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. 演算法
  • 2. 範例
  1. 4 - Graph
  2. 4.4 - Shortest Path Length

4.4.1 - Dijkstra's algorithm

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Last updated 6 years ago

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1. 演算法

Cost Matrix C[i][j]:為一個nxn矩陣，n=|V|

C[i,j]={邊長,if (i, j) exists∞,if (i, j) doesn’t exist0,if i = jC[i,j]= \begin{cases} 邊長, & \textrm{if (i, j) exists}\\ ∞, & \textrm{if (i, j) doesn't exist}\\ 0, & \textrm{if i = j} \end{cases}C[i,j]=⎩⎨⎧​邊長,∞,0,​if (i, j) existsif (i, j) doesn’t existif i = j​

S[i]={0,未確定起點到i的shortest path length1,已確定起點到i的shortest path lengthS[i]= \begin{cases} 0, & \textrm{未確定起點到i的shortest path length}\\ 1, & \textrm{已確定起點到i的shortest path length} \end{cases}S[i]={0,1,​未確定起點到i的shortest path length已確定起點到i的shortest path length​

Dist[i]:起點到i的shortest path lengthDist[i]:\textrm{起點到i的shortest path length}Dist[i]:起點到i的shortest path length

Dist[w]=min{Dist[w], Dist[u]+C[u,w]}Dist[w]=min\{Dist[w],\ Dist[u]+C[u,w]\}Dist[w]=min{Dist[w], Dist[u]+C[u,w]}

2. 範例

求5到各點的最短路徑？