Data Structure
  • 資料結構自學筆記
  • 1 - Stack & Queue
    • 1.1 - Stack
    • 1.2 - Queue
    • 1.3 - Stack and Queue
  • 2 - Tree & Binary Tree
    • 2.1 - Tree
    • 2.2 - Binary Tree
    • 2.3 - Binary Tree Traversal
    • 2.4 - Binary Search Tree
    • 2.5 - Heap
    • 2.6 - Thread Binary Tree
    • 2.7 - Tree and Binary Tree Conversion
    • 2.8 Advanced Trees
      • 2.8.1 - Min-Max Heap
      • 2.8.2 - Deap
      • 2.8.3 - Symmetric Min-Max Heap
      • 2.8.4 - Extended Binary Tree
      • 2.8.5 - AVL Tree
      • 2.8.6 - M-Way Search Tree
      • 2.8.7 - B Tree
      • 2.8.8 - Red-Black Tree
      • 2.8.9 - Optimal Binary Search Tree
      • 2.8.10 - Splay Tree
      • 2.8.11 - Leftest Heap
      • 2.8.12 - Binomial Heap
  • 3 - Search & Sort
    • 3.1 - Searching
    • 3.2 - Elementary Sorting
      • 3.2.1 - Insertion Sort
      • 3.2.2 - Selection Sort
      • 3.2.3 - Bubble Sort
      • 3.2.4 - Shell Sort
    • 3.3 - Sophisticated Sorting
      • 3.3.1 - Quick Sort
      • 3.3.2 - Merge Sort
      • 3.3.3 - Heap Sort
      • 3.3.4 - Radix Sort
      • 3.3.5 - Bucket Sort
      • 3.3.6 - Counting Sort
    • 3.4 - Summary
  • 4 - Graph
    • 4.1 - Intro
    • 4.2 - Graph Traversal
    • 4.3 - Spanning Tree
      • 4.3.1 - Kruskal's algorithm
      • 4.3.2 - Prim's algorithm
      • 4.3.3 - Sollin's algorithm
    • 4.4 - Shortest Path Length
      • 4.4.1 - Dijkstra's algorithm
      • 4.4.2 - Bellman-Ford algorithm
      • 4.4.3 - Floyd-Warshall algorithm
    • 4.5 - AOV Network
    • 4.6 - AOE Network
    • 4.7 - Others
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  1. 4 - Graph
  2. 4.4 - Shortest Path Length

4.4.3 - Floyd-Warshall algorithm

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Last updated 6 years ago

  • 可有負邊存在

1. 演算法

AkA^kAk為一個n*n矩陣， Ak[i.j]A^{k}[i.j]Ak[i.j]為 iii 到 jjj 的shortest path length，途中經過的頂點編號必須≤kkk。

A0A^0A0為Cost Matrix，依序求出 A1,A2,...,AnA^1, A^2,...,A^nA1,A2,...,An， AnA^nAn及為所求。

e.g.

A0=[04116023∞0]A^0= \begin{bmatrix} 0&4&11\\ 6&0&2\\ 3&∞&0 \end{bmatrix}A0=​063​40∞​1120​​

A1=[0411602370]A^1= \begin{bmatrix} 0&4&11\\ 6&0&2\\ 3&7&0 \end{bmatrix}A1=​063​407​1120​​

A2=[046602370]A^2= \begin{bmatrix} 0&4&6\\ 6&0&2\\ 3&7&0 \end{bmatrix}A2=​063​407​620​​

A3=[046502370]A^3= \begin{bmatrix} 0&4&6\\ 5&0&2\\ 3&7&0 \end{bmatrix}A3=​053​407​620​​