Single Source, Shortest Path Problem

Transcription

1 Lecture : From ijkstra to Prim Today s Topics: ijkstra s Shortest Path lgorithm epth First Search Spanning Trees Minimum Spanning Trees Prim s lgorithm overed in hapter 9 in the textbook Some slides based on: S by S. Wolfman, 000 Single Source, Shortest Path Problem Given a graph G = (V, ) and a source vertex s in V, find the minimum cost paths from s to every vertex in V 9 8 Source

2 ijkstra s Shortest Path lgorithm. Initialize the cost of each node to. Initialize the cost of the source to 0. While there are unknown nodes left in the graph. Select the unknown node N with the lowest cost (greedy choice). Mark N as known. For each node adjacent to N If (N s cost + cost of (N, )) < s cost s cost = N s cost + cost of (N, ) Prev[] =N //store preceding node 9 8 (Prev allows paths to be reconstructed) ijkstra s lgorithm (greed in action) vertex known cost No No Yes 0 No No Initial Prev vertex known cost Yes 8 Yes 0 Yes 0 Yes Yes Final Prev -

3 nalysis of ijkstra s lgorithm Main loop: While there are unknown nodes left in the graph. Select the unknown node N with the lowest cost. Mark N as known. For each node adjacent to N If (N s cost + cost of (N, )) < s cost s cost = N s cost + cost of (N, ) V times O( V ) O( ) total Total time = V (O( V )) +O( ) = O( V + ) ense graph: =Θ( V )! Totaltime=O( V )=O( ) Sparse graph: =Θ( V )! Totaltime=O( V )=O( ) χ Quadratic! an we do better? nalysis of ijkstra s lgorithm Yes! Use a priority queue to store vertices with key = cost V times: Select the unknown node N with the lowest cost times: deletemin s cost = N s cost + cost of (N, ) decreasekey 7 Total run time =?

4 nalysis of ijkstra s lgorithm Yes! Use a priority queue to store vertices with key = cost V times: Select the unknown node N with the lowest cost times: deletemin s cost = N s cost + cost of (N, ) decreasekey 7 Total run time = O( V log V + log V ) 7 oes ijkstra s lgorithm lways Work? ijkstra s algorithm is an example of a greedy algorithm Greedy algorithms always make choices that currently seem the best Short-sighted no consideration of long-term or global issues Locally optimal does not always mean globally optimal In ijkstra s case choose the least cost node, but what if there is another path through other vertices that is cheaper? an prove: Never happens if all edge weights are positive 8

5 The loudy Proof of ijkstra s orrectness Least cost node G Next shortest path from inside the known cloud P TH KNOWN LOU Source If the path to G is the next shortest path, the path to P must be at least as long. Therefore, any path through P to G cannot be shorter! 9 Inside the loud (Proof) verything inside the cloud has the correct shortest path Proof is by induction on the # of nodes in the cloud: ase case: Initial cloud is just the source with shortest path 0 Inductive hypothesis: cloud of k- nodes all have shortest paths Inductive step: choose the least cost node G! has to be the shortest path to G (previous slide). dd k th node G to the cloud 0

6 Inside the loud (Proof) verything inside the cloud has the correct shortest path Proof is by induction on the # of nodes in the cloud: ase case: Initial cloud is just the source with shortest path 0 Inductive hypothesis: cloud of k- nodes all have shortest paths Inductive step: choose the least cost node G! has to be the shortest path to G (previous slide). dd k th node G to the cloud ut waitaminute!! What about negative weights?? Kevin acon Gotcha!! Negative Weights: ijkstra s chilles Heel ijkstra:! (cost=-) Leastcostpath:!! (cost=-8) Negative cycles: What s the shortest path from to?(orto,,or, for that matter)

7 epth First Search (FS) We used readth First Search for finding shortest paths in an unweighted graph Use a queue to explore neighbors of source vertex, neighbors of each neighbor, and so on: edge away, two edges away, etc. Its counterpart: epth First Search second way to explore all nodes in a graph FS searches down one path as deep as possible When no new nodes available, it backtracks When backtracking, we explore side-paths that weren t taken FS allows an easy recursive implementation So, FS uses a stack while FS uses a queue FS Pseudocode Pseudocode for FS: FS(v) If v is unvisited mark v as visited print v (or process v) for each edge (v,w) FS(w) Works for directed or undirected graphs FS() FS() Running time = O( V + )

8 What about FS on this graph? What happens when you do FS( )? 70 Go as deep as possible, Then backtrack 78 0 We get a spanning tree

9 FS and FS may give different trees FS() FS() 7 Spanning Tree efinition Spanning tree: a subset of edges from a connected graph that: touches all vertices in the graph (spans the graph) forms a tree (is connected and contains no cycles) 7 9 Minimum spanning tree: the spanning tree with the least total edge cost 8

10 Minimum Spanning Tree (MST) We are given a weighted, undirected graph G =(V, ), with weight function w:! R mapping edges to real valued weights Problem: Findthe minimum cost spanning tree Why minimum spanning trees? Lots of applications Minimize length of gas pipelines between cities Find cheapest way to wire a house (with minimum cable) Find a way to connect various routers on a network that minimizes total delay tc 0

11 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={} v0 0 8 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!) v0 0 v 8

12 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!). dd v to V and the edge to if no cycle is created v0 0 v 8 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!). dd v to V and the edge to if no cycle is created. Repeat until all vertices have been added v0 0 8 v

13 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!). dd v to V and the edge to if no cycle is created. Repeat until all vertices have been added v0 0 8 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!). dd v to V and the edge to if no cycle is created. Repeat until all vertices have been added v0 0 8

14 Prim s lgorithm for Finding the MST. Starting from an empty tree, T,pickavertex,v0, at random and initialize: V ={v0} and ={}. hoose a vertex v not in V such that edge weight from v to a vertex in V is minimal (greedy again!). dd v to V and the edge to if no cycle is created. Repeat until all vertices have been added v Prim s lgorithm for Finding the MST one! Totalcost=++++ =0 8

15 Next lass: nalysis of Prim s lgorithm Kruskal takes a bow faster MST To o: Programming ssignment # (on t wait until the last few days!!!) ontinue chapter 9 9