Shortest Paths and Minimum Spanning Trees

Transcription

1 /9/ hortest Paths and Minimum panning Trees dmin avid Kauchak cs0 pring 0 ijkstra s algorithm What about ijkstra s on? 0-0

2 /9/ What about ijkstra s on? ijkstra s algorithm only works for positive edge weights 0 ounding the distance l nother invariant: or each vertex v, v] is an upper bound on the actual shortest distance l start off at l only update the value if we find a shorter distance l n update procedure = min{ = min{ = min{ an we ever go wrong applying this update rule? l We can apply this rule as many times as we want and will never underestimate v] l v] will be right if u is along the shortest path to v and is correct l onsider the shortest path from s to v When will v] be right? l If u is along the shortest path to v and is correct s p p p p k v

3 /9/ = min{ = min{ l v] will be right if u is along the shortest path to v and is correct l What happens if we update all of the vertices with the above update? l v] will be right if u is along the shortest path to v and is correct l What happens if we update all of the vertices with the above update? s p p p p k v s p p p p k v correct = min{ = min{ l v] will be right if u is along the shortest path to v and is correct l What happens if we update all of the vertices with the above update? l v] will be right if u is along the shortest path to v and is correct l oes the order that we update the vertices matter? s p p p p k v s p p p p k v correct correct correct correct

4 /9/ = min{ = min{ l v] will be right if u is along the shortest path to v and is correct l How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times l v] will be right if u is along the shortest path to v and is correct l How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times s p p p p k v s p p p p k v correct correct = min{ = min{ l v] will be right if u is along the shortest path to v and is correct l How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times l v] will be right if u is along the shortest path to v and is correct l How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times s p p p p k v s p p p p k v correct correct correct correct correct correct correct

5 /9/ = min{ = min{ l v] will be right if u is along the shortest path to v and is correct l How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times l v] will be right if u is along the shortest path to v and is correct l What is the longest (vetex-wise) the path from s to any node v can be? l V - edges/vertices s p p p p k v correct correct correct correct s p p p p k v correct correct correct correct ellman-ord algorithm ellman-ord algorithm Initialize all the distances do it V - times iterate over all edges/vertices and apply update rule

6 /9/ ellman-ord algorithm Negative cycles What is the shortest path from a to e? check for negative cycles 0-0 ellman-ord algorithm ellman-ord algorithm 0 How many edges is the shortest path from s to: - : - - -

7 /9/ ellman-ord algorithm ellman-ord algorithm 0 How many edges is the shortest path from s to: 0 How many edges is the shortest path from s to: - - : - - : : ellman-ord algorithm ellman-ord algorithm 0 How many edges is the shortest path from s to: 0 How many edges is the shortest path from s to: - - : : - - : : : 7

8 /9/ ellman-ord algorithm ellman-ord algorithm How many edges is the shortest path from s to: : : : Iteration: 0 ellman-ord algorithm ellman-ord algorithm Iteration: Iteration:

9 /9/ ellman-ord algorithm ellman-ord algorithm Iteration: has the correct distance and path Iteration: ellman-ord algorithm ellman-ord algorithm Iteration: has the correct distance and path Iteration:

10 /9/ ellman-ord algorithm orrectness of ellman-ord Iteration: 7 (and all other nodes) have the correct distance and path Loop invariant: orrectness of ellman-ord Runtime of ellman-ord Loop invariant: fter iteration i, all vertices with shortest paths from s of length i edges or less have correct distances O( V ) 0

11 /9/ Runtime of ellman-ord ll pairs shortest paths l imple approach l all ellman-ord V times l O( V ) l loyd-warshall Θ( V ) l Johnson s algorithm O( V log V + V ) an you modify the algorithm to run faster (in some circumstances)? Minimum spanning trees example l What is the lowest weight set of edges that connects all vertices of an undirected graph with positive weights l Input: n undirected, positive weight graph, =(V,) l Output: tree T=(V, ) where that minimizes weight ( T) = w e e '

12 /9/ s an an have a cycle? s an an have a cycle? pplications? lgorithm ideas? l onnectivity l Networks (e.g. communications) l ircuit design/wiring l hub/spoke models (e.g. flights, transportation) l Traveling salesman problem?

13 /9/ uts l cut is a partitioning of the vertices into two sets and V- l n edge crosses the cut if it connects a vertex u V and v V- Minimum cut property iven a partion, let edge e be the minimum cost edge that crosses the partition. very minimum spanning tree contains edge e. Prove this! Minimum cut property iven a partion, let edge e be the minimum cost edge that crosses the partition. very minimum spanning tree contains edge e. V- Minimum cut property iven a partion, let edge e be the minimum cost edge that crosses the partition. very minimum spanning tree contains edge e. V- e e e onsider an with edge e that is not the minimum edge e Using e instead of e, still connects the graph, but produces a tree with smaller weights

14 /9/ Kruskal s algorithm iven a partition, let edge e be the minimum cost edge that crosses the partition. very minimum spanning tree contains edge e. Kruskal s algorithm dd smallest edge that connects two sets not already connected Kruskal s algorithm dd smallest edge that connects two sets not already connected Kruskal s algorithm dd smallest edge that connects two sets not already connected

15 /9/ Kruskal s algorithm dd smallest edge that connects two sets not already connected Kruskal s algorithm dd smallest edge that connects two sets not already connected Kruskal s algorithm dd smallest edge that connects two sets not already connected orrectness of Kruskal s l Never adds an edge that connects already connected vertices l lways adds lowest cost edge to connect two sets. y min cut property, that edge must be part of the

16 /9/ Running time of Kruskal s Running time of Kruskal s V calls to Makeet O( log ) calls to indet V calls to Union Running time of Kruskal s algorithm isjoint set data structure O( log ) + Makeet indet calls Union V calls Total Linked lists V O( V ) V O( V + log ) O( V ) Linked lists + heuristics V O( log V ) V O( log V + log ) O( log )

17 /9/ algorithm algorithm algorithm tart at some root node and build out the by adding the lowest weighted edge at the frontier 7

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20 /9/ 0 0 orrectness of? l an we use the min-cut property? l iven a partion, let edge e be the minimum cost edge that crosses the partition. very minimum spanning tree contains edge e. l Let be the set of vertices visited so far l The only time we add a new edge is if it s the lowest weight edge from to V- 0 0

21 /9/ Running time of Prim s Running time of Prim s Θ( V ) Θ( V ) V calls to xtract-min calls to ecrease-key Running time of Prim s ame as ijksta s algorithm MakeHeap V xtractmin ecreasekey Total rray O( V ) O( V ) O( ) O( V ) in heap O( V ) O( V log V ) O( log V ) O(( V + ) log V ) O( log V ) ib heap O( V ) O( V log V ) O( ) O( V log V + ) Kruskal s: O( log )