Introduction to Algorithms
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1 Introduction to Algorithms 6.046J/18.401J LECTURE 17 Shortest Paths I Properties of shortest paths Dijkstra s algorithm Correctness Analysis Breadth-first search Prof. Erik Demaine November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.1
2 Paths in graphs Consider a digraph G = (V, E) with edge-weight function w : E R. The weight of path p = v 1 v L v k is defined to be k 1 i= 1 w ( p) = w( v i, v i + 1). November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.
3 Paths in graphs Consider a digraph G = (V, E) with edge-weight function w : E R. The weight of path p = v 1 v L v k is defined to be Example: k 1 i= 1 w ( p) = w( v i, v i + 1). v 1 v v 3 v 5 v v w(p) = November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.3
4 Shortest paths A shortest path from u to v is a path of minimum weight from u to v. The shortestpath weight from u to v is defined as δ(u, v) = min{w(p) : p is a path from u to v}. Note: δ(u, v) = if no path from u to v exists. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.4
5 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.5
6 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Proof. Cut and paste: November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.6
7 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Proof. Cut and paste: November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.7
8 Triangle inequality Theorem. For all u, v, x V, we have δ(u, v) δ(u, x) + δ(x, v). November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.8
9 Triangle inequality Theorem. For all u, v, x V, we have δ(u, v) δ(u, x) + δ(x, v). Proof. uu δ(u, v) vv δ(u, x) δ(x, v) xx November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.9
10 Well-definedness of shortest paths If a graph G contains a negative-weight cycle, then some shortest paths may not exist. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.10
11 Well-definedness of shortest paths If a graph G contains a negative-weight cycle, then some shortest paths may not exist. Example: < 0 uu vv November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.11
12 Single-source shortest paths Problem. From a given source vertex s V, find the shortest-path weights δ(s, v) for all v V. If all edge weights w(u, v) are nonnegative, all shortest-path weights must exist. IDEA: Greedy. 1. Maintain a set S of vertices whose shortestpath distances from s are known.. At each step add to S the vertex v V S whose distance estimate from s is minimal. 3. Update the distance estimates of vertices adjacent to v. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.1
13 Dijkstra s algorithm d[s] 0 for each v V {s} do d[v] S Q V Q is a priority queue maintaining V S November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.13
14 Dijkstra s algorithm d[s] 0 for each v V {s} do d[v] S Q V Q is a priority queue maintaining V S while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.14
15 Dijkstra s algorithm d[s] 0 for each v V {s} do d[v] S Q V while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) Q is a priority queue maintaining V S relaxation step Implicit DECREASE-KEY November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.15
16 Example of Dijkstra s algorithm Graph with nonnegative edge weights: AA 10 BB D CC EE November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.16
17 Initialize: Example of Dijkstra s algorithm 0 Q: A B C D E AA 10 3 BB CC D S: {} EE November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.17
18 Example of Dijkstra s algorithm A EXTRACT-MIN(Q): 0 Q: A B C D E AA 10 3 BB CC D S: { A } EE November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.18
19 Example of Dijkstra s algorithm Relax all edges leaving A: 0 Q: A B C D E AA BB D CC S: { A } EE 3 November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.19
20 Example of Dijkstra s algorithm C EXTRACT-MIN(Q): Q: 0 A B C D E AA BB D CC S: { A, C } EE 3 November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.0
21 Example of Dijkstra s algorithm Relax all edges leaving C: Q: 0 A B C D E AA BB D CC S: { A, C } EE 3 5 November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.1
22 Example of Dijkstra s algorithm E EXTRACT-MIN(Q): Q: 0 A B C D E AA BB D CC EE 3 5 S: { A, C, E } November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.
23 Example of Dijkstra s algorithm Relax all edges leaving E: Q: 0 A B C D E AA BB D CC S: { A, C, E } EE 3 5 November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.3
24 Example of Dijkstra s algorithm B EXTRACT-MIN(Q): Q: 0 A B C D E AA BB D CC EE 3 5 S: { A, C, E, B } November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.4
25 Example of Dijkstra s algorithm Relax all edges leaving B: Q: 0 A B C D E AA 7 9 BB D November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L CC S: { A, C, E, B } EE 3 5
26 Example of Dijkstra s algorithm D EXTRACT-MIN(Q): Q: 0 A B C D E AA 7 9 BB D November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L CC EE 3 5 S: { A, C, E, B, D }
27 Correctness Part I Lemma. Initializing d[s] 0 and d[v] for all v V {s} establishes d[v] δ(s, v) for all v V, and this invariant is maintained over any sequence of relaxation steps. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.7
28 Correctness Part I Lemma. Initializing d[s] 0 and d[v] for all v V {s} establishes d[v] δ(s, v) for all v V, and this invariant is maintained over any sequence of relaxation steps. Proof. Suppose not. Let v be the first vertex for which d[v] < δ(s, v), and let u be the vertex that caused d[v] to change: d[v] = d[u] + w(u, v). Then, d[v] < δ(s, v) supposition δ(s, u) + δ(u, v) triangle inequality δ(s,u) + w(u, v) sh. path specific path d[u] + w(u, v) v is first violation Contradiction. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.8
29 Correctness Part II Lemma. Let u be v s predecessor on a shortest path from s to v. Then, if d[u] = δ(s, u) and edge (u, v) is relaxed, we have d[v] = δ(s, v) after the relaxation. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.9
30 Correctness Part II Lemma. Let u be v s predecessor on a shortest path from s to v. Then, if d[u] = δ(s, u) and edge (u, v) is relaxed, we have d[v] = δ(s, v) after the relaxation. Proof. Observe that δ(s, v) = δ(s, u)+ w(u, v). Suppose that d[v] > δ(s, v) before the relaxation. (Otherwise, we re done.) Then, the test d[v] > d[u] + w(u, v) succeeds, because d[v] > δ(s, v) = δ(s, u)+ w(u, v) = d[u] + w(u, v), and the algorithm sets d[v] = d[u] + w(u, v) = δ(s, v). November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.30
31 Correctness Part III Theorem. Dijkstra s algorithm terminates with d[v] = δ(s, v) for all v V. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.31
32 Correctness Part III Theorem. Dijkstra s algorithm terminates with d[v] = δ(s, v) for all v V. Proof. It suffices to show that d[v] = δ(s, v) for every v V when v is added to S. Suppose u is the first vertex added to S for which d[u] >δ(s, u). Let y be the first vertex in V S along a shortest path from s to u, and let x be its predecessor: S, just before adding u. ss xx yy uu November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.3
33 Correctness Part III (continued) ss S xx yy uu Since u is the first vertex violating the claimed invariant, we have d[x] = δ(s, x). When x was added to S, the edge (x, y) was relaxed, which implies that d[y] =δ(s, y) δ(s, u) < d[u]. But, d[u] d[y] by our choice of u. Contradiction. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.33
34 Analysis of Dijkstra while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.34
35 Analysis of Dijkstra V times while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.35
36 Analysis of Dijkstra V times degree(u) times while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.36
37 Analysis of Dijkstra V times degree(u) times while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) Handshaking Lemma Θ(E) implicit DECREASE-KEY s. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.37
38 Analysis of Dijkstra V times degree(u) times while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) Handshaking Lemma Θ(E) implicit DECREASE-KEY s. Time = Θ(V T EXTRACT -MIN + E T DECREASE-KEY ) Note: Same formula as in the analysis of Prim s minimum spanning tree algorithm. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.38
39 Analysis of Dijkstra (continued) Time = Θ(V) T EXTRACT -MIN + Θ(E) T DECREASE-KEY Q TEXTRACT-MIN T DECREASE -KEY Total November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.39
40 Analysis of Dijkstra (continued) Time = Θ(V) T EXTRACT -MIN + Θ(E) T DECREASE-KEY Q TEXTRACT-MIN T DECREASE -KEY Total array O(V) O(1) O(V ) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.40
41 Analysis of Dijkstra (continued) Time = Θ(V) T EXTRACT -MIN + Θ(E) T DECREASE-KEY Q TEXTRACT-MIN T DECREASE -KEY Total array O(V) O(1) O(V ) binary heap O(lg V) O(lg V) O(E lg V) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.41
42 Analysis of Dijkstra (continued) Time = Θ(V) T EXTRACT -MIN + Θ(E) T DECREASE-KEY Q TEXTRACT-MIN T DECREASE -KEY Total array O(V) O(1) O(V ) binary heap Fibonacci heap O(lg V) O(lg V) O(E lg V) O(lg V) amortized O(1) amortized O(E + V lg V) worst case November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.4
43 Unweighted graphs Suppose that w(u, v) = 1 for all (u, v) E. Can Dijkstra s algorithm be improved? November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.43
44 Unweighted graphs Suppose that w(u, v) = 1 for all (u, v) E. Can Dijkstra s algorithm be improved? Use a simple FIFO queue instead of a priority queue. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.44
45 Unweighted graphs Suppose that w(u, v) = 1 for all (u, v) E. Can Dijkstra s algorithm 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) November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.45
46 Unweighted graphs Suppose that w(u, v) = 1 for all (u, v) E. Can Dijkstra s algorithm 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). November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.46
47 Example of breadth-first search aa bb cc dd ee f gg hh ii Q: November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.47
48 Example of breadth-first search 0 aa bb cc dd ee f gg hh ii 0 Q: a November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.48
49 Example of breadth-first search aa bb cc dd ee 1 1 Q: a b d f gg hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.49
50 Example of breadth-first search aa bb cc dd ee 1 Q: a b d c e f gg hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.50
51 Example of breadth-first search aa bb cc dd ee Q: a bdc e f gg hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.51
52 Example of breadth-first search aa bb cc dd ee Q: a bdce f gg hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.5
53 Example of breadth-first search aa bb cc dd ee 3 f gg Q: a b d c e g i hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.53
54 Example of breadth-first search 0 1 aa bb cc 1 dd ee 3 4 f gg Q: a b d c e g i f hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.54
55 Example of breadth-first search 0 1 aa bb cc 1 dd ee f gg Q: a b d c e g i f h hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.55
56 Example of breadth-first search 0 1 aa bb cc 1 dd ee f gg 3 4 Q: a b d c e g i f h hh ii November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.56
57 Example of breadth-first search 0 1 aa bb cc 1 dd ee f gg 3 hh ii Q: a b d c e g i f h November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.57
58 Example of breadth-first search 0 1 aa bb cc 1 dd ee f gg 3 hh ii Q: a b d c e g i f h November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.58
59 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] + 1. November 14, 005 Copyright by Erik D. Demaine and Charles E. Leiserson L17.59
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