Understanding Dijkstra’s Algorithm for Shortest Paths
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This document provides an in-depth exploration of Dijkstra's Algorithm, focusing on its application to finding the shortest paths in graphs. Key concepts such as optimal substructure, overlapping subproblems, and greedy choice properties are examined. The correctness and analysis of the algorithm are discussed, including initialization, relaxation of edges, and the handling of negative cycles. Additionally, implications of using non-negative edge weights and potential optimizations for Dijkstra’s algorithm are examined to enhance algorithm performance in various scenarios.
Understanding Dijkstra’s Algorithm for Shortest Paths
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Presentation Transcript
Introduction to Algorithms Jiafen Liu Sept. 2013
Today’s Tasks • Shortest Paths • Properties of shortest paths • Dijkstra’s algorithm • Correctness • Analysis • Breadth-first search
Paths in graphs • Consider a digraph G= (V, E) with edge-weight function w: E→R. The weight of path p = v1→v2→…→vk is defined to be • Example: w(p)=?
Shortest paths • A shortest path from u to v is a path of minimum weight from u to v. • The shortest path weight from u to v is defined as δ(u, v) = min{w(p): p is a path from u to v}. • In what case there is no shortest path? • Note: δ(u, v) = ∞ if no path from u to v exists. • There is a negative cycle. • In other cases, usually a finite set.
Well-definedness of shortest paths • If a graph G contains a negative-weight cycle, then some shortest paths may not exist. • Example:
Thought of solving • Dynamic Programming • Greedy Algorithms • Characteristics： • Optimal Substructure • Overlapping Subproblems • Greedy choice property
Optimal substructure • What is the optimal substructure for this problem? • Theorem. A subpath of a shortest path is a shortest path. • How to prove? • Cut and paste
What’s the subproblems？ • Shortest path going from u to v, maybe involves computing shortest path from u to some intermediate point x, and then from x to v. • Different choosing of x1,x2…xn consists overlapping subproblems. • How many subproblems are there? • |v|2
Triangle inequality • Important for us to make greedy choice • Theorem: For all u, v, x∈V, we have • Figure Illustration:
Single-source shortest paths • Problem. From a given source vertex s∈V, find the shortest-path weights δ(s, v) for all v∈V. • If all edge weights w(u, v) are nonnegative, all shortest-path weights must exist. • IDEA: Greedy. 1.Maintain a set S of vertices whose shortest-path distances from s are known. (s∈S) 2.At each step add the vertex v∈V–S whose distance estimate from s is minimal to S . 3.Update the distance estimates of vertices adjacent to v.
Correctness —Part I • Lemma. Initializing d[s]←0 and d[v]←∞ for all v∈V–{s} establishes d[v] ≥ δ(s,v) for all v∈V, and this invariant is maintained over any sequence of relaxation steps. • Any suggestions about how to prove? • Induction • Proof. • Base case: check the initializing term • Induction: suppose that this fails to hold at some point.
Correctness —Part I • Suppose not. Let v be the first vertex for which d[v] < δ(s, v), and let u be the vertex that caused d[v] to change: d[v] = d[u] + w(u, v). Then how to induct a contradition? d[v]< δ(s, v) ≤ δ(s, u) + δ(u, v) ≤ δ(s,u) + w(u, v) d[v] = d[u] + w(u, v) • Contradiction. supposition triangle inequality sh. path ≤ specific path v is first violation
Correctness —Part II • Lemma. Let u be v’s predecessor on a shortest path from s to v: s→…u→v. if d[u]=δ(s, u) and if we relax edge(u, v), Then, we have d[v] = δ(s, v) after the relaxation. • Proof. • s→…u→v is the shortest path from s to v, so δ(s, v) = w(s→… →u)+ w(u, v) = δ(s, u)+ w(u, v).
Correctness —Part II • Suppose that d[v] > δ(s, v) before the relaxation. • By Correctness I, d[v] ≥ δ(s,v) • Otherwise d[v] = δ(s,v) , we’re done. • Then test d[v] > d[u] + w(u, v) succeeds • because d[v] > δ(s, v) = δ(s, u)+ w(u, v) = d[u] + w(u, v) • Then the algorithm sets d[v] = d[u]+ w(u,v) = δ(s, v).
Correctness —Part III • Theorem. Dijkstra’s algorithm terminates with d[v] = δ(s, v) for all v∈V. • Something to be addressed: • Once we add a vertex to S, we never change its weight • Proof. • It suffices to show that d[v] = δ(s, v) for every v∈V when v is added to S.
Correctness —Part III • Suppose for contradiction: Suppose u is the first vertex added to S for which d[u] ≠ δ(s, u). • d[u] > δ(s, u). • Let p be a shortest path from s to u. • Now look at the first place here where the path p exits S
Correctness —Part III • Let y be the first vertex in V–S along a shortest path from s to u, and let x be its predecessor. • Since u is the first vertex violating the claimed invariant, we have d[x] = δ(s, x) .
Correctness —Part III • When x was added to S, the edge (x, y) was relaxed, which implies that d[y] = δ(s, y) ≤ δ(s, u) < d[u] • But, d[u] ≤ d[y] by our choice of u. Contradiction.
Analysis of Dijkstra • Note: Same formula as in the analysis of Prim’s minimum spanning tree algorithm.
Unweighted graphs • Suppose that w(u, v) =1 for all (u, v) ∈E . Can Dijkstra’s algorithm be improved? • Use a simple FIFO queue instead of a priority queue. • Analysis: Time = O(V+ E).
Correctness of BFS • IDEA: The FIFO Q in breadth-first search mimics the priority queue Q in Dijkstra. • Invariant: v comes after u in Q implies that d[v] = d[u] or d[v] = d[u] + 1.
