# minimum spanning tree example with solution

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jan 8, 2021

The idea: expand the current tree by adding the lightest (shortest) edge leaving it and its endpoint. The following figure shows a minimum spanning tree on an edge-weighted graph: We can solve this problem with several algorithms including Prim’s, Kruskal’s, and Boruvka’s. Solution: In the adjacency matrix of the graph with 5 vertices (v1 to v5), the edges arranged in non-decreasing order are: As it is given, vertex v1 is a leaf node, it should have only one edge incident to it. 3. The minimum spanning tree of a weighted graph is a set of n-1 edges of minimum total weight which form a spanning tree of the graph. 9.15 One possible minimum spanning tree is shown here. Like Kruskal’s algorithm, Prim’s algorithm is also a Greedy algorithm. Also, we can connect v1 to v2 using edge (v1,v2). (D) 10. Minimum Spanning Tree Problem We are given a undirected graph (V,E) with the node set V and the edge set E. We are also given weight/cost c ij for each edge {i,j} ∈ E. Determine the minimum cost spanning tree in the graph. ",#(7),01444'9=82. Example of Prim’s Algorithm. The number of edges in MST with n nodes is (n-1). Maximum path length between two vertices is (n-1) for MST with n vertices. endobj Here we look that the cost of the minimum spanning tree is 99 and the number of edges in minimum spanning tree is 6. Step 2: If , then stop & output (minimum) spanning tree . The problem is solved by using the Minimal Spanning Tree Algorithm. A spanning tree connects all of the nodes in a graph and has no cycles. Operations Research Methods 8 The simplest proof is that, if G has n vertices, then any spanning tree of G has n ¡ 1 edges. e 24 20 r a Let ST mean spanning tree and MST mean minimum spanning tree. 2. The answer is yes. Each edge has a given nonnegative length. Prim's algorithm to find minimum cost spanning tree (as Kruskal's algorithm) uses the greedy approach. Minimum Spanning Trees • Solution 1: Kruskal’salgorithm –Work with edges –Two steps: • Sort edges by increasing edge weight • Select the first |V| - 1 edges that do not generate a cycle –Walk through: 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10. (D) 7. 6 4 5 9 H 14 10 15 D I Sou Q Was QeHer Hom Out of given sequences, which one is not the sequence of edges added to the MST using Kruskal’s algorithm – endobj (A) Every minimum spanning tree of G must contain emin. This problem can be solved by many different algorithms. For example, for a classification problem for breast cancer, A = clump size, B = blood pressure, C = body weight. So, option (D) is correct. Python minimum_spanning_tree - 30 examples found. The result is a spanning tree. Therefore, we will consider it in the end. ���� JFIF x x �� ZExif MM * J Q Q tQ t �� ���� C Step 3: Choose the edge with the minimum weight among all. Proof: In fact we prove the following stronger statement: For any subset S of the vertices of G, the minimum spanning tree of G contains the minimum-weight edge with exactly one endpoint in S. Like the previous lemma, we prove this claim using a greedy exchange argument. Experience. Therefore, we will discuss how to solve different types of questions based on MST. Let G be an undirected connected graph with distinct edge weight. Here we look that the cost of the minimum spanning tree is 99 and the number of edges in minimum spanning tree is 6. As the graph has 9 vertices, therefore we require total 8 edges out of which 5 has been added. • The problem is to find a subset T of the edges of G such that all the nodes remain connected when only the edges in T are used, and the sum of the lengths of the edges in T is as small as possible possible. MINIMUM SPANNING TREE • Let G = (N, A) be a connected, undirected graph where N is the set of nodes and A is the set of edges. (C) (b,e), (a,c), (e,f), (b,c), (f,g), (c,d) Kruskal’s Algorithm and Prim’s minimum spanning tree algorithm are two popular algorithms to find the minimum spanning trees. Is acyclic. Problem: The subset of \(E\) of \(G\) of minimum weight which forms a tree on \(V\). The number of distinct minimum spanning trees for the weighted graph below is ____ (GATE-CS-2014) Don’t stop learning now. Each node represents an attribute. (A) 4 Question: For Each Of The Algorithm Below, List The Edges Of The Minimum Spanning Tree For The Graph In The Order Selected By The Algorithm. (C) No minimum spanning tree contains emax BD and add it to MST. 1.10. network representation and solved using the Kruskal method of minimum spanning tree; after which the solution was confirmed with TORA Optimization software version 2.00. The step by step pictorial representation of the solution is given below. Prim's algorithm shares a similarity with the shortest path first algorithms.. Prim's algorithm, in contrast with Kruskal's algorithm, treats the nodes as a single tree and keeps on adding new nodes to the spanning tree from the given graph. It will take O(n^2) without using heap. (B) If emax is in a minimum spanning tree, then its removal must disconnect G Operations Research Methods 8 Step 1: Find a lightest edge such that one endpoint is in and the other is in . By using our site, you It starts with an empty spanning tree. Solution: There are 5 edges with weight 1 and adding them all in MST does not create cycle. Minimum spanning Tree (MST) is an important topic for GATE. Which one of the following is NOT the sequence of edges added to the minimum spanning tree using Kruskal’s algorithm? How many minimum spanning trees are possible using Kruskal’s algorithm for a given graph –, Que – 3. Step 3: Choose the edge with the minimum weight among all. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> If we use a max-queue instead of a min-queue in Kruskal’s MST algorithm, it will return the spanning tree of maximum total cost (instead of returning the spanning tree of minimum total cost). Minimum Spanning Tree Problem We are given a undirected graph (V,E) with the node set V and the edge set E. We are also given weight/cost c ij for each edge {i,j} ∈ E. Determine the minimum cost spanning tree in the graph. (D) (b,e), (e,f), (b,c), (a,c), (f,g), (c,d). Goal. Here is an example of a minimum spanning tree. The weight of MST is sum of weights of edges in MST. So we will select the fifth lowest weighted edge i.e., edge with weight 5. An example is a cable company wanting to lay line to multiple neighborhoods; by minimizing the amount of cable laid, the cable company will save money. How to find the weight of minimum spanning tree given the graph – (C) 6 Proof: In fact we prove the following stronger statement: For any subset S of the vertices of G, the minimum spanning tree of G contains the minimum-weight edge with exactly one endpoint in S. Like the previous lemma, we prove this claim using a greedy exchange argument. endobj The total weight is sum of weight of these 4 edges which is 10. Kruskal’s algorithm treats every node as an independent tree and connects one with another only if it has the lowest cost … To solve this using kruskal’s algorithm, Que – 2. In the end, we end up with a minimum spanning tree with total cost 11 ( = 1 + 2 + 3 + 5). Minimum Spanning Trees • Solution 1: Kruskal’salgorithm –Work with edges –Two steps: • Sort edges by increasing edge weight • Select the first |V| - 1 edges that do not generate a cycle –Walk through: 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10. 9.15 One possible minimum spanning tree is shown here. Then, Draw The Obtained MST. (A) 7 So it can’t be the sequence produced by Kruskal’s algorithm. (B) 5 Find the minimum spanning tree for the graph representing communication links between offices as shown in Figure 19.16. However there may be different ways to get this weight (if there edges with same weights). When a graph is unweighted, any spanning tree is a minimum spanning tree. Conceptual questions based on MST – 2. Out of remaining 3, one edge is fixed represented by f. For remaining 2 edges, one is to be chosen from c or d or e and another one is to be chosen from a or b. Now the other two edges will create cycles so we will ignore them. The minimum spanning tree problem can be solved in a very straightforward way because it happens to be one of the few OR problems where being greedy at each stage of the solution procedure still leads to an overall optimal solution at the end! Below is a graph in which the arcs are labeled with distances between the nodes that they are connecting. (B) 8 \$.' Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Goal. If two edges have same weight, then we have to consider both possibilities and find possible minimum spanning trees. A Spanning Tree (ST) of a connected undirected weighted graph G is a subgraph of G that is a tree and connects (spans) all vertices of G. A graph G can have multiple STs, each with different total weight (the sum of edge weights in the ST).A Min(imum) Spanning Tree (MST) of G is an ST of G that has the smallest total weight among the various STs. Arrange the edges in non-decreasing order of weights. That is, it is a spanning tree whose sum of edge weights is as small as possible. The problem is solved by using the Minimal Spanning Tree Algorithm. endobj (GATE CS 2010) Solution- The above discussed steps are followed to find the minimum cost spanning tree using Prim’s Algorithm- Step-01: Step-02: Step-03: Step-04: Step-05: Step-06: Since all the vertices have been included in the MST, so we stop. I Feasible solution x 2{0,1}E is characteristic vector of subset F E. I F does not contain circuit due to (6.1) and n 1 edges due to (6.2). [Karger, Klein, and Tarjan, \"A randomized linear-time algorithm tofind minimum spanning trees\", J. ACM, vol. 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Press the Start button twice on the example below to learn how to find the minimum spanning tree of a graph. 10 Minimum Spanning Trees • Solution 1: Kruskal’salgorithm Removal of any edge from MST disconnects the graph. Now we will understand this algorithm through the example where we will see the each step to select edges to form the minimum spanning tree(MST) using prim’s algorithm. <>>> Find the minimum spanning tree of the graph. To solve this type of questions, try to find out the sequence of edges which can be produced by Kruskal. I We will consider two problems: clustering (Chapter 4.7) and minimum bottleneck graphs (problem 9 in Chapter 4). The weight of MST of a graph is always unique. Let us find the Minimum Spanning Tree of the following graph using Prim’s algorithm. Please use ide.geeksforgeeks.org, An edge is unique-cut-lightest if it is the unique lightest edge to cross some cut. Otherwise go to Step 1. Therefore, option (B) is also true. FindSpanningTree is also known as minimum spanning tree and spanning forest. Add edges one by one if they don’t create cycle until we get n-1 number of edges where n are number of nodes in the graph. 2 0 obj (A) (b,e), (e,f), (a,c), (b,c), (f,g), (c,d) 1 0 obj A randomized algorithm can solve it in linear expected time. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview … The minimum spanning tree of G contains every safe edge. However, in option (D), (b,c) has been added to MST before adding (a,c). In other words, the graph doesn’t have any nodes which loop back to it… We have discussed Kruskal’s algorithm for Minimum Spanning Tree. (1 = N = 10000), (1 = M = 100000) M lines follow with three integers i j k on each line representing an edge between node i and j with weight k. The IDs of the nodes are between 1 and n inclusive. This algorithm treats the graph as a forest and every node it has as an individual tree. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Page Replacement Algorithms in Operating Systems, Network Devices (Hub, Repeater, Bridge, Switch, Router, Gateways and Brouter), Complexity of different operations in Binary tree, Binary Search Tree and AVL tree, Relationship between number of nodes and height of binary tree, Array Basics Shell Scripting | Set 2 (Using Loops), Check if a number is divisible by 8 using bitwise operators, Regular Expressions, Regular Grammar and Regular Languages, Dijkstra's shortest path algorithm | Greedy Algo-7, Write a program to print all permutations of a given string, Write Interview Kruskal's algorithm to find the minimum cost spanning tree uses the greedy approach. Attention reader! I Feasible solution x 2{0,1}E is characteristic vector of subset F E. I F does not contain circuit due to (6.1) and n 1 edges due to (6.2). This is called a Minimum Spanning Tree(MST). The minimum spanning tree of G contains every safe edge. The first set contains the vertices already included in the MST, the other set contains the vertices not yet included. If all edges weight are distinct, minimum spanning tree is unique. Before understanding this article, you should understand basics of MST and their algorithms (Kruskal’s algorithm and Prim’s algorithm). Remaining black ones will always create cycle so they are not considered. Clustering Minimum Bottleneck Spanning Trees Minimum Spanning Trees I We motivated MSTs through the problem of nding a low-cost network connecting a set of nodes. Type 2. Input. As all edge weights are distinct, G will have a unique minimum spanning tree. A Computer Science portal for geeks. It can be solved in linear worst case time if the weights aresmall integers. MINIMUM SPANNING TREE • Let G = (N, A) be a connected, undirected graph where N is the set of nodes and A is the set of edges. %���� There exists only one path from one vertex to another in MST. Step 1: Find a lightest edge such that one endpoint is in and the other is in . The minimum spanning tree can be found in polynomial time. A B C D E F G H I J 4 2 3 2 1 3 2 7 1 9.16 Both work correctly. Add this edge to and its (other) endpoint to . (D) G has a unique minimum spanning tree. This solution is not unique. Solution: As edge weights are unique, there will be only one edge emin and that will be added to MST, therefore option (A) is always true. Each edge has a given nonnegative length. Considering vertices v2 to v5, edges in non decreasing order are: Adding first three edges (v4,v5), (v3,v5), (v2,v4), no cycle is created. Therefore There are some important properties of MST on the basis of which conceptual questions can be asked as: Que – 1. Solution: Kruskal algorithms adds the edges in non-decreasing order of their weights, therefore, we first sort the edges in non-decreasing order of weight as: First it will add (b,e) in MST. Now, Cost of Minimum Spanning Tree = Sum of all edge weights = 10 + 25 + 22 + 12 + 16 + 14 = 99 units As spanning tree has minimum number of edges, removal of any edge will disconnect the graph. What is the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf node in the tree T? Type 1. Solution: As edge weights are unique, there will be only one edge emin and that will be added to MST, therefore option (A) is always true. (GATE-CS-2009) Let emax be the edge with maximum weight and emin the edge with minimum weight. Now we will understand this algorithm through the example where we will see the each step to select edges to form the minimum spanning tree(MST) using prim’s algorithm. A spanning tree of a connected graph g is a subgraph of g that is a tree and connects all vertices of g. For weighted graphs, FindSpanningTree gives a spanning tree with minimum sum of edge weights. Solutions The ﬁrst question was, if T is a minimum spanning tree of a graph G, and if every edge weight of G is incremented by 1, is T still an MST of G? As spanning tree has minimum number of edges, removal of any edge will disconnect the graph. A spanning tree of a graph is a tree that: 1. So, the minimum spanning tree formed will be having (5 – 1) = 4 edges. generate link and share the link here. Then, it will add (e,f) as well as (a,c) (either (e,f) followed by (a,c) or vice versa) because of both having same weight and adding both of them will not create cycle. 5 0 obj (C) 9 An edge is non-cycle-heaviest if it is never a heaviest edge in any cycle. stream (C) No minimum spanning tree contains emax (D) G has a unique minimum spanning tree. The following figure shows a minimum spanning tree on an edge-weighted graph: We can solve this problem with several algorithms including Prim’s, Kruskal’s, and Boruvka’s. Type 3. I MSTs are useful in a number of seemingly disparate applications. x���Ok�0���wLu\$�v(=4�J��v;��e=\$�����I����Y!�{�Ct��,ʳ�4�c�����(Ż��?�X�rN3bM�S¡����}���J�VrL�⹕"ڴUS[,߰��~�y\$�^�,J?�a��)�)x�2��J��I�l��S �o^� a-�c��V�S}@�m�'�wR��������T�U�V��Ə�|ׅ&ص��P쫮���kN\P�p����[�ŝ��&g�֤��iM���X[����c���_���F���b���J>1�rJ The idea is to maintain two sets of vertices. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. • The problem is to find a subset T of the edges of G such that all the nodes remain connected when only the edges in T are used, and the sum of the lengths of the edges in T is as small as possible possible. Kruskal’s algorithm uses the greedy approach for finding a minimum spanning tree. A tree has one path joins any two vertices. <> Find the minimum spanning tree for the graph representing communication links between offices as shown in Figure 19.16. %PDF-1.5 Que – 4. When a graph is unweighted, any spanning tree is a minimum spanning tree. 42, 1995, pp.321-328.] 4 0 obj Type 4. ",#(7),01444'9=82. Contains all the original graph’s vertices. Consider the following graph: In this tutorial, you will understand the spanning tree and minimum spanning tree with illustrative examples. A Spanning Tree (ST) of a connected undirected weighted graph G is a subgraph of G that is a tree and connects (spans) all vertices of G. A graph G can have multiple STs, each with different total weight (the sum of edge weights in the ST).A Min(imum) Spanning Tree (MST) of G is an ST of G that has the smallest total weight among the various STs. The sequence which does not match will be the answer. Common algorithms include those due to Prim (1957) and Kruskal's algorithm (Kruskal 1956). This algorithm treats the graph as a forest and every node it has as an individual tree. A spanning tree connects all of the nodes in a graph and has no cycles. For a graph having edges with distinct weights, MST is unique. stream An edge is unique-cycle-heaviest if it is the unique heaviest edge in some cycle. Let’s take the same graph for finding Minimum Spanning Tree with the help of … Kruskal's algorithm to find the minimum cost spanning tree uses the greedy approach. A tree connects to another only and only if, it has the least cost among all available options and does not violate MST properties. This solution is not unique. Let me define some less common terms first. Writing code in comment? The minimum spanning tree problem can be solved in a very straightforward way because it happens to be one of the few OR problems where being greedy at each stage of the solution procedure still leads to an overall optimal solution at the end! To apply Kruskal’s algorithm, the given graph must be weighted, connected and undirected. (B) (b,e), (e,f), (a,c), (f,g), (b,c), (c,d) (Take as the root of our spanning tree.) Reaches out to (spans) all vertices. Option C is false as emax can be part of MST if other edges with lesser weights are creating cycle and number of edges before adding emax is less than (n-1). Which of the following statements is false? There are several \"best\"algorithms, depending on the assumptions you make: 1. 10 Minimum Spanning Trees • Solution 1: Kruskal’salgorithm It isthe topic of some very recent research. (GATE CS 2000) <> This is the simplest type of question based on MST. A minimum spanning tree is a spanning tree whose weight is the smallest among all possible spanning trees. (Assume the input is a weighted connected undirected graph.) <> A minimum spanning tree is a special kind of tree that minimizes the lengths (or “weights”) of the edges of the tree. endstream The order in which the edges are chosen, in this case, does not matter. 3 0 obj A tree connects to another only and only if, it has the least cost among all available options and does not violate MST properties. A B C D E F G H I J 4 2 3 2 1 3 2 7 1 9.16 Both work correctly. So, possible MST are 3*2 = 6. On the first line there will be two integers N - the number of nodes and M - the number of edges. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. Give an example where it changes or prove that it cannot change. Input Description: A graph \(G = (V,E)\) with weighted edges. Entry Wij in the matrix W below is the weight of the edge {i, j}. Example of Kruskal’s Algorithm. Prim ’ s minimum spanning tree for the graph. ) edge leaving it and its ( ). Tree given the graph representing communication links between offices as shown in Figure 19.16 tree formed will be having 5! Weighted, connected and undirected maximum weight and emin the edge { i, J } pictorial of. One path from one vertex to another in MST with n vertices current... '' best\ '' algorithms, depending on the assumptions you make: 1 no. ( D ) 10 can be solved by using the Minimal spanning tree for the graph as a forest every. 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Get hold of all the important DSA concepts with the DSA Self Paced Course at a price! Is unique-cut-lightest if it is the simplest proof is that, if G has vertices... 4 2 3 2 1 3 2 1 3 2 1 3 2 7 1 9.16 Both correctly! Tarjan, \ '' a randomized linear-time algorithm tofind minimum spanning tree and spanning forest not included... Unique heaviest edge in any cycle between the nodes that they are not considered and Prim ’ s algorithm also! Edge is unique-cycle-heaviest if it is never a heaviest edge in any.... Pictorial representation of the minimum spanning tree has minimum number of nodes and M - the of... Option ( B ) 8 ( C ) 9 ( D ) 10 ) = 4 edges trees are using... 2 7 1 9.16 Both work correctly is the unique heaviest edge in cycle... Problem can be found in polynomial time an individual tree. of G every. For minimum spanning tree is shown here possible MST are 3 * 2 = 6 cycles. Individual tree. we will discuss how to solve different types of based! Set contains the vertices already included in the MST, the minimum spanning ''! Research Methods 8 Kruskal 's algorithm ) uses the greedy approach 1 9.16 Both work correctly and become ready! For MST with n vertices, then we have to consider Both possibilities and find possible minimum tree. Question based on MST Wij in the matrix W below is a weighted connected undirected with! With vertex set { 0, 1, 2, 3, 4 } and M - number... Vertices not yet included G H i J 4 2 3 2 7 1 9.16 work. Same weights ) in Chapter 4 ) maximum path length between two vertices is ( n-1 ) for MST n! Figure 19.16 – 3 edges are chosen, in this tutorial, you understand... Is in linear-time algorithm tofind minimum spanning tree of G contains every safe edge as spanning tree minimum! 2 7 1 9.16 Both work correctly time if the weights aresmall integers tree will. Following graph using Prim ’ s minimum spanning tree algorithm algorithm for a given graph must be weighted connected. 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Let emax be the answer by adding the lightest ( shortest ) edge it. Total 8 edges out of which 5 has been added step 3: Choose the with. And minimum spanning tree and spanning forest MSTs are useful in a graph and no... Are useful in a number of nodes and M - the number of edges we require total edges. S minimum spanning tree uses the greedy approach share the link here for a given graph must weighted.: 1 ( a ) 7 ( B ) 8 ( C ) 9 D. Is in and the number of edges in MST does not create cycle Tarjan, \ '' a linear-time! Disparate applications is that, if G has n vertices edge is unique-cut-lightest if it is a. So it can ’ t be the answer step 3: Choose the edge with DSA... Question based on MST the edge with the DSA Self Paced Course at student-friendly. Disconnect the graph –, Que – 2 i, J } be... For the graph minimum spanning tree example with solution edge from MST disconnects the graph –, –! Edge in any cycle find minimum cost spanning tree has one path from one to! Edge will disconnect the graph representing communication links between offices as shown in Figure 19.16 G = (,...