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CS 332: Algorithms Single-Source Shortest Path
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],14 10 3 6 4 5 2 9 15 8 Run on example graph
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],        14 10 3 6 4 5 2 9 15 8 Run on example graph
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],   0     14 10 3 6 4 5 2 9 15 8 Pick a start vertex r r
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],   0     14 10 3 6 4 5 2 9 15 8 Red vertices have been removed from Q u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],   0   3  14 10 3 6 4 5 2 9 15 8 Red arrows indicate parent pointers u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],14   0   3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],14   0   3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],14   0 8  3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10   0 8  3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10   0 8  3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 2  0 8  3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 2  0 8 15 3  14 10 3 6 4 5 2 9 15 8 u
Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 2  0 8 15 3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 2 9 0 8 15 3  14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],5 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],5 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],5 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],5 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],5 2 9 0 8 15 3 4 14 10 3 6 4 5 2 9 15 8 u
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What is the hidden cost in this code?
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],How often is ExtractMin() called? How often is DecreaseKey() called?
Review: Prim’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time? A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)
Single-Source Shortest Path ,[object Object],[object Object],[object Object],[object Object]
Shortest Path Properties ,[object Object],[object Object],[object Object],[object Object],[object Object]
Shortest Path Properties ,[object Object],[object Object],[object Object],x u v This path is no longer than any other path
Shortest Path Properties ,[object Object],< 0
Relaxation ,[object Object],[object Object],[object Object],[object Object],[object Object],9 5 2 7 5 2 Relax 6 5 2 6 5 2 Relax
Bellman-Ford Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Initialize d[], which will converge to shortest-path value  Relaxation: Make |V|-1 passes, relaxing each edge Test for solution Under what condition do we get a solution?
Bellman-Ford Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time?
Bellman-Ford Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time? A: O(VE)
Bellman-Ford Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],B E D C A -1 2 2 1 -3 5 3 4 Ex: work on board s
Bellman-Ford ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Bellman-Ford ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DAG Shortest Paths ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Dijkstra’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object]
Dijkstra’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Relaxation Step Note: this is really a call to Q->DecreaseKey() B C D A 10 4 3 2 1 5 Ex: run the algorithm
Dijkstra’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],How many times is ExtractMin() called? How many times is DecraseKey() called? What will be the total running time?
Dijkstra’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],How many times is ExtractMin() called? How many times is DecraseKey() called? A: O(E lg V) using binary heap for Q Can acheive O(V lg V + E) with Fibonacci heaps
Dijkstra’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Correctness: we must show that when u is removed from Q, it has already converged
Correctness Of Dijkstra's Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],s x y u p 2 p 2
Correctness Of Dijkstra's Algorithm ,[object Object],[object Object],[object Object],[object Object],s x y u p 2 p 2
Disjoint-Set Union Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1? 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2? 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5? 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8? 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9? 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13? 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14? 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17? 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19? 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21? Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25? 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:
Correctness Of Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } What will affect the running time?
Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } What will affect the running time? 1 Sort O(V) MakeSet() calls O(E) FindSet() calls O(V) Union() calls ( Exactly how many Union()s? )
Kruskal’s Algorithm: Running Time ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Disjoint Set Union ,[object Object],[object Object],[object Object],[object Object],[object Object]
Disjoint Set Union ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Disjoint Set Union: Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Amortized Analysis of Disjoint Sets ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Amortized Analysis of Disjoint Sets ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
The End

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lecture 20

  • 1. CS 332: Algorithms Single-Source Shortest Path
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  • 72. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } What will affect the running time?
  • 73. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } What will affect the running time? 1 Sort O(V) MakeSet() calls O(E) FindSet() calls O(V) Union() calls ( Exactly how many Union()s? )
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