Minimum Spanning Tree: Algorithms [Boruvka(Sollin s), Prim & Kruskal] Presented by Saki Billah University of Applied Sciences, Mittweida

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1 Minimum Spanning Tree: Algorithms [Boruvka(Sollin s), Prim & Kruskal] Presented by Saki University of Applied Sciences, Mittweida

2 Summary of Presentation Definition of Minimum Spanning Tree (MST) Short History Lemmas of MST Pseudocode for MST MST Solution with Algorithms Burovka(Sollin s), Kruskal, and Prim Implementation with Python Algorithm Correctness Proof Uses, Advantages and Limitation Conclusion 2

3 Definition of Minimum Spanning Tree (MST) A minimum spanning tree of a weighted graph (with weights w e 0 for e E ) is a spanning tree which minimizes the quantity = w(t) = e T we 3

4 Definition of MST Say for, we are given a connected, undirected, weighted graph. This is a graph G=(V, E) together with a function w : E R that assigns a real weight w(e) to each edge e. Our task is to find the minimum spanning tree of G, that is, the spanning tree T that minimizes the function; So we said, w(t) = e T we 4

5 Short History Czech scientist Otakar Borůvka developed the first known algorithm for finding a minimum spanning tree. Another Algorithm of MST was developed by Vojtěch Jarník and put in practice by Robert Clay Prim. The other algorithm is called Kruskal's algorithm, and was pulbished by Joseph Kruskal. 5

6 Lemmas for MST There are several different methods for computing minimum spanning trees and All minimum spanning tree algorithms are based on two simple observations or Lemmas. 6

7 Lemmas for MST Lemma 1. The minimum spanning tree contains every safe edge: 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. We prove this claim using a greedy exchange argument. 7

8 Lemmas for MST Lemma 2. The minimum spanning tree contains no useless edge. Adding any useless edge to F would introduce a cycle. Our generic minimum spanning tree algorithm repeatedly adds one or more safe edges to the evolving forest F. Whenever we add new edges to F, some undecided edges become safe, and others become useless. To specify a particular algorithm, we must decide which safe edges to add, and we must describe how to identify new safe and new useless edges, at each iteration of our generic template. 8

9 Proof Example for Lemmas The Graph below follows Lemma-1 and Lemma-2, as in Lemmas-1, Every safe edge is in MST and in Lemma-2, If e is connected then there would be a cycle 9

10 Different Algorithm to Find MST: Algorithms (Burovka(Sollin s), Prim and Kruskal) & Time taken to Find MST 10

11 1. Boruvka's algorithm takes O(E log V) time 2. Prim run-time is either O(E log V) or O(E + V logv) or O(V+E log V) depending on the data-structures used. 3. Kruskal's algorithm, which also takes O(E log V) time 11

12 Faster Algorithm: 1. Linear time randomized algorithm by Karger, Klein & Tarjan 2. The fastest non-randomized comparison-based algorithm with known complexity, by Bernard Chazelle, is based on the soft heap. Is required O(E α(e,v)) time. 12

13 Pseudocode for MST Basic Pseudocode for MST: function MST(G,W): T = {} while T does not form a spanning tree: find the minimum weighted edge in E that is safe for T T = T union {(u,v)} return T 13

14 Pseudocode in Boruvka, Prim & Kruskal and Their Implementation 14

15 Boruvka(Sollin s Pseudocode for MST F Ø While F is disconnected do for all components Ci do F F U {ei} for ei= the min-weight edge leaving Ci end for end while 15

16 Boruvka: How to Find MST Start with Minimum weight vertex of any Tree. Find the next minimum weight in any vertex. Consider only edges that leave the connected component. Add smallest considered edge to your connected component. Continue until a spanning tree is created. 16

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23 Prim Pseudocode for MST T Ø While T is not a spanning tree do T T U {e} for e= the min-weight edge extending the tree T to a new vertex. end while 23

24 Prim: How to Find MST Start with one (any) vertex. Branch outwards to grow your connected component. Consider only edges that leave the connected component. Add smallest considered edge to your connected component. Continue until a spanning tree is created. 24

25 MST: Prim Example 25

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34 Kruskal Pseudocode for MST S E F Ø ; while S= Ø and F is not spanning do Remove the min-weight edge from S. if F U {e} [does not create a cycle then F F U {e} else Discard e end if end while 34

35 Kruskal: How to Find MST Start with V disjoint components Consider lesser weight edges first to incrementally connect components Make certain to avoid cycles. Continue until spanning tree is created. 35

36 MST: Kruskal Example 36

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44 Proof of Correctness: Boruvka, Prim and Kruskal Algorithm 44

45 Proof Of Correctness: Boruvka To Prove the Correctness of Boruvka We have to follow 2 Lemmas, these are: Lemma-1: Let v V be any vertex in G, the MST of G must contain edge (v, w) that is the minimum weight edge incident on v. Lemma-2: The set of edges marked for contraction during a Boruvka phase induces a forest in G. 45

46 Proof Of Correctness: Prim Induction: If we take any spanning tree and add an edge to it, we get a cycle. That's because there was already one path between the endpoints, and now there are two. Contradiction: Say it fails. Consider the first edge "e" chosen by the algorithm that is not consistent with an MST. Let T be the tree we have built just before adding in e, and let M be the MST consistent with T. 46

47 Proof Of Correctness: Kruskal Spanning Tree Validity: By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality: Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output. 47

48 Uses, Advantages and Limitation of MST 48

49 Uses, Advantages and Limitation Uses and Advantages: Can be Used to find the solutions for MST of any electric power, water, telephone lines, transporting route etc to minimize the cost. In Network sensor specially in Plane a set of sensor nodes and transmission data is measured by MST in terms of Euclidian Distance and it is sink with all nodes of sensor transmission and transmission power of all nodes are the minimum value. Limitation: MST can not be used to solve the Travel salesman problem (TSP) because he needs to return home to take rest, more over TSP is a cycle which is not follow the lemma of MST. 49

50 Conclusion Kruskal is better than Boruvka and Prim in a low numbered edges where Prim and Boruvka takes fewer time than Kruskal in a big set of edges. Among all three Algorithm Prim is taken less time in Binary search tree or adjacency list or Fibonacci. 50

51 Sources Book: Algorithm Design, by Jon Kleinberg and Éva Tardos 3. Book: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein: Introduction to Algorithms, third edition

52 Thank You Any Questions? 52

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