lecture 33: directed graph connectivity

LECTURE 33:DIRECTED GRAPH CONNECTIVITY

CSC 213 – Large Scale Programming

Today’s Goals

Examine new properties of DirectedGraphs What reaching & reachable mean for a Graph

How humans go about computing these properties

Algorithms for computers to compute these Examine what meant by DAG and why

you care Simple ways to develop schedules will be

examined How can algorithm tell when a schedule

impossible

Directed Graph

Only directed edges Replace undirected

edge with 2 directed edges

Relationships go in only one direction One-way streets Flights Scheduling

A

C

E

B

D

Directed Graph

Only directed edges Replace undirected

edge with 2 directed edges

Relationships go in only one direction One-way streets Flights Scheduling Talking to your ex

Directed Graph Properties

Each edge goes one-way (a,b) connects a to b (a,b) does not connect b to a

Can discuss in-edges & out-edges (a,b) is out-edge for a (a,b) is in-edge for b

Adjacency-based Graph classes can change Use 2 Sequences for adjacency-list vertices Define source & target dimension in adjacency-

matrixincidentEdges returns both in-edges & out-edges

a

c

e

b

d

Reachability

e, a, d reachable from c

a

c

e

b

d

f

Reachability

e, a, d reachable from c c reaches e

a

c

e

b

d

f

Reachability

e, a, d reachable from c c reaches e c reaches e & e incident upon d, a

a

c

e

b

d

f

Reachability

e, a, d reachable from c c reaches e c reaches e & e is incident upon d, a d, a out-edges to c only

a

c

e

b

d

f

Reachability

a, c, d, e, f reachable from b

a

c

e

b

d

f

Reachability

a, c, d, e, f reachable from b Path exists from b to every vertex

a

c

e

b

d

f

Reachability

a, c, d, e, f reachable from b Path exists from b to every vertex Actually have multiple paths to most

vertices

a

c

e

b

d

f

Transitive Closure of G

Transitive closure of G usually written as G*

b

a

d

c

e

G G*

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

b

a

d

c

e

G G*

b

a

d

c

e

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

Edge in G* if target reachable from source in G

b

a

d

c

e

G G*

b

a

d

c

e

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

Edge in G* if target reachable from source in G

b

a

d

c

e

G G*

b

a

d

c

e

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

Edge in G* if target reachable from source in G

b

a

d

c

e

G G*

b

a

d

c

e

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

Edge in G* if target reachable from source in G

b

a

d

c

e

G G*

b

a

d

c

e

Transitive Closure of G

Transitive closure of G usually written as G* Identical vertex sets in G & G* G & G* have different edge sets, however

Edge in G* if target reachable from source in G

b

a

d

c

e

G G*

b

a

d

c

e

Computing Transitive Closure Use dynamic programming to compute

this This solution known as Floyd-Warshall

Algorithm But how fast is it?

Floyd-Warshall’s Algorithm

Number G’s vertices from 1 to n Algorithm will compute n directed graphs Set G0=G to initialize this algorithm Graph of transitive closure is end result (Gn

= G*) All n directed graphs have same

vertices Gk contains all edges in Gk-1 (and Gk-2, Gk-3 ,…,

G0) Gk also has edge (vi,vj) if edges (vi,vk) & (vk,vj)

in Gk-1

Takes O(n3) time with adjacency matrix Better to use “brute force” if few edges

exist

Floyd-Warshall’s Algorithm

Number G’s vertices from 1 to n Algorithm will compute n directed graphs Set G0=G to initialize this algorithm Graph of transitive closure is end result (Gn

= G*) All n directed graphs have same

vertices Gk contains all edges in Gk-1 (and Gk-2, Gk-3 ,…,

G0) Gk also has edge (vi,vj) if edges (vi,vk) & (vk,vj)

in Gk-1

Takes O(n3) time with adjacency matrix Better to use “brute force” if few edges

exist

Floyd-Warshall Example – G0

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Floyd-Warshall Example – G1

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Floyd-Warshall Example – G1

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Floyd-Warshall Example – G1

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Floyd-Warshall Example – G1

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Floyd-Warshall Example – G1

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Floyd-Warshall Example – G2

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Floyd-Warshall Example – G2

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Floyd-Warshall Example – G3

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Floyd-Warshall Example – G3

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Floyd-Warshall Example – G3

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Floyd-Warshall Example – G3

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Floyd-Warshall Example – G3

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G4

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Floyd-Warshall Example – G5

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Floyd-Warshall Example – G5

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Floyd-Warshall Example – G5

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Floyd-Warshall Example – G5

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Floyd-Warshall Example – G6

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Floyd-Warshall Example – G6

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Floyd-Warshall Example – G7

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Floyd-Warshall Example – G*

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Directed Acyclic Graph

Often called a DAG Number & sort

vertices Topological order

found Each edge (vi,vj) has

i < j Finds valid

schedules… …or proves cannot

exist!

b

a

d

c

e

G (is a DAG)

b

a

d

c

e

A valid ordering of G

1

2

3

4 5

Directed Acyclic Graph

Often called a DAG Number & sort

vertices Topological order

found Each edge (vi,vj) has

i < j Finds valid

schedules… …or proves cannot

exist!

b

a

d

c

e

G (is a DAG)

b

a

d

c

e

Another valid ordering of G

2

1

3

4 5

Topological Sorting

Edges connect smaller to larger vertex numberswake up

studyeat

class study

homeworkwork

study

go to bed dream about classwork

Professor’s expectation of student’s day

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2 3

4 5

67

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Topological Sort Algorithm

Algorithm topologicalSort(Graph<V,E> G) H // Make a copy of Gm new Vertex[G.numVertices()]stack new …Stack<Vertex>() // Push onto stack any vertex in H with no outgoing edges n G.numVertices() while n ≥ 1 do

v stack.pop() n n – 1m[n] v

foreach e in v.incidentEdges() ≠ 0 do w H.opposite(e, v)

// Remove e from H if w has no outgoing edges then stack.push(w) return m

Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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Topological Sorting Example

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For Next Lecture

Weekly assignment out & due tomorrow

Programming assignment #3 plans due Friday Please do not wait to start working on

these ideas For Wednesday, read 13.5.2-13.5.3 &

13.6.2 Find quickest way to get someplace, can it

be done? How can we find the spanning tree that

weighs least?