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Graph Algorithms Shortest path problems. Graph Algorithms Shortest path problems.

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Presentation on theme: "Graph Algorithms Shortest path problems. Graph Algorithms Shortest path problems."— Presentation transcript:

1 Graph Algorithms Shortest path problems

2 Graph Algorithms Shortest path problems

3 Graph Algorithms Shortest path problems

4 Graph Algorithms Single-Source Shortest Paths Given graph (directed or undirected) G = (V,E) with weight function w: E  R and a vertex s  V, find for all vertices v  V the minimum possible weight for path from s to v. We will discuss two general case algorithms: Dijkstra's (positive edge weights only) Bellman-Ford (positive end negative edge weights) If all edge weights are equal (let's say 1), the problem is solved by BFS in  (V+E) time.

5 Graph Algorithms Dijkstra’s Algorithm - Relax Relax(vertex u, vertex v, weight w) if d[v] > d[u] + w(u,v) then d[v]  d[u] + w(u,v) p[v]  u

6 Graph Algorithms Dijkstra’s Algorithm - Idea

7 Graph Algorithms Dijkstra’s Algorithm - SSSP-Dijkstra SSSP-Dijkstra(graph (G,w), vertex s) InitializeSingleSource(G, s) S   Q  V[G] while Q  0 do u  ExtractMin(Q) S  S  {u} for v  Adj[u] do Relax(u,v,w) InitializeSingleSource(graph G, vertex s) for v  V[G] do d[v]   p[v]  0 d[s]  0

8 Graph Algorithms Dijkstra’s Algorithm - Example 10 1 5 2 6 4 9 7 2 3

9 Graph Algorithms Dijkstra’s Algorithm - Example 0     10 1 5 2 6 4 9 7 2 3

10 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 10   1 5 2 6 4 9 7 2 3

11 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 10   1 5 2 6 4 9 7 2 3

12 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 14 10 1 5 2 6 4 9 7 2 3

13 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 14 10 1 5 2 6 4 9 7 2 3

14 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 13 10 1 5 2 6 4 9 7 2 3

15 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 13 10 1 5 2 6 4 9 7 2 3

16 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 9 10 1 5 2 6 4 9 7 2 3

17 Graph Algorithms Dijkstra’s Algorithm - Example 0 5 8 7 9 10 1 5 2 6 4 9 7 2 3

18 Graph Algorithms Dijkstra’s Algorithm - Complexity SSSP-Dijkstra(graph (G,w), vertex s) InitializeSingleSource(G, s) S   Q  V[G] while Q  0 do u  ExtractMin(Q) S  S  {u} for u  Adj[u] do Relax(u,v,w) InitializeSingleSource(graph G, vertex s) for v  V[G] do d[v]   p[v]  0 d[s]  0 Relax(vertex u, vertex v, weight w) if d[v] > d[u] + w(u,v) then d[v]  d[u] + w(u,v) p[v]  u  (V)  (1) ?  (E) times in total executed  (V) times

19 Graph Algorithms Dijkstra’s Algorithm - Complexity InitializeSingleSource T I (V,E) =  (V) Relax T R (V,E) =  (1)? SSSP-Dijkstra T(V,E) = T I (V,E) +  (V) + V  (log V) + E T R (V,E) = =  (V) +  (V) + V  (log V) + E  (1) =  (E + V log V)

20 Graph Algorithms Dijkstra’s Algorithm - Complexity

21 Graph Algorithms Dijkstra’s Algorithm - Correctness

22 Graph Algorithms Dijkstra’s Algorithm - negative weights?

23 Graph Algorithms Bellman-Ford Algorithm - negative cycles?

24 Graph Algorithms Bellman-Ford Algorithm - Idea

25 Graph Algorithms Bellman-Ford Algorithm - SSSP-BellmanFord SSSP-BellmanFord(graph (G,w), vertex s) InitializeSingleSource(G, s) for i  1 to |V[G]  1| do for (u,v)  E[G] do Relax(u,v,w) for (u,v)  E[G] do if d[v] > d[u] + w(u,v) then return false return true

26 Graph Algorithms Bellman-Ford Algorithm - Example 6 7 7 -3 2 8 -4 9 5 -2

27 Graph Algorithms Bellman-Ford Algorithm - Example 0     6 7 7 -3 2 8 -4 9 5 -2

28 Graph Algorithms Bellman-Ford Algorithm - Example 0 7 6   6 7 7 -3 2 8 -4 9 5 -2

29 Graph Algorithms Bellman-Ford Algorithm - Example 0 7 6 2 4 6 7 7 -3 2 8 -4 9 5 -2

30 Graph Algorithms Bellman-Ford Algorithm - Example 0 7 2 2 4 6 5 7 7 -3 2 8 -2 -4 9

31 Graph Algorithms Bellman-Ford Algorithm - Example 0 7 2 -2 4 6 5 7 7 -3 2 8 -2 -4 9

32 Graph Algorithms Bellman-Ford Algorithm - Complexity SSSP-BellmanFord(graph (G,w), vertex s) InitializeSingleSource(G, s) for i  1 to |V[G]  1| do for (u,v)  E[G] do Relax(u,v,w) for (u,v)  E[G] do if d[v] > d[u] + w(u,v) then return false return true executed  (V) times  (E)  (1)

33 Graph Algorithms Bellman-Ford Algorithm - Complexity InitializeSingleSource T I (V,E) =  (V) Relax T R (V,E) =  (1)? SSSP-BellmanFord T(V,E) = T I (V,E) + V E T R (V,E) + E = =  (V) + V E  (1) + E = =  (V E)

34 Graph Algorithms Bellman-Ford Algorithm - Correctness

35 Graph Algorithms Bellman-Ford Algorithm - Correctness

36 Graph Algorithms Bellman-Ford Algorithm - Correctness

37 Graph Algorithms Bellman-Ford Algorithm - Correctness

38 Graph Algorithms Shortest Paths in DAGs - SSSP-DAG SSSP-DAG(graph (G,w), vertex s) topologically sort vertices of G InitializeSingleSource(G, s) for each vertex u taken in topologically sorted order do for each vertex v  Adj[u] do Relax(u,v,w)

39 Graph Algorithms Shortest Paths in DAGs - Example 527-2 61 3 4 2

40 Graph Algorithms Shortest Paths in DAGs - Example  0  527-2 61 3 4 2

41 Graph Algorithms Shortest Paths in DAGs - Example  0  527-2 61 3 4 2

42 Graph Algorithms Shortest Paths in DAGs - Example  0  527-2 61 3 4 2

43 Graph Algorithms Shortest Paths in DAGs - Example 6  02 527-2 61 3 4 2

44 Graph Algorithms Shortest Paths in DAGs - Example 664  02 527-2 61 3 4 2

45 Graph Algorithms Shortest Paths in DAGs - Example 664  02 527-2 61 3 4 2

46 Graph Algorithms Shortest Paths in DAGs - Example 654  02 527-2 61 3 4 2

47 Graph Algorithms Shortest Paths in DAGs - Example 654  02 527-2 61 3 4 2

48 Graph Algorithms Shortest Paths in DAGs - Example 653  02 527-2 61 3 4 2

49 Graph Algorithms Shortest Paths in DAGs - Example 653  02 527-2 61 3 4 2

50 Graph Algorithms Shortest Paths in DAGs - Complexity T(V,E) =  (V + E) +  (V) +  (V) + E  (1) =  (V + E) SSSP-DAG(graph (G,w), vertex s) topologically sort vertices of G InitializeSingleSource(G, s) for each vertex u taken in topologically sorted order do for each vertex v  Adj[u] do Relax(u,v,w)

51 Graph Algorithms Application of SSSP - currency conversion

52 Graph Algorithms Application of SSSP - currency conversion

53 Graph Algorithms Application of SSSP - currency conversion

54 Graph Algorithms Application of SSSP - constraint satisfaction

55 Graph Algorithms Application of SSSP - constraint satisfaction

56 Graph Algorithms Application of SSSP - constraint satisfaction

57 Graph Algorithms Application of SSSP - constraint satisfaction

58 Graph Algorithms Application of SSSP - constraint satisfaction

59 Graph Algorithms Application of SSSP - constraint satisfaction

60 Graph Algorithms All-Pairs Shortest Paths Given graph (directed or undirected) G = (V,E) with weight function w: E  R find for all pairs of vertices u,v  V the minimum possible weight for path from u to v.

61 Graph Algorithms Floyd-Warshall Algorithm - Idea

62 Graph Algorithms Floyd-Warshall Algorithm - Idea

63 Graph Algorithms Floyd-Warshall Algorithm - Idea d s,t (i) – the shortest path from s to t containing only vertices v 1,..., v i d s,t (0) = w(s,t) d s,t (k) = w(s,t)if k = 0 min{d s,t (k-1), d s,k (k-1) + d k,t (k-1) }if k > 0

64 Graph Algorithms Floyd-Warshall Algorithm - Algorithm FloydWarshall(matrix W, integer n) for k  1 to n do for i  1 to n do for j  1 to n do d ij (k)  min(d ij (k-1), d ik (k-1) + d kj (k-1) ) return D (n)

65 Graph Algorithms Floyd-Warshall Algorithm - Example 2 45 13 3 4 -4 -5 6 7 1 8 2 038  -4  0  17  40  2  -50   60 W

66 Graph Algorithms Floyd-Warshall Algorithm - Example 038  -4  0  17  40  2  -50   60 0000 000 00 000 00 D (0)  (0)

67 Graph Algorithms Floyd-Warshall Algorithm - Example 038  -4  0  17  40  25-50-2  60 0000 000 00 01001 00 D (1)  (1)

68 Graph Algorithms Floyd-Warshall Algorithm - Example 0384-4  0  17  40511 25-50-2  60 00020 000 0022 01001 00 D (2)  (2)

69 Graph Algorithms Floyd-Warshall Algorithm - Example 0384-4  0  17  40511 2-50-2  60 00020 000 0022 03001 00 D (3)  (3)

70 Graph Algorithms Floyd-Warshall Algorithm - Example 034-4 30 1 74053 2 -50-2 85160 00420 40401 40021 03001 43400 D (4)  (4)

71 Graph Algorithms Floyd-Warshall Algorithm - Example 032-4 30 1 74053 2 -50-2 85160 00450 40401 40021 03001 43400 D (5)  (5)

72 Graph Algorithms Floyd-Warshall Algorithm - Extracting the shortest paths

73 Graph Algorithms Floyd-Warshall Algorithm - Complexity T(V,E) =  (n 3 ) =  (V 3 ) FloydWarshall(matrix W, integer n) for k  1 to n do for i  1 to n do for j  1 to n do d ij (k)  min(d ij (k-1), d ik (k-1) + d kj (k-1) ) return D (n) 3 for cycles, each executed exactly n times

74 Graph Algorithms All-Pairs Shortest Paths -Johnson's algorithm

75 Graph Algorithms All-Pairs Shortest Paths - Reweighting

76 Graph Algorithms All-Pairs Shortest Paths - Reweighting

77 Graph Algorithms All-Pairs Shortest Paths - Reweighting

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