数据结构笔记——Dijkstra算法

秒速五厘米 2023-02-18 08:22 23阅读 0赞

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**目录**

一、BFS算法的局限性

二、Dijkstra算法

三、如何使用数组信息

四、用于负权值带权图

--------------------

## **一、BFS算法的局限性** ##

带权路径长度——当图是带权图时，一条路径上所有边的权值之和，称为该路径的带权路径长度

BFS算法求单源最短路径只适用于无权图，或所有边的权值都相同的图

## **二、Dijkstra算法** ##

![20200616190853199.png][]

初始：从V0开始，初始化三个数组信息如下

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70][]

第1轮：循环遍历所有结点，找到还没确定最短路径，且dist最小的顶点Vi,零final\[j\] = true

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 1][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 2][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 3][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 4][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 5][]

检查所有邻接自Vi的顶点，若其final值为false，则更新dist和path信息

第2轮：循环遍历所有结点，找到还没确定最短路径，且dist最小的顶点Vi,零final\[j\] = true

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 6][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 7][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 8][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 9][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 10][]

第3轮：循环遍历所有结点，找到还没确定最短路径，且dist最小的顶点Vi,零final\[j\] = true

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 11][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 12][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 13][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 14][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 15][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 16][]

第4轮：循环遍历所有结点，找到还没确定最短路径，且dist最小的顶点Vi,零final\[j\] = true

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 17][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 18][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 19][]

## **三、如何使用数组信息** ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 20][]

V0到V2的最短（带权）路径长度为：dist\[2\] = 9

通过path\[\]可知，V0到V2的最短（带权）路径：V2<- -- V1<-----V4<------V0

**时间复杂度：O（n^2）即O（|V|^2）**

## **四、用于负权值带权图** ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 21][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 22][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 23][]

**结论：Dijkstra算法不适用于有负权值的带权图**

[20200616190853199.png]: https://img-blog.csdnimg.cn/20200616190853199.png
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