[Data structure] Minimum spanning tree (Prim algorithm, Prim algorithm, Prim), shortest path (Dijkstra algorithm, Dijkstra algorithm, single-source shortest path)

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FAQ

1. Basic concepts: the definition and properties of the minimum spanning tree

(1) Definition of Minimal Spanning Tree

• 生成树的代价：任$G(V,E)$ is an undirected connected network graph, andthe sum of the weights of each edgeis calledthe cost of the spanning tree.
• 最小生成树: in graph GGAmong all spanning trees in$G$ , the spanning tree with the least costisthe minimum spanning tree.

(2) The nature of the minimum spanning tree (MST)

Setting $G=(V,E)$ is an undirected connected network graph,$U$ is a non-empty subset of the vertex set. If$(u,v)$ is an edge with minimum weight, where$u\in U,v\in V-U$ , then there must be a tree containing edges$u,The\; minimum\; spanning\; tree\; for\; v$ .

2. How to use the MST property to find the minimum spanning tree

• Find the edge ( u , v ) of minimum weight between two point sets (u,v)$(u,v)$ , let ( u , v ) (u,v)with minimum weight$(u,v)$ becomes part of the minimum spanning tree and will be greater than the minimum weight$(u,v)$ delete.

Next, there are two ideas:

• Starting from one point, adding points at a time to form a point set（Prim算法）
• Starting from the edge, merge the point sets to avoid forming a cycle（Kruskal算法）

3. Prim algorithm

(1) Prim algorithm idea

Do operations on points, maintain a vertex set A of points in the minimum spanning tree, and a vertex set B of points to be processed, find the shortest edge connecting these two sets each time, and add both vertices to it Set A until all vertices are processed.

Abstract description: (think abstract skip)

(2) The detailed process of Prim algorithm to form the minimum spanning tree

Legend:

• The red line represents the minimum spanning tree
• The blue circle represents the set $U$ , the set of other vertices is$V-U$
• The blue line represents $UV$ sum$V-$Adjacent side of$U$

$Calculate\; the\; cost\; between\; each\; point\; inU\; and\; its\; adjacent\; points,\; find\; out\; the\; pointV5with\; the\; smallest\; cost,\; and\; includeV5intothe\; Uset.$

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(3) C++ and python implementation of Prim's algorithm

4. Dijkstra algorithm

(1) Connection with Prim algorithm

Dijkstra's algorithm and Prim's algorithm are both shortest path algorithms , which are mainly used to find the shortest path of a graph .

The difference is that Dijkstra's algorithm is suitable for the shortest path from the starting point of a directed graph to other points , while Prim's algorithm is suitable for finding the minimum spanning tree of an undirected graph . Their solution procedures are also slightly different. The Dijkstra algorithm selects the point closest to the starting point each time as a new access point, and updates the shortest distance from other points to the starting point until all points are visited. Prim's algorithm starts from a starting point and continuously selects the point connected to the visited point and has the smallest edge weight until all points on the graph are visited.

(2) Dijkstra algorithm idea

Calculate the path length from point A to every point in the graph, select a shortest path: A->B, and add vertex B to set S.

After adding a shortest path, is there a shorter path from vertex A to other points?

Update the shortest path:

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A->C,A->D,A->ESelect the shortest path in: , A->Dand add the D vertices to the S set. Update all shortest paths.

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