lecture 20: shortest paths shang-hua teng. weighted directed graphs weight on edges for distance 400...

Lecture 20:Shortest Paths

Shang-Hua Teng

Weighted Directed Graphs• Weight on edges for distance

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Shortest Paths• Given a weighted, directed graph G=(V, E) with

weight function w: ER. The weight of path p=<v0, v1,..., vk> is the sum of the weights of its edges:

• We define the shortest-path weight from u to v by

• A shortest path from vertex u to vertex v is any path with w(p)=(u, v)

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Otherwise.

Variants of Shortest Path Problem• Single-source shortest paths problem

– Finds all the shortest path of vertices reachable from a single source vertex s

• Single-destination shortest-path problem– By reversing the direction of each edge in the graph, we can

reduce this problem to a single-source problem

• Single-pair shortest-path problem– No algorithm for this problem are known that run

asymptotically faster than the best single-source algorithm in the worst case

• All-pairs shortest-path problem

Relaxation• For each vertex vV, we maintain an

attribute d[v], which is an upper bound on the weight of a shortest path from source s to v. We call d[v] a shortest-path estimate.

Possible Predecessor of v in the shortest path

Relaxation

• Relaxing an edge (u, v) consists of testing whether we can improve the shortest path found so far by going through u and, if so, update d[v] and [v]

By Triangle Inequality

Dijkstra’s Algorithm

Coping with Negative Weights

• Deciding whether there is a negative circle

• If the graph does not have a negative circle reachable from the source s, for the shortest path tree.

Bellman-Ford Algorithm

BellmanFord() for each v V d[v] = ; d[s] = 0; for i=1 to |V|-1 for each edge (u,v) E Relax(u,v, w(u,v)); for each edge (u,v) E if (d[v] > d[u] + w(u,v)) return “no solution”;

Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w

Initialize d[], whichwill converge to shortest-path value

Relaxation: Make |V|-1 passes, relaxing each edge

Test for solution Under what conditiondo we get a solution?




What will be the running time?




What will be the running time?

A: O(VE)




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Ex: work on board

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Bellman-Ford• Note that order in which edges are processed affects

how quickly it converges• Correctness: show d[v] = (s,v) after |V|-1 passes

– Lemma: d[v] (s,v) always• Initially true• Let v be first vertex for which d[v] < (s,v)• Let u be the vertex that caused d[v] to change:

d[v] = d[u] + w(u,v)• Then d[v] < (s,v)

(s,v) (s,u) + w(u,v) (Why?) (s,u) + w(u,v) d[u] + w(u,v) (Why?)

• So d[v] < d[u] + w(u,v). Contradiction.

Bellman-Ford

• Prove: after |V|-1 passes, all d values correct– Consider shortest path from s to v:

s v1 v2 v3 v4 v• Initially, d[s] = 0 is correct, and doesn’t change (Why?)

• After 1 pass through edges, d[v1] is correct (Why?) and doesn’t change

• After 2 passes, d[v2] is correct and doesn’t change

• …

• Terminates in |V| - 1 passes: (Why?)

• What if it doesn’t?

Properties of Shortest Paths

• Triangle inequality– For any edge (u,v) in E, (s,v) <= (s,u) + w(u,v)

• Upper bound property– d[v] >= (s,v)

• Monotonic property – d[v] never increase

• No-path property– If v is not reachable then d[v] = (s,v) = infty

Properties of Shortest Paths• Convergence property

– If (u,v) is on the shortest path from s to v and if d[u] = (s,u) at any time prior to relaxing (u,v), then d[v] = (s,v) at all time afterward

• Path-relaxation property– If p=< s=v0,v1, v2,..., vk> is the shortest path from s to vk

and edges of p are relaxed in order in the index, then d[vk] = (s, vk). This property holds regardless of any other relaxation steps that occur, even if they are intermixed with relaxations of the edges of p

Properties of Shortest Paths

• Predecessor-subgraph property– Once d[v] = (s,v), the predecessor subgraph is

a shortest-paths tree rooted at s

Matrix Basic

• Vector: array of numbers; unit vector

• Inner product, outer product, norm

• Matrix: rectangular table of numbers, square matrix; Matrix transpose

• All zero matrix and all one matrix

• Identity matrix

• 0-1 matrix, Boolean matrix, matrix of graphs

Matrix Operations

• Matrix-vector operation– System of linear equations– Eigenvalues and Eigenvectors

• Matrix matrix operations

1. Matrix Addition:

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2. Scalar Multiplication:

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3. Matrix Multiplication

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lecture 20: shortest paths shang-hua teng. weighted directed graphs weight on edges for distance 400...

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