CSE 780 Algorithms

Advanced Algorithms

SSSP Dijkstra’s algorithm

SSSP in DAGs

CSE 780 Algorithms

Review

Given edge (u,v) (s,v) ≤ (s,u) + w(u,v)

Relax(u,v,w)

If d[u] = (s,u) before the operation Then d[v] = (s,v)

Path relaxation property

CSE 780 Algorithms

Directed Acyclic Graphs

SSSPs in DAGs, allowing negative weights

Can be done in O(|V| + |E|)

CSE 780 Algorithms

Pseudo-code

Time complexity:O(|V| + |E|)

CSE 780 Algorithms

Example

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Correctness

Claim: d[v] = (s, v) at termination Assume

is the shortest path Relax happens in order of path Therefore claim is true

by Path-Relaxation property

= v= s

CSE 780 Algorithms

Shortest-Paths Problem

A shortest-paths problem, given a weighted, directed graph G=(V,E) with weight function w: E→ R, is to find the shortest path between two arbitrary vertices, u and v.

This problem is a generalization of BFS to handle weighted graphs.

Edge weights can be interpreted, instead of as distances, as time, cost, etc.

CSE 780 Algorithms

Dijkstra’s Algorithm

Given directed graph G=(V,E), with only non-negative weights

Bear similarity to breadth-first search and Prim’s algorithm

Maintain a set S V whose shortest-path from source s have already been determined.

Use a min-priority queue Q to store V-S, keys are the shortest path weight estimate d[v]

Repeatedly select next vertex from V-S, and add it to S.

CSE 780 Algorithms

INIT-SINGLE-SOURCE(V,s)

For each vertex v ε V

do d[v] ←∞

л[v] ← NIL

d[s] ← 0

CSE 780 Algorithms

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms

Pseudo-code

Only d[s] = 0in Q

If d[v] improves during the Relax operation, do Decrease-key(Q, v, d[v])

CSE 780 Algorithms

Example

CSE 780 Algorithms

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

CSE 780 Algorithms

Analysis

Time complexity: #Extract-min: O(|V|) #Decrease-key = #Relax: O(|E|) Use binary heap for priority heap:

Time complexity: O((|V| + |E|) log |V|) Use Fibonacci heap for priority heap:

Amortized time complexity: O(|V| log |V| + |E|)

Greedy algorithm: Any moment, select the lightest vertex in V-S to add to

set S

CSE 780 Algorithms

Correctness

Claim: At the start of each iteration of the while loop, d[v] =

(s, v) for each vertex v S. Proof: (by contradiction)

Let u be the first vertex d[u] (s,u) when added to S Must be a shortest path from s to u

CSE 780 Algorithms

Remarks

Similar to breadth-first search, It adds nodes into S in order of their distance to s

The algorithm also induces a shortest-paths tree

Why is non-negative weights required? Example?