Dijkstra s Algorithm

Transcription

1 Dijkstra s Algorithm Weighted Graphs In a weighted graph, each edge has an associated numerical value, called the weight of the edge Edge weights may represent, distances, costs, etc. Example: In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports HNL SFO 337 LAX ORD DFW LGA PVD MIA 1 1

2 Problem Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v. Length of a path is the sum of the weights of its edges. Example: Shortest path between Providence and Honolulu Applications Internet packet routing Flight reservations Driving directions HNL SFO 337 LAX ORD DFW LGA PVD MIA 1 3 Properties Property 1: A subpath of a shortest path is itself a shortest path Property : There is a tree of shortest paths from a start vertex to all the other vertices Example: Tree of shortest paths from Providence HNL SFO 337 LAX ORD DFW LGA PVD MIA 1

3 Dijkstra s Algorithm The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: the graph is connected the edge weights are non-negative We grow a cloud of vertices, beginning with s and eventually covering all the vertices We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u) We update the labels of the vertices adjacent to u Edge Relaxation Consider an edge e = (u,z) such that u is the vertex most recently added to the cloud z is not in the cloud s d(u) = u e 1 d(z) = 7 z The relaxation of edge e updates distance d(z) as follows: d(z) min{d(z),d(u) + weight(e)} s d(u) = u e 1 d(z) = 6 z 6 3

4 Example B A 7 1 C Greedy algorithms: solve a problem in stages by doing what appears to be the best at each stage. D B A 7 1 C 3 D Example (cont.)

5 Data Structures Additional data structures needed are: d [ ]: array of best estimates of shortest path to each vertex prev [ ]: array of predecessors for each vertex PQ: priority queue (e.g., a min heap) Dijkstra's algorithm keeps two sets of vertices: K = set of vertices whose shortest paths from the source have already been computed/finalized (the cloud) V K = the remaining vertices (have not been finalized) At each stage, we pick the vertex u with the smallest distance label d [u] from set V K 9 Initialization for every vertex v { d[v] = ; } prev[v] = 1; d[s] = ; for every vertex v insert v into PQ; // K = { }; empty set Which node is now the root of the heap? 1

6 Main Algorithm while PQ is not empty { v = PQ.deleteMin( ); // K = K + {v }; for every vertex u adjacent to v relax (v, u, weight(v, u)) } Before and after the while loop, V = K PQ Optimization: use array flag[] to reduce the number of calls to method relax( ) flag[u] is set to true when u is added to the cloud if (flag[u] == false) then relax (v, u, weight(v, u)) 11 Relaxation Consider an edge e = (u,z) such that u is the vertex most recently added to the cloud z is not in the cloud s d(u) = u e 1 d(z) = 7 z The relaxation of edge e updates distance d(z) as follows: d(z) min{d(z),d(u) + weight(e)} s d(u) = u e 1 d(z) = 6 z 1 6

7 Relaxation Code The path {s v u} is shorter than the current path from s to u. So update the path from s to u to make it go through v. relax (v, u, weight(v, u)) { if d[u] > ( d[v] + weight(v, u) ) { d[u] = d[v] + weight(v, u); prev[u] = v; } } If d[u] is updated, the position of vertex u in the PQ must be updated accordingly (upheap percolation) 13 More Examples

8 Running Time Analysis Initialization takes O(V) time Priority queue operations Each vertex is (inserted once and) removed once from the priority queue, where each removal takes at most O(log V) time O(V log V) The d value of a vertex w in the priority queue is modified at most deg(w) times, where each value change takes at most O(log V) time O(E log V) Recall that Σ v deg(v) = E The total running time is O((V + E) log V), assuming the adjacency list structure The running time can also be expressed as O(E log V) since the graph is connected 1 Why It Doesn t Work for Negative-Weight Edges Dijkstra s algorithm is based on the greedy method. It adds vertices by increasing distance. If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud C s true distance is 1, but it is already in the cloud with d(c)=! 16

9 Graphs with Negative-Weight Edges Use Bellman-Ford algorithm 17 Final Exam Materials to be studied have been posted on the course web site under link FINAL. Office hours: Thursday, April, PM PM Tuesday, April 1, PM PM 1 9