Shortest Paths and Minimum Spanning Trees

Transcription

1 // hortest Paths and Minimum panning Trees avid Kauchak cs pring dmin an resubmit homeworks - for up to half credit back l ue by the end of the week Read book

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6 // Is ijkstra s algorithm correct? Invariant: Is ijkstra s algorithm correct? Invariant: or every vertex removed from the heap, dist[v] is the actual shortest distance from s to v Is ijkstra s algorithm correct? Invariant: or every vertex removed from the heap, dist[v] is the actual shortest distance from s to v proof? l The only time a vertex gets visited is when the distance from s to that vertex is smaller than the distance to any remaining vertex l Therefore, there cannot be any other path that hasn t been visited already that would result in a shorter path

7 // Running time? Running time? call to Make Running time? Running time? V iterations V calls 7

8 // Running time? Running time? epends on the heap implementation Make V xtractmin ecreasekey Total rray O( V ) O( V ) O( ) O( V ) O( ) calls in heap O( V ) O( V log V ) O( log V ) O(( V + ) log V ) O( log V ) Running time? epends on the heap implementation Make V xtractmin ecreasekey Total rray O( V ) O( V ) O( ) O( V ) Running time? epends on the heap implementation Make 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 ) in heap O( V ) O( V log V ) O( log V ) O(( V + ) log V ) O( log V ) Is this an improvement? If < V / log V ib heap O( V ) O( V log V ) O( ) O( V log V + )

9 // What about ijkstra s on? What about ijkstra s on? ijkstra s algorithm only works for positive edge weights - ounding the distance nother invariant: or each vertex v, dist[v] is an upper bound on the actual shortest distance ounding the distance nother invariant: or each vertex v, dist[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 n update procedure dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} Is this a valid invariant? 9

10 // dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} 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 dist[v] When will dist[v] be right? l If u is along the shortest path to v and dist[u] is correct dist[v] will be right if u is along the shortest path to v and dist[u] is correct onsider the shortest path from s to v s p p p p k v dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist[v] will be right if u is along the shortest path to v and dist[u] is correct dist[v] will be right if u is along the shortest path to v and dist[u] is correct What happens if we update all of the vertices with the above update? 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

11 // dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist[v] will be right if u is along the shortest path to v and dist[u] is correct What happens if we update all of the vertices with the above update? dist[v] will be right if u is along the shortest path to v and dist[u] is correct 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 dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist[v] will be right if u is along the shortest path to v and dist[u] is correct How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times dist[v] will be right if u is along the shortest path to v and dist[u] is correct 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

12 // dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist[v] will be right if u is along the shortest path to v and dist[u] is correct How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times dist[v] will be right if u is along the shortest path to v and dist[u] is correct 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 dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist [ v] = min{ dist[ v], dist[ u] + w( u, v)} dist[v] will be right if u is along the shortest path to v and dist[u] is correct How many times do we have to do this for vertex p i to have the correct shortest path from s? l i times dist[v] will be right if u is along the shortest path to v and dist[u] is correct What is the longest (vertex-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 v p p p k correct correct correct correct

13 // ellman-ord algorithm ellman-ord algorithm Initialize all the distances do it V - times iterate over all edges/vertices and apply update rule ellman-ord algorithm Negative cycles What is the shortest path from a to e? check for negative cycles -

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

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

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

17 // ellman-ord algorithm ellman-ord algorithm - Iteration: has the correct distance and path - Iteration: ellman-ord algorithm orrectness of ellman-ord Iteration: 7 (and all other nodes) have the correct distance and path Loop invariant: 7

18 // 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 ) Runtime of ellman-ord ingle source shortest paths ll of the shortest path algorithms we ve looked at today are call single source shortest paths algorithms Why? an you modify the algorithm to run faster (in some circumstances)?

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

20 // s an an have a cycle? pplications? onnectivity l Networks (e.g. communications) l ircuit design/wiring hub/spoke models (e.g. flights, transportation) Traveling salesman problem? lgorithm ideas? uts cut is a partitioning of the vertices into two sets and V- n edge crosses the cut if it connects a vertex u V and v V-

21 // 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- e e onsider an with edge e that is not the minimum edge 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- 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. e e Using e instead of e, still connects the graph, but produces a tree with smaller weights

22 // Kruskal s algorithm dd smallest edge that connects two sets not already connected 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 Kruskal s algorithm dd smallest edge that connects two sets not already connected

23 // 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 Never adds an edge that connects already connected vertices Running time of Kruskal s lways adds lowest cost edge to connect two sets. y min cut property, that edge must be part of the

24 // Running time of Kruskal s Running time of Kruskal s isjoint set data structure O( log ) + V calls to Makeet Makeet indet calls Union V calls Total O( log ) calls to indet V calls to Union 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 ) Prim s algorithm Prim s algorithm

25 // Prim s algorithm Prim s algorithm tart at some root node and build out the by adding the lowest weighted edge at the frontier Prim s Prim s

26 // Prim s Prim s Prim s Prim s

27 // 7 Prim s Prim s Prim s Prim s

28 // Prim s Prim s orrectness of Prim s? Running time of Prim s 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. Let be the set of vertices visited so far The only time we add a new edge is if it s the lowest weight edge from to V-

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