Chapter25

25.All-Pairs Shortest Paths ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],25.All-Pairs Shortest Paths

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.All-Pairs Shortest Paths

[object Object],[object Object],[object Object],[object Object],[object Object],25.1 Shortest paths and matrix multiplication

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.1 Shortest paths and matrix multiplication 需要花 Θ (n 3 ) 的時間

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.1 Shortest paths and matrix multiplication

25.1 Shortest paths and matrix multiplication 3 4 8 1 -5 7 2 6 -4

[object Object],[object Object],25.1 Shortest paths and matrix multiplication

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.1 Shortest paths and matrix multiplication

[object Object],[object Object],[object Object],Exercise 25.1

[object Object],[object Object],[object Object],[object Object],[object Object],25.2 The Floyd-Warshall algorithm

[object Object],[object Object],25.2 The Floyd-Warshall algorithm p 1 p 2

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.2 The Floyd-Warshall algorithm

25.2 The Floyd-Warshall algorithm 3 4 8 1 -5 7 2 6 -4 π ij (0) = NIL if i = j or w ij = ∞ π ij (0) = i if i ≠ j and w ij < ∞ π ij (k) = π ij (k-1) if d ij (k-1) ≤ d ik (k-1) + d kj (k-1) π ij (k) = π kj (k-1) if d ij (k-1) > d ik (k-1) + d kj (k-1)

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25.2 The Floyd-Warshall 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],25.2 The Floyd-Warshall algorithm

[object Object],[object Object],[object Object],Exercise 25.2

[object Object],[object Object],[object Object],[object Object],[object Object],25.3 Johnson's algorithm for sparse graphs

[object Object],[object Object],25.3 Johnson's algorithm for sparse graphs

[object Object],[object Object],25.3 Johnson's algorithm for sparse graphs 3 4 8 1 -5 7 2 6 -4 1 2 3 4 5 0 0 0 0 0 0 -4 0 -1 -5 0

[object Object],25.3 Johnson's algorithm for sparse graphs 3 4 8 1 -5 7 2 6 -4 0 0 0 0 0 1 2 3 4 5 -4 0 -1 -5 0 4 0 13 0 0 10 2 2 0 0 5 1 0 4 0 1 2 3 4 5 -4 0 -1 -5 0

[object Object],25.3 Johnson's algorithm for sparse graphs 4 0 13 0 0 10 2 2 0 1 2 3 4 5 4 0 13 0 0 10 2 2 0 0 5 1 0 4 0 1 2 3 4 5 -4 0 -1 -5 0 0/0 2/1 2/-3 2/2 0/-4 2/3 0/0 0/-4 2/6 2/-1 2/7 0/4 0/0 0/5 2/3 2/2 0/-1 0/-5 0/0 2/-2 4/8 2/5 2/1 2/6 0/0

25.3 Johnson's algorithm for sparse graphs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Chapter25

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Chapter25