cs 8833 algorithms algorithms shortest path problems

CS 8833 Algorithms

Algorithms

Shortest Path Problems

CS 8833 Algorithms

G = (V, E) weighted directed graphw: ER weight function

Weight of a path p = <v0, v1,. . ., vn>

Shortest path weight from u to v

Shortest path from u to v: Any path from u to v with w(p) = (u,v)

[v] predecessor of v on a path

n

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11 )()(

otherwise

exitspath vu, a if}:)(min{),(p

vupwvu

CS 8833 Algorithms

CS 8833 Algorithms

Variants Single-source shortest paths:

find shortest paths from source vertex to every other vertex

Single-destination shortest paths:find shortest paths to a destination from every vertex

Single-pair shortest-pathfind shortest path from u to v

All pairs shortest paths

CS 8833 Algorithms

Lemma 25.1 Subpaths of shortest paths are shortest

paths.Given G=(G,E) w: E R

Let p = p = <v1, v2,. . ., vk> be a shortest path from v1 to vk

For any i,j such that 1 i j k, let pij be a subpath from vi to vj.. Then pij is a shortest path from vi to vj.

CS 8833 Algorithms

1 i j k

p

CS 8833 Algorithms

Corollary 25.2Let G = (V,E) w: E R

Suppose shortest path p from a source s to vertex v can be decomposed into

p’

s u v

for vertex u and path p’.

Then weight of the shortest path from s to v is

(s,v) = (s,u) + w(u,v)

CS 8833 Algorithms

Lemma 25.3

Let G = (V,E) w: E R

Source vertex s

For all edges (u,v)E

(s,v) (s,u) + w(u,v)

CS 8833 Algorithms

s v

u1

u2

u4

u3

un

CS 8833 Algorithms

Relaxation Shortest path estimate

d[v] is an attribute of each vertex which is an upper bound on the weight of the shortest path from s to v

Relaxation is the process of incrementally reducing d[v] until it is an exact weight of the shortest path from s to v

CS 8833 Algorithms

INITIALIZE-SINGLE-SOURCE(G, s)

1. for each vertex v V(G)

2. do d[v]

3. [v] nil

4. d[s] 0

CS 8833 Algorithms

CS 8833 Algorithms

Relaxing an Edge (u,v) Question: Can we improve the shortest

path to v found so far by going through u?

If yes, update d[v] and [v]

CS 8833 Algorithms

RELAX(u,v,w)

1. if d[v] > d[u] + w(u,v)

2. then d[v] d[u] + w(u,v)

3. [v] u

CS 8833 Algorithms

EXAMPLE 1

s

u

v

s

u

v

Relax

CS 8833 Algorithms

EXAMPLE 2

s

u

v

s

u

v

Relax

CS 8833 Algorithms

Dijkstra’s Algorithm Problem:

– Solve the single source shortest-path problem on a weighted, directed graph G(V,E) for the cases in which edge weights are non-negative

CS 8833 Algorithms

Dijkstra’s Algorithm Approach

– maintain a set S of vertices whose final shortest path weights from the source s have been determined.

– repeat» select vertex from V-S with the minimum

shortest path estimate» insert u in S» relax all edges leaving u

CS 8833 Algorithms

DIJKSTRA(G,w,s)

1. INITIALIZE-SINGLE-SOURCE(G,s)

2. S

3. Q V[G]

4. while Q

5. do u EXTRACT-MIN(Q)

6. S S {u}

7. for each vertex v Adj[u]

8. do RELAX(u,v,w)

CS 8833 Algorithms

CS 8833 Algorithms

Analysis of Dijkstra’s Algorithm

Suppose priority Q is:– an ordered (by d) linked list

» Building the Q O(V lg V)» Each EXTRACT-MIN O(V)» This is done V times O(V2)» Each edge is relaxed one time O(E)» Total time O(V2 + E) = O(V2)

CS 8833 Algorithms

Analysis of Dijkstra’s Algorithm

Suppose priority Q is:– a binary heap

» BUILD-HEAP O(V)» Each EXTRACT-MIN O(lg V)» This is done V times O(V lg

V)» Each edge is relaxation O(lg V)» Each edge relaxed one time O(E lg

V)» Total time O(V lg V + E lg V))

CS 8833 Algorithms

Properties of Relaxation Lemma 25.4

G=(V,E) w: E R (u,v) E

After relaxing edge (u,v) by executing RELAX(u,v,w) we have

d[v] d[u] + w(u,v)

CS 8833 Algorithms

Lemma 25.5– Given:

G=(V,E) w: E R source s V

Graph initialized by

INITIALIZE-SINGLE-SOURCE(G,s)– then

d[v] (s,v) for all v V

and this invariant is maintained over all relaxation steps

Once d[v] achieves a lower bound (s,v), it never changes

CS 8833 Algorithms

Corollary 25.6– Given:


No path connects s to given v– then

after initialization d[v] (s,v)

and this inequality is maintained over all relaxation steps.

CS 8833 Algorithms



Let s - - u v be the shortest path in G for all vertices u and v.

Suppose G initialized by INITIALIZE-SINGLE-SOURCE is followed by a sequence of relaxations including RELAX(u,v,w)

– Then d[u] = (s,u) prior to call implies that d[u] = (s,u) after the call

CS 8833 Algorithms

Bottom Line Therefore, relaxation causes the

shortest path estimates to descend monotonically toward the actual shortest-path weights.

CS 8833 Algorithms

Shortest-Paths Tree of G(V,E) The shortest-paths tree at S of G(V,E)

is a directed subgraph G’-(V’,E’), where V’ V, E’E, such that– V’ is the set of vertices reachable from S in

G– G’ forms a rooted tree with root s, and– for all v V’, the unique simple path from s

to v in G’ is a shortest path from s to v in G

CS 8833 Algorithms

Goal We want to show that successive

relaxations will yield a shortest-path tree

CS 8833 Algorithms



Assume that G contains no negative-weight cycles reachable from s.

– Then after the graph is initialized with INITIALIZE-SINGLE-SOURCE

• the predecessor subgraph G forms a rooted tree with root s, and

• any sequence of relaxation steps on edges in G maintains this property as an invariant.

CS 8833 Algorithms

Algorithms

Bellman-Ford Algorithm

Directed-Acyclic Graphs

All Pairs-Shortest Path Algorithm

CS 8833 Algorithms

Why does Dijkstra’s greedy algorithm work?

Because we know that when we add a node u to the set S, the value d is the length of the shortest path from s to u.

But, this only works if the edges of the graph are nonnegative.

CS 8833 Algorithms

a

b c

7 10

-4

Would Dikjstra’s Algorithm work with this graph?

CS 8833 Algorithms

a

b c

7 6

-14

What is the length of the shortest path from a to c in this graph?

CS 8833 Algorithms

Bellman-Ford Algorithm The Bellman-Ford algorithm can be

used to solve the general single source shortest path problem

Negative weights are allowed The algorithm detects negative cycles

and returns false if the graph contains one.

CS 8833 Algorithms

BELLMAN-FORD(G,w,s)

1 INITIALIZE-SINGLE-SOURCE(G,s)

2 for i 1 to |V[G]| -1

3 do for each edge (u,v) E[G]

4 do RELAX(u,v,w)

5 for each edge (u,v) E[G]

6 do if d[v] > d[u] + w(u,v)

7 then return false

8 return true

CS 8833 Algorithms

When there are no cycles of negative length, what is maximum number of edges in a shortest path when the graph has |V[G]| vertices and |E[G]| edges?

CS 8833 Algorithms

Dynamic Programming Formulation

The following recurrence shows how the Bellman Ford algorithm computes the d values for paths of length k.

)}},(][{min ],[min{][ 1i

1 uiwidistudud kkk

CS 8833 Algorithms

a

b c

7 10

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Processing order

1 2 3

(a,c)

(b,a)

(c,b)

(c,d)

(d,b)

d

-59

CS 8833 Algorithms

Complexity of the Bellman Ford Algorithm

Time complexity:

Performance can be improved by– adding a test to the loop to see if any d values

were updated on the previous iteration, or– maintain a queue of vertices whose d value

changed on the previous iteration-only process these on the next iteration

CS 8833 Algorithms

Single-source shortest paths in directed acyclic graphs

Topological sorting is the key to efficient algorithms for many DAG applications.

A topological sort of a DAG is a linear ordering of all of its vertices such that if G contains an edge (u,v), then u appears before v in the ordering.

CS 8833 Algorithms

b

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f

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CS 8833 Algorithms

DAG-SHORTEST-PATHS(G,w,s)

1 topologically sort the vertices of G

2 INITIALIZE-SINGLE-SOURCE(G,s)

3 for each vertex u taken in topological order

4 do for each vertex v Adj[u]

5 do RELAX(u,v,w)

CS 8833 Algorithms

Topological Sorting

TOPOLOGICAL-SORT(G)

1 call DFS(G) to compute finishing times f[v] for each vertex v

2 as each vertex is finished, insert it onto the front of a linked list

3 return the linked list of vertices

CS 8833 Algorithms

Depth-first search Goal: search all edges in the graph one

time Strategy: Search deeper in the graph

whenever possible Edges are explored out of the most

recently discovered vertex v that still has unexplored edges leaving it.

Backtrack when a dead end is encountered

CS 8833 Algorithms

Predecessor subgraph The predecessor subgraph of a depth

first search– forms a depth-first forest– composed of depth-first trees

The edges in E are called tree edges

CS 8833 Algorithms

Vertex coloring scheme All vertices are initially white A vertex is colored gray when it is

discovered A vertex is colored black when it is

finished (all vertices adjacent to the vertex have been examined completely)

CS 8833 Algorithms

Time Stamps Each vertex v has two time-stamps

– d[v] records when v is first discovered (and grayed)

– f[v] records when the search finishes examining its adjacency list (and is blackened)

For every vertex u– d[u] < f[u]

CS 8833 Algorithms

Color and Time Stamp Summary

Vertex u is – white before d[u]– gray between d[u] and f[u]– black after f[u]

Time is a global variable in the pseudocode

CS 8833 Algorithms

DFS(G)

1 for each vertex u V[G]

2 do color[u] white

3 [u] nil

4 time 0

5 do for each vertex u Adj[u]

6 do if color[u] = white

7 then DFS-VISIT(u)

CS 8833 Algorithms

DFS-VISIT(u)

1 color[u] gray

2 d[u] time time time +1

3 for each vertex v Adj[u]

4 do if color[v] = white

5 then [v] u

6 DFS-VISIT(v)

7 color[u] black

8 f[u] time time time +1

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CS 8833 Algorithms

Running time of DFS lines 1-3 of DFS lines 5-7 of DFS lines 2-6 of DFS-VISIT

CS 8833 Algorithms

Running Time of Topological Sort

DFS Insertion in linked list

CS 8833 Algorithms

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CS 8833 Algorithms

Running Time for DAG-SHORTEST-PATHS

Topological sort Initialize-single source 3-5 each edge examined one time

cs 8833 algorithms algorithms shortest path problems

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