE GREEDY ALGORITHM

Step 1 P1is connected Q1 Step 2 After Pi is connected to Qj we check

whether Pi+1 can be connected to Qj+1

If the density is increased to d+1 try to connect Pi+1 to Qj+2

Step 3 Repeat Step2 until all Pirsquos are connected

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

How greedy method helps

We determine a minimum value parameter (local density)

This is the value we try to optimize and then use the greedy method straightforwardly

The greedy algorithm connects the terminals greedily

At any time our algorithm ensures that the resulting density does not exceed d

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Minimum cooperative guards problem for 1-spiral polygons

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

ART GALLERY PROBLEM

Solution of the art gallery problem

We are given a polygon which represents an art gallery and we are required to place a minimum number of guards in the polygon such that every point of the polygon is visible to at least one guard

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Minimum cooperative guards problem

The minimum cooperative guards problem puts more constraint on the art gallery problem

1048708Note that it may be quite dangerous for a guard to be stationed in an art gallery if he cannot be seen by any other guards

We represent the relationship among guards by a visibility graph G of the guards

1048708In G every guard is represented by a vertex and there is an edge between two vertices if and only if the corresponding guards can see each other

1048708In addition to finding a minimum set of guards who can see the given polygon we further require that the visibility graph of these guards is connected

1048708In other words we require that no guard is isolated and there is a path between every pair of guards We call such a problem the minimum cooperative guards problem

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Minimum cooperative guards problemhellip

A solution of the minimum cooperative guards problem for the polygon shown earlier

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

NOTATION A reflex (convex) chain denoted as

RC (CC) of a simple polygon is a chain of edges of this polygon if all of the vertices on this chain are reflex (convex) with respect to the interior of the polygon except the vertices at the end of the chain

The term ldquoreflex chainrdquo always refers to a maximal reflex chain that is it is not contained in any other reflex chain

A 1-spiral polygon P is a simple polygon whose boundary contains exactly one reflex chain (Convex

polygon is 0-spiral)

Vs Ve

Ve-1

CC

RC

Vs+1

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

A typical 1-spiral polygon

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

NOTATION Traversing the boundary of P counterclockwise the starting

(ending) vertex of the reflex chain is called vs (ve) and the indices of vertices of P are numbered in increasing order

A subchain of the boundary of P from point p to point q in counterclockwise order is denoted as C[p q]

Vs Ve

Ve-1

CC

RC

Vs+1

p q

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

STARTING REGION ENDING REGION

Vs Ve

Ve-1

CC

RC

Vs+1

Vs

Vs

Vs+1

Vs+1

Ve

Ve

CC

CCRC

RCVe-1

Ve-1

starting region

ending region

starting region

ending region

pq

p

q

For two points p and q on the convex (reflex) chain we define p lt q if p is closer to vs on the convex (reflex) chain than is q

There must be a guard standing in both the starting and ending regions

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Reaching the solution

Place a guard at l1 We as how much can this guard

see To answer this draw a tangent

line of RC starting from l1 and hitting at l2 on CC

Place a guard at l2 In the adjacent figure

l1l2l3l4l5 constitute an optimal solution for the minimum cooperative guards problem

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Supporting line segment Let a be a point in P and b a point on CC of P A line

segment ab tangent at a reflex vertex c is called a left (right) supporting line segment with respect to a if there are exterior points of P to the right (left) of ab in every neighborhood of c We call vertex c the contact vertex of ab

If a is in the starting (ending) region we define the right (left) supporting line segment with respect to a as avs (ave) with vs (ve) as its contact vertex

abc

a

bc

c

b

RC

CC

RC

CCleft supporting line with respect to a

right supporting line with respect to a c

b

Vs Ve

a a

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

ALGORITHM Input a 1-spiral polygon Output a set of points which correspond to the solution of the

minimum co-operative guards problem Step1 find the reflex chain RC and the convex chain CC of P Step 2 find the intersection points l1 and l2 of CC with the

directed lines starting from vs and ve along the first and last edges of RC respectively

Step 3 let lk=1 while lk is not in the ending region do draw the left tangent of lk with respect to RC until it hits CC at lk+1 (lklk+1 is

the left supporting line segment with respect to lk) end while

step 4 report (l1l2helliplk)

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

Analysishellip Since the number of guards is minimal we

conclude that our algorithm produces optimal solution

Time- complexity Analysis step 1 and step 2 can be done in O(n) time by a

linear scan of the boundary of P Step3- we conduct a linear scan on the reflex

chain counterclockwise and on the convex chain clockwise to find all the required left-supporting line-segments This can also be done in O(n) time

Thus this algorithm is linear

THANK YOU

THANK YOU