Graph Algorithms: Shortest Path We are given a weighted, directed graph G = (V, E), with weight function w: E  R mapping edges to real valued weights. 1 5 10 1 8 3 2 4 3 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path We are given a weighted, directed graph G = (V, E), with weight function w: E  R mapping edges to real valued weights. 1 The weight of a path p = (v , v , . . ., v ) is  w(v , v ). 5 10 1 1 2 k k-1 8 3 2 4 3 i i+1 i=1 1 1 3 The weight of the path along the red edges is 1 + 6 + 1 + 4 = 12. 1 4 6 6 5 2

Graph Algorithms: Shortest Path We are given a weighted, directed graph G = (V, E), with weight function w: E  R mapping edges to real valued weights. 1 The weight of a path p = (v , v , . . ., v ) is  w(v , v ). 5 10 1 1 2 k k-1 8 3 2 4 3 i i+1 i=1 1 Single source shortest path problem: given a vertex s, for every vertex v  V find a shortest path from s to v. 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Single source shortest path problem: given vertex s, for every vertex v  V find a shortest path from s to v. 1 5 10 1 Dijkstra’s algorithm solves this problem efficiently for the case in which all weights are nonnegative (as in the example graph). 8 3 2 4 3 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. 1 It also maintains, for each vertex v not in S, an upper bound d[v] on the weight of a shortest path from source s to v. 5 10 1 8 3 2 4 3 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. 1 It also maintains, for each vertex v not in S, an upper bound d[v] on the weight of a shortest path from source s to v. 5 10 1 8 3 2 4 3 1 The algorithm repeatedly selects the vertex u  V – S with minimum bound d[v], inserts u into S, and relaxes all edges leaving u. 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 Set S = { } initially. 5 10 1 v 1 2 3 4 5 6 8 3 2 4 3 d      1 1 3 1 4 6 Also initialize a queue with all the vertices and their upper bounds. 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 S = { } 5 10 1 v 1 2 3 4 5 6 8 3 2 4 3 d      1 Select the vertex u  V – S with minimum bound d[v], insert u into S, . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 (here we keep the bound with the vertex in S) S = 5 10 1 v 2 3 4 5 6 8 3 2 4 3 d      1 . . . remove it from the queue, insert it in S,. . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 S = 5 10 1 v 2 3 4 5 6 8 3 2 4 3 10 1 5  d  1 . . . and relax the edges from this vertex. 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 S = 5 10 1 v 3 4 2 5 6 8 3 2 4 3 1 5 10  d  1 . . . reordering the queue according to the new upper bounds. 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 S = 5 10 1 v 3 4 2 5 6 8 3 2 4 3 1 5 10  d  1 Repeat . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 S = 5 1 10 1 v 4 2 5 6 8 3 2 4 3 5 10  d  1 . . . remove from queue and put in S 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 S = 5 1 10 1 v 4 2 5 6 8 3 2 4 3 4 9 2 d  1 . . . relax the edges . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 S = 5 1 10 1 v 5 4 2 6 8 3 2 4 3 d 2 4 9  1 . . . and reorder the queue . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 S = 5 1 10 1 v 5 4 2 6 8 3 2 4 3 d 2 4 9  1 Repeat . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 5 S = 5 1 2 10 1 v 4 2 6 8 3 2 4 3 4 9 d  1 . . . remove from queue and put in S 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 5 S = 5 1 2 10 1 v 4 2 6 8 3 2 4 3 4 9 d 4 1 . . . relax the edges . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 5 S = 5 1 2 10 1 v 4 6 2 8 3 2 4 3 d 4 4 9 1 . . . and reorder the queue . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 5 S = 5 1 2 10 1 v 4 6 2 8 3 2 4 3 d 4 4 9 1 Repeat . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 4 5 S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1 . . . remove from queue and put in S 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 4 5 S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1 . . . relax the edges (no change in this case) . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 4 5 S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1 . . . and reorder the queue (no change in this case) . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 1 3 4 5 S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1 Repeat . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 6 1 3 4 5 S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 d 9 1 . . . remove from queue and put in S 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 6 1 3 4 5 S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 d 9 1 . . . relax the edges (no change in this case) . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 6 1 3 4 5 S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 9 d 1 Repeat . . . 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s 1 6 1 2 3 4 5 S = 5 9 1 4 2 4 10 1 v 8 3 2 4 3 d 1 Done! 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. Suppose vertex 1 is the source. s v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 8 3 2 4 3 The result is the bottom row which contains the length of the shortest path from s to the vertex above it. 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 To compute the corresponding paths, we augment the data structure with an additional attribute, p(v), which is the vertex that precedes v in a shortest path. 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 Using the predecessor attribute, p(v), it’s easy to construct the path to v, working backwards from v to s. 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 E.g., v = 6. 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 E.g., v = 6. p(6) = 5, 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 E.g., v = 6. p(6) = 5, p(5) = 3, 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 E.g., v = 6. p(6) = 5, p(5) = 3, p(3) = 1. 1 1 3 1 4 6 6 5 2

Graph Algorithms: Shortest Path v 1 6 1 2 3 4 5 S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3 E.g., v = 6. p(6) = 5, p(5) = 3, p(3) = 1. 1 1 3 1 4 6 6 5 Path: 1, 3, 5, 6. Weight: 4. 2

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed how many times?

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times.

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed how many times?

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed |E| times. Total time = |V| T + |E| T Delete_min Decrease_key

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed |E| times. Total time = |V| T + |E| T Delete_min Decrease_key Priority queue T T Total time Array Binary heap Decrease_key Delete_min

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed |E| times. Total time = |V| T + |E| T Delete_min Decrease_key Priority queue T T Total time Array O(|V|) O(1) Binary heap Decrease_key Delete_min

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed |E| times. Total time = |V| T + |E| T Delete_min Decrease_key Priority queue T T Total time Array O(|V|) O(1) O(|V| ) Binary heap O(log |V|) O(log |V|) Decrease_key Delete_min 2

Graph Algorithms: Shortest Path Computing time analysis: look at different implementations of the priority queue, with different costs for the queue operations. Delete_min executed |V| times. Decrease_key executed |E| times. Total time = |V| T + |E| T Delete_min Decrease_key Priority queue T T Total time Unsorted Array O(|V|) O(1) O(|V| ) Binary heap O(log |V|) O(log |V|) O(E log |V|) Decrease_key Delete_min 2