最小生成樹(shù) (Minimum Spanning Tree)

• An MST is a subset of the edges of the connected, undirected graph that connect all the vertices together, in which there is no forming of a cycle and there should be minimum possible total edge weight.

  MST是已連接的無(wú)向圖的邊的子集，該邊將所有頂點(diǎn)連接在一起，其中不形成循環(huán)，因此總邊的權(quán)重應(yīng)最小。

• In this weight of a tree is defined as the sum of the weight of all its edges which are connected but no formation of the cycle is there.

  在該樹(shù)中，樹(shù)的權(quán)重定義為樹(shù)的所有相連邊的權(quán)重之和，但沒(méi)有形成循環(huán)。

• A tree T is said to be a spanning tree of a connected graph X if T is a subgraph of X and T contains all vertices of X.

  如果T是X的子圖并且T包含X的所有頂點(diǎn)，則將樹(shù)T稱(chēng)為連通圖X的生成樹(shù)。

生成樹(shù)的應(yīng)用 (Application of Spanning Tree)

Spanning tree has wide applications in many areas like network design.

生成樹(shù)在網(wǎng)絡(luò)設(shè)計(jì)等許多領(lǐng)域都有廣泛的應(yīng)用。

Spanning tree is important in designing routing algorithms.

生成樹(shù)在設(shè)計(jì)路由算法時(shí)很重要。

Practical application based on minimum spanning tree includes taxonomy and cluster analysis.

基于最小生成樹(shù)的實(shí)際應(yīng)用包括分類(lèi)法和聚類(lèi)分析。

1)Kruskal算法 (1) Kruskal’s Algorithm)

• It is an application of a greedy algorithm.

  它是貪婪算法的一種應(yīng)用。

• In this edges are selected with minimum weight and added to MST till no cycle is formed.

  在這種情況下，以最小的重量選擇邊緣，并將其添加到MST中，直到?jīng)]有循環(huán)形成為止。

• It is used to find a minimum cost.

  它用于查找最低成本。

• It finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.

  它找到形成樹(shù)的邊緣子集，該樹(shù)包括每個(gè)頂點(diǎn)，樹(shù)中所有邊緣的總權(quán)重最小。

• Kruskal algorithm does not form a tree at each step.

  Kruskal算法并非在每個(gè)步驟都形成一棵樹(shù)。

Steps for the kruskal’s algorithm are as follows:

kruskal算法的步驟如下：

Firstly arrange all the edges in increasing order of their weight.

首先，以重量增加的順序排列所有邊緣。

Then the edges should be added if it does not form a circuit.

如果沒(méi)有形成電路，則應(yīng)添加邊緣。

Continue these steps till all the edges are visited and MST is formed.

繼續(xù)這些步驟，直到訪問(wèn)了所有邊緣并形成了MST。

Add the cost of all edges in MST to get a minimum cost of a spanning tree.

在MST中添加所有邊的成本，以獲取生成樹(shù)的最低成本。

2)Prim的算法 (2) Prim’s algorithm)

• This algorithm generally focused on vertices.

  該算法通常集中在頂點(diǎn)上。

• Prim's algorithm always forms a tree at every step.

  Prim的算法總是在每一步都形成一棵樹(shù)。

• It applies the nearest neighbor method to select new edges.

  它應(yīng)用最近鄰居方法來(lái)選擇新邊。

• This algorithm is generally used when we have to find a minimum cost of a dense graph because this number of edges will be high.

  當(dāng)我們必須找到密集圖的最低成本時(shí)，通常會(huì)使用此算法，因?yàn)樵撨吘墧?shù)量很高。

• Basically, Prim's algorithm is faster than the Kruskal's algorithm in the case of the complex graph.

  基本上，在復(fù)雜圖的情況下，Prim算法比Kruskal算法更快。

Steps for the Prim’s algorithms are as follows:

Prim算法的步驟如下：

Start with a vertex, say u.

從頂點(diǎn)開(kāi)始，說(shuō)u 。

Select another vertex v such that edges are formed from u and v and are of minimum weight, connect uv and add it to set of MST for edges A.

選擇另一個(gè)頂點(diǎn)v ，以使邊緣由u和v形成并具有最小權(quán)重，連接uv并將其添加到邊緣A的MST集。

Now among the set of all vertices find other vertex v_i that is not included in A such that (v_i, v_j) is minimum labeled and is the nearest neighbor of all vertices in set A and it does not form a cycle, add it to A.

現(xiàn)在，集所有頂點(diǎn)中找到其他頂點(diǎn)v _我是不包含在使得(V _{I，V j)}為最小的標(biāo)記，是集合A中的所有頂點(diǎn)的近鄰，并沒(méi)有形成一個(gè)周期，加它到A。

Continue this process till we get an MST, then the MST formed will be of minimum cost.

繼續(xù)執(zhí)行此過(guò)程，直到獲得MST為止，然后形成的MST成本最低。

Reference: Kruskal's algorithm

參考： Kruskal算法

翻譯自: https://www.includehelp.com/algorithms/p-and-k-algorithms.aspx

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