Single Source Shortest Paths
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1 CMPS 00 Fall 015 Single Source Shortest Paths Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 1
2 Paths in graphs Consider a digraph G = (V, E) with an edge-weight function w : E R. The weight of path p = v 1 v... v k is defined to be Example: k 1 i 1 w ( p) w( v i, v i 1). v 1 4 v 5 1 v 5 v v 4 w(p) =
3 Shortest paths A shortest path from u to v is a path of minimum weight from u to v. The shortest-path weight from u to v is defined as (u, v) = min{w(p) : p is a path from u to v}. Note: (u, v) = if no path from u to v exists.
4 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Proof. Cut and paste: 4
5 Triangle inequality Theorem. For all u, v, x V, we have (u, v) (u, x) + (x, v). Proof. (u,v) minimizes over all paths from u to v u (u, v) v Concatenating two shortest paths from u to x and from x to v yields one specific path from u to v (u, x) (x, v) x 5
6 Well-definedness of shortest paths If a graph G contains a negative-weight cycle, then some shortest paths may not exist. Example: u -6 5 < 0-8 v 6
7 Single-source shortest paths Problem. From a given source vertex s V, find the shortest-path weights (s, v) for all v V. Assumption: All edge weights w(u, v) are non-negative. It follows that all shortest-path weights must exist. IDEA: Greedy. 1. Maintain a set S of vertices whose shortest-path weights from s are known, i.e., d[v]= (s,v). At each step add to S the vertex u V S whose distance estimate d[u] from s is minimal.. Update the distance estimates d[v] of vertices v adjacent to u. 7
8 Dijkstra s algorithm d[s] 0 for each v V {s} do d[v] S Vertices for which d[v]=d(s,v) Q V Q is a priority queue maintaining V S sorted by d-values d[v] while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) relaxation step implicit DECREASE-KEY in Q 8
9 Example of Dijkstra s algorithm Graph with nonnegative edge weights: A B D C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 9
10 Example of Dijkstra s Initialize: algorithm B D S: {} 0 A Q: A B C D E 0 C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v)
11 Example of Dijkstra s algorithm A EXTRACT-MIN(Q): S: { A } 0 A B D Q: A B C D E 0 C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 11
12 Example of Dijkstra s algorithm Relax all edges leaving A: S: { A } 0 A B D Q: A B C D E 0 C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 1
13 Example of Dijkstra s algorithm C EXTRACT-MIN(Q): S: { A, C } 0 A B D Q: A B C D E 0 C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 1
14 Example of Dijkstra s algorithm Relax all edges leaving C: S: { A, C } 0 A 7 B 11 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 14
15 Example of Dijkstra s algorithm E EXTRACT-MIN(Q): S: { A, C, E } 0 A 7 B 11 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 15
16 Example of Dijkstra s algorithm Relax all edges leaving E: S: { A, C, E } 0 A 7 B 11 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 16
17 Example of Dijkstra s algorithm B EXTRACT-MIN(Q): S: { A, C, E, B } 0 A 7 B 11 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 17
18 Example of Dijkstra s algorithm Relax all edges leaving B: S: { A, C, E, B } 0 A 7 B 9 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 18
19 Example of Dijkstra s algorithm D EXTRACT-MIN(Q): S: { A, C, E, B, D}0 A 7 B 9 D Q: A B C D E C E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 19
20 Analysis of Dijkstra V times degree(u) times while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) Handshaking Lemma ( E ) implicit DECREASE-KEY s. Time = ( V ) T EXTRACT -MIN + ( E ) T DECREASE-KEY 0
21 Analysis of Dijkstra (continued) Time = ( V ) T E XTRACT-MIN + ( E ) T D ECREASE-KEY Q T EXTRACT -MIN T DECREASE-KEY Total array O( V ) O(1) O( V ) binary heap Fibonacci heap O(log V ) O(log V ) O( E log V ) O(log V ) amortized O(1) O( E + V log V ) amortized worst case 1
22 Correctness Theorem. (i) For all v S: d[v] = (s, v) (ii) For all v S: d[v] = weight of shortest path from s to v that uses only (besides v itself) vertices in S. Corollary. Dijkstra s algorithm terminates with d[v] = (s, v) for all v V.
23 Correctness Theorem. (i) For all v S: d[v] = (s, v) (ii) For all v S: d[v] = weight of shortest path from s to v that uses only (besides v itself) vertices in S. Proof. By induction. Base: Before the while loop, d[s]=0 and d[v]= for all v s, so (i) and (ii) are true. Step: Assume (i) and (ii) are true before an iteration; now we need to show they remain true after another iteration. Let u be the vertex added to S, so d[u] d[v] for all other v S.
24 Correctness Theorem. (i) For all v S: d[v] = (s, v) (ii) For all v S: d[v] = weight of shortest path from s to v that uses only (besides v itself) vertices in S. (i) Need to show that d[u] = (s, u). Assume the contrary. There is a path p from s to u with w(p) < d[u]. Because of (ii) that path uses vertices S, in addition to u. Let y be first vertex on p such that y S. S, just before adding u. s x y u Path p from s to u 4
25 Correctness Theorem. (i) For all v S: d[v] = (s, v) (ii) For all v S: d[v] = weight of shortest path from s to v that uses only (besides v itself) vertices in S. u S, just before adding u. s x y Path p from s to u d[y] w(p) < d[u]. Contradiction to the choice of u. weights are nonnegative assumption about path 5
26 Correctness Theorem. (i) For all v S: d[v] = (s, v) (ii) For all v S: d[v] = weight of shortest path from s to v that uses only (besides v itself) vertices in S. (ii) Let v S. Let p be a shortest path from s to v that uses only (besides v itself) vertices in S. p does not contain u: (ii) true by inductive hypothesis p contains u: p consists of vertices in S\{u} and ends with an edge from u to v. w(p)=d[u]+w(u,v), which is the value of d[v] after adding u. So (ii) is true. 6
27 Unweighted graphs Suppose w(u, v) = 1 for all (u, v) E. Can the code for Dijkstra be improved? Use a simple FIFO queue instead of a priority queue. Breadth-first search while Q do u DEQUEUE(Q) for each v Adj[u] do if d[v] = then d[v] d[u] + 1 ENQUEUE(Q, v) Analysis: Time = O( V + E ). while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) 7
28 Example of breadth-first search a b c d e f g h i Q: 8
29 Example of breadth-first search 0 a f h d b g e i c 0 Q: a 9
30 Example of breadth-first search 0 a 1 d f h 1 b g e i c 1 1 Q: a b d 0
31 Example of breadth-first search 0 1 a b c 1 d e 1 Q: a b d c e f g h i 1
32 Example of breadth-first search 0 1 a b c 1 d e Q: a b dc e f g h i
33 Example of breadth-first search 0 1 a b c 1 d e Q: a b dce f g h i
34 Example of breadth-first search 0 1 a b c 1 d e Q: a b d c e g i f g h i 4
35 Example of breadth-first search 0 1 a b c 1 d e 4 f g 4 Q: a b d c e g i f h i 5
36 0 1 Example of breadth-first search a b c 1 d e f g Q: a b d c e g i f h h i 6
37 0 1 Example of breadth-first search a b c 1 d e f g Q: a b d c e g i f h h i 7
38 Example of breadth-first search 0 1 a b c 1 d e 4 4 f h g i Q: a b d c e g i f h 8
39 Example of breadth-first search 0 1 a b c 1 d e 4 4 f h g i Q: a b d c e g i f h 9
40 Correctness of BFS while Q do u DEQUEUE(Q) for each v Adj[u] do if d[v] = then d[v] d[u] + 1 ENQUEUE(Q, v) Key idea: The FIFO Q in breadth-first search mimics the priority queue Q in Dijkstra. Invariant: v comes after u in Q implies that d[v] = d[u] or d[v] = d[u]
41 How to find the actual shortest paths? Store a predecessor tree: d[s] 0 for each v V {s} do d[v] S Q V while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u Q is a priority queue maintaining V S 41
42 Example of Dijkstra s algorithm Graph with nonnegative edge weights: A B D C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 4
43 Example of Dijkstra s Initialize: algorithm B D S: {} 0 A Q: A B C D E 0 C E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 4
44 Example of Dijkstra s algorithm A EXTRACT-MIN(Q): S: { A } 0 : A B C D E Q: A B C D E 0 A B C D E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 44
45 Example of Dijkstra s algorithm Relax all edges leaving A: S: { A } 0 : A B C D E Q: - A B C D E 0 A B C D E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 45
46 Example of Dijkstra s algorithm Relax all edges leaving A: S: { A } 0 : A B C D E Q: - A B C D E 0 A B C D E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 46
47 Example of Dijkstra s algorithm C EXTRACT-MIN(Q): S: { A, C } 0 : A B C D E Q: - A B C D E 0 A B C D E while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 47
48 Example of Dijkstra s algorithm Relax all edges leaving C: S: { A, C } 0 : A B C D E Q: - A B C D E A 7 B C 11 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 48
49 Example of Dijkstra s algorithm Relax all edges leaving C: S: { A, C } 0 : A B C D E Q: - A B C D E A 7 B C 11 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 49
50 Example of Dijkstra s algorithm E EXTRACT-MIN(Q): S: { A, C, E } 0 : A B C D E Q: C C C A B C D E A 7 B C 11 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 50
51 Example of Dijkstra s algorithm Relax all edges leaving E: S: { A, C, E } 0 : A B C D E Q: C C C A B C D E A 7 B C 11 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 51
52 Example of Dijkstra s algorithm B EXTRACT-MIN(Q): S: { A, C, E, B } 0 : A B C D E Q: C C C A B C D E A 7 B C 11 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 5
53 Example of Dijkstra s algorithm Relax all edges leaving B: S: { A, C, E, B } 0 : A B C D E Q: C B C A B C D E A 7 B C 9 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 5
54 Example of Dijkstra s algorithm D EXTRACT-MIN(Q): S: { A, C, E, B, D}0 : A B C D E Q: C B C A B C D E A 7 B C 9 D E 5 while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) [v] u 54
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