Today’s Outline CSE 326: Data Structures Shortest path algorithms Graphs, Paths & Dijkstra’s 1. Unweighted graphs: BFS 2. Weighted graphs without negative cost edges: Algorithm Dijkstra’s Algorithm 3. Negative cost edges but no negative cost Hal Perkins cycles Spring 2007 Lectures 22-23 Reading: Weiss, Ch. 9 2 Graph Connectivity Graph Traversals Undirected graphs are connected if there is a path between any two vertices • Breadth-first search (and depth-first search) work for arbitrary (directed or undirected) graphs - not just mazes! Directed graphs are strongly connected if there is a path from any – Must mark visited vertices so you do not go into an infinite one vertex to any other loop! • Either can be used to determine connectivity : – Is there a path between two given vertices? Directed graphs are weakly connected if there is a path between – Is the graph (weakly) connected? any two vertices, ignoring direction • Which one: – Uses a queue? A complete graph has an edge between every pair of vertices – Uses a stack? – Always finds the shortest path (for unweighted graphs)? 3 4 1
Single Source Shortest Paths (SSSP) The Shortest Path Problem Given a graph G, edge costs c i,j , and vertices s Given a graph G, edge costs c i,j , and vertex s , and t in G , find the shortest path from s to t . find the shortest paths from s to all vertices in G. For a path p = v 0 v 1 v 2 … v k – Is this harder or easier than the previous problem? – unweighted length of path p = k (a.k.a. length ) – weighted length of path p = ∑ i =0.. k -1 c i,i +1 (a.k.a cost ) Path length equals path cost when ? 5 6 All Pairs Shortest Paths (APSP) Variations of SSSP Given a graph G and edge costs c i,j , find the shortest paths between all pairs of vertices in G. – Weighted vs. unweighted – Directed vs undirected – Is this harder or easier than SSSP? – Cyclic vs. acyclic – Positive weights only vs. negative weights allowed – Shortest path vs. longest path – Could we use SSSP as a subroutine to solve this? – … 7 8 2
Applications SSSP: Unweighted Version Ideas? – Network routing – Driving directions – Cheap flight tickets – Critical paths in project management (see textbook) – … 9 10 Weighted SSSP: void Graph::unweighted (Vertex s){ The Quest For Food Queue q(NUM_VERTICES); Vertex v, w; Home q.enqueue(s); Ben & Jerry’s Cedars s.dist = 0; Neelam’s 40 U Village 285 Delfino’s 70 75 365 25 while (!q.isEmpty()){ 375 Coke Closet v = q.dequeue(); each edge examined 350 The Ave 350 10 at most once – if adjacency for each w adjacent to v 25 HUB lists are used ALLEN if (w.dist == INFINITY){ 35 35 15 Café Allegro w.dist = v.dist + 1; Schultzy’s Vending Machine in EE1 w.path = v; each vertex enqueued 15,356 q.enqueue(w); at most once } Parent’s Home } } total running time: O( ) Can we calculate shortest distance to all nodes from Allen Center? 11 12 3
Dijkstra, Edsger Wybe Dijkstra’s Algorithm: Idea Adapt BFS to handle Legendary figure in computer science; weighted graphs was a professor at University of Texas. Supported teaching introductory Two kinds of vertices: computer courses without computers – Finished or known (pencil and paper programming) vertices • Shortest distance has Supposedly wouldn’t (until very late in been computed life) read his e-mail; so, his staff had to E.W. Dijkstra (1930-2002) – Unknown vertices print out messages and put them in his • Have tentative box. distance 1972 Turning Award Winner, Programming Languages, semaphores, and … 13 14 Dijkstra’s Algorithm: Idea Dijkstra’s Algorithm: Pseudocode Initialize the cost of each node to ∞ At each step: Initialize the cost of the source to 0 1) Pick closest unknown vertex 2) Add it to known vertices While there are unknown nodes left in the graph 3) Update distances Select an unknown node b with the lowest cost Mark b as known For each node a adjacent to b a ’s cost = min( a ’s old cost, b ’s cost + cost of ( b , a )) 15 16 4
2 v 1 s v 0 void Graph::dijkstra(Vertex s){ Vertex v,w; 5 1 1 2 1 1 v 2 v 3 v 4 Initialize s.dist = 0 and set dist of all other vertices to infinity 6 2 while (there exist unknown vertices, find the 5 3 one b with the smallest distance) V Known Dist path b.known = true; 10 v 5 v 6 v0 for each a adjacent to b v1 if (!a.known) if (b.dist + Cost_ba < a.dist){ v2 decrease(a.dist to= b.dist + Cost_ba); a.path = b; v3 } v4 } } v5 v6 17 18 Dijkstra’s Alg: Implementation Dijkstra’s Algorithm: Summary Initialize the cost of each node to ∞ • Classic algorithm for solving SSSP in weighted graphs without negative weights Initialize the cost of the source to 0 While there are unknown nodes left in the graph • A greedy algorithm (irrevocably makes decisions Select the unknown node b with the lowest cost without considering future consequences) Mark b as known For each node a adjacent to b a ’s cost = min( a ’s old cost, b ’s cost + cost of ( b , a )) • Intuition for correctness: – shortest path from source vertex to itself is 0 What data structures should we use? – cost of going to adjacent nodes is at most edge weights – cheapest of these must be shortest path to that node – update paths for new node and continue picking cheapest Running time? path 19 20 5
Correctness: The Cloud Proof Correctness: Inside the Cloud Next shortest path from Prove by induction on # of nodes in the cloud: inside the known cloud Initial cloud is just the source with shortest path 0 V Assume: Everything inside the cloud has the correct Better path shortest path to V? No! The Known Inductive step: Only when we prove the shortest Cloud path to some node v (which is not in the cloud) is W correct, we add it to the cloud Source How does Dijkstra’s decide which vertex to add to the Known set next? When does Dijkstra’s algorithm not work? • If path to V is shortest, path to W must be at least as long (or else we would have picked W as the next vertex) 21 22 • So the path through W to V cannot be any shorter! The Trouble with Dijkstra’s vs BFS Negative Weight Cycles At each step: At each step: 1) Pick vertex from queue 1) Pick closest unknown vertex 2) Add it to visited vertices 2) Add it to finished vertices 3) Update queue with neighbors 3) Update distances 2 A B 10 Breadth-first Search Dijkstra’s Algorithm -5 E 1 2 Some Similarities : C D What’s the shortest path from A to E? Problem? 23 24 6
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