Outline. (single-source) shortest path. (all-pairs) shortest path. minimum spanning tree. Dijkstra (Section 4.4) Bellman-Ford (Section 4.
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1 Weighted Graphs 1
2 Outline (single-source) shortest path Dijkstra (Section 4.4) Bellman-Ford (Section 4.6) (all-pairs) shortest path Floyd-Warshall (Section 6.6) minimum spanning tree Kruskal (Section 5.1.3) Prim (Section 5.1.5)
3 Shortest Path Problems How can we find the shortest route between two points on a road map? Model the problem as a graph problem: Road map is a weighted graph: vertices = cities edges = road segments between cities edge weights = road distances Goal: find a shortest path between two vertices (cities) 3
4 Shortest Path Problem Input: Directed graph G = (V, E) Weight function w : E R Weight of path p = v 0, v 1,..., v k w( p) k i 1 w( v i 1, v i ) Shortest-path weight from u to v: s 0 t y x z 7 δ(u, v) = min w(p) : u p v if there exists a path from u to v otherwise Note: there might be multiple shortest paths from u to v 4
5 Variants of Shortest Path Single-source shortest paths G = (V, E) find a shortest path from a given source vertex s to each vertex v V Single-destination shortest paths Find a shortest path to a given destination vertex t from each vertex v Question: How to solve it? 5
6 Variants of Shortest Path Single-source shortest paths G = (V, E) find a shortest path from a given source vertex s to each vertex v V Single-destination shortest paths Find a shortest path to a given destination vertex t from each vertex v Reversing the direction of each edge single-source 6
7 Variants of Shortest Paths (cont d) Single-pair shortest path Find a shortest path from u to v for given vertices u and v All-pairs shortest-paths Find a shortest path from u to v for every pair of vertices u and v 7
8 Shortest-Path Problems Single-source (single-destination). Find a shortest path from a given source (vertex s) to all the other vertices positive weights greedy algorithm pos. & neg. weights dynamic programming All-pairs. Find shortest-paths for every pair of vertices pos. & neg. weights dynamic programming 8
9 Optimal Substructure Theorem Given: v j A weighted, directed graph G = (V, E) v 1 p jk A weight function w: E R, p 1i p ij p ij v k A shortest path p = v 1, v,..., v k from v 1 to v k v i A subpath of p: p ij = v i, v i+1,..., v j, with 1 i j k Then: p ij is a shortest path from v i to v j p 1i p ij p jk Proof: p = v 1 v i v j v k w(p) = w(p 1i ) + w(p ij ) + w(p jk ) Assume p ij from v i to v j with w(p ij ) < w(p ij ) w(p ) = w(p 1i ) + w(p ij ) + w(p jk ) < w(p) contradiction! 9
10 Triangle Inequality for positive weights s For all (u, v) E, we have: δ (s, v) δ (s, u) + δ (u, v) u v - If u is on the shortest path to v we have the equality sign s u v 10
11 Algorithms Operations common in both algorithms: Initialization Relaxation Dijkstra s algorithm Negative weights are not allowed Bellman-Ford algorithm Negative weights are allowed Negative cycles reachable from the source are not allowed. 11
12 Shortest-Paths Notation For each vertex v V: δ(s, v): shortest-path weight d[v]: shortest-path weight estimate Initially, d[v]= d[v] δ(s,v) as algorithm progresses [v] = predecessor of v on a shortest path from s If no predecessor, [v] = NIL induces a tree shortest-path tree s 0 t y x z 7 1
13 Initialization Alg.: INITIALIZE-SINGLE-SOURCE(V, s) 1. for each v V. do d[v] 3. [v] NIL 4. d[s] 0 All the shortest-paths algorithms start with INITIALIZE-SINGLE-SOURCE 13
14 Relaxation Step Relaxing an edge (u, v) = testing whether we can improve the shortest path to v found so far by going through u If d[v] > d[u] + w(u, v) we can improve the shortest path to v s u d[v]=d[u]+w(u,v) [v] u v 5 9 s u v 5 6 After relaxation: d[v] d[u] + w(u, v) RELAX(u, v, w) RELAX(u, v, w) u v 5 7 u v 5 6 no change 14
15 Dijkstra s algorithm 15
16 Dijkstra s Algorithm Single-source shortest path problem: positive-weight edges: w(u, v) > 0, (u, v) E Each edge is relaxed only once! Maintains two sets of vertices: d[v]=δ (s, v) d[v]>δ (s, v) 16
17 Dijkstra s Algorithm (cont.) Vertices in V S reside in a min-priority queue Keys in Q are estimates of shortest-path weights d[u] Repeatedly select a vertex u V S, with the minimum shortest-path estimate d[u] Relax all edges leaving u 17
18 Dijkstra (G, w, s) s 0 S=<> Q=<s,t,x,z,y> S=<s> Q=<y,t,x,z> 10 t 3 5 y x z s 0 10 t y x z 18
19 Example (cont.) 10 t x 10 t x s s y 7 z y 7 z S=<s,y> Q=<z,t,x> S=<s,y,z> Q=<t,x> 19
20 Example (cont.) S=<s,y,z,t> Q=<x> S=<s,y,z,t,x> Q=<> s t y 9 x z s t y 9 x z 0
21 Dijkstra (G, w, s) 1. INITIALIZE-SINGLE-SOURCE(V, s). S 3. Q V[G] 4. while Q 5. do u EXTRACT-MIN(Q) 6. S S {u} O(V) build min-heap 7. for each vertex v Adj[u] 8. do RELAX(u, v, w) Executed O(V) times O(lgV) 9. Update Q (DECREASE_KEY) (V) O(VlgV) O(E) times (total) O(lgV) O(ElgV) Running time: O(VlgV + ElgV) = O(ElgV) 1
22 Binary Heap vs Fibonacci Heap Running time depends on the implementation of the heap
23 Correctness Dijkstra s algorithm is a greedy algorithm make choices that currently seem the best locally optimal does not always mean globally optimal Correct because maintains following two properties: for every known vertex, recorded distance is shortest distance to that vertex from source vertex for every unknown vertex v, its recorded distance is shortest path distance to v from source vertex, considering only currently known vertices and v Let s prove the Correctness 3
24 Correctness of Dijskstra s Algorithm For each vertex u V, we have d[u] = δ(s, u) at the time when u is added to S. Proof: Let u be the first vertex for which d[u] δ(s, u) when added to S Let s look at a true shortest path p from s to u: 4
25 Correctness of Dijskstra s Algorithm What is the value of d[u]? d[u] d[v]+w(v,u)= δ(s,v)+w(v,u) What is the value of d[u ]? d[u ] d[v ]+w(v,u )= δ(s,v )+w(v,u ) Since u is in the shortest path of u: d[u ]<δ(s,u) Using the upper bound property: d[u]>δ(s,u) d[u ]<d[u] Contradiction! Priority Queue Q: <u,, u,.> (i.e., d[u]< <d[u ]< ) 5
26 Consider the graph: Example the distances are appropriately initialized all vertices are marked as being unvisited
27 Example Visit vertex 1 and update its neighbours, marking it as visited the shortest paths to, 4, and 5 are updated
28 Example The next vertex we visit is vertex 4 vertex don t update vertex < update vertex < update
29 Next, visit vertex Example vertex < update vertex 4 already visited vertex don t update vertex < update
30 Example Next, we have a choice of either 3 or 6 We will choose to visit 3 vertex < 8 update vertex don t update
31 We then visit 6 Example vertex don t update vertex < update
32 Example Next, we finally visit vertex 5: vertices 4 and 6 have already been visited vertex < 10 update vertex < 9 update vertex don t update
33 Example Given a choice between vertices 7 and 8, we choose vertex 7 vertices 5 has already been visited vertex don t update
34 Next, we visit vertex 8: Example vertex < 13 update
35 Example Finally, we visit the end vertex Therefore, the shortest path from 1 to 9 has length 11
36 Example We can find the shortest path by working back from the final vertex: 9, 8, 5, 3,, 1 Thus, the shortest path is (1,, 3, 5, 8, 9)
37 Example 3 37
38 Example 3 38
39 Example 3 39
40 Example 3 40
41 Example 3 41
42 Example 3 4
43 Example 3 43
44 Example 3 44
45 Example 4: Initialization Distance(source) = 0 0 A B Distance (all vertices but source) = C D E F 1 G Pick vertex in List with minimum distance. 45
46 Example 4: Update neighbors' distance 0 A B C D E Distance(B) = Distance(D) = 1 F 1 G 46
47 Example 4: Remove vertex with minimum distance 0 A B C D E F 1 G Pick vertex in List with minimum distance, i.e., D 47
48 Example 4: Update neighbors 0 A B C D E Distance(C) = 1 + = 3 Distance(E) = 1 + = 3 Distance(F) = = 9 Distance(G) = = 5 F 1 G
49 Example 4: Continued... Pick vertex in List with minimum distance (B) and update neighbors 0 A B C D E 3 5 F G Note : distance(d) not updated since D is already known and distance(e) not updated since it is larger than previously computed 49
50 Example 4: Continued... Pick vertex List with minimum distance (E) and update neighbors 0 A B C D E F 1 G 9 5 No updating 50
51 Example 4: Continued... Pick vertex List with minimum distance (C) and update neighbors 0 A B C D E Distance(F) = = 8 F 1 G
52 Example 4: Continued... Pick vertex List with minimum distance (G) and update neighbors 0 A B C D E Previous distance Distance(F) = min (8, 5+1) = 6 F 1 G 6 5 5
53 Example 4 (end) 0 A B C D E F 1 G 6 5 Pick vertex not in S with lowest cost (F) and update neighbors 53
54 Example 5: s=bwi 54
55 Example 5 55
56 Example 5 56
57 Example 5 57
58 Example 5 58
59 Negative weights Dijkstra fails on graphs with negative edges Example: Bringing z into S and performing edge relaxation invalidates the previously computed shorted path distance (14) to x S 59
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