Prove that prims algorithm produces a minimum spanning tree, Mathematics

Assignment Help:

Prove that Prim's algorithm produces a minimum spanning tree of a connected weighted graph.

Ans: Suppose G be a connected, weighted graph. At each iteration of Prim's algorithm, an edge should be found that connects a vertex in a subgraph to a vertex outside the subgraph. As G is connected, there will all time be a path to each vertex. The output T of Prim's algorithm is a tree, as the edge and vertex added to T are connected. Suppose T1 be a minimum spanning tree of G. If T1=T then T is a minimum spanning tree. If not, let e be the first edge added throughout the construction of T that is not in T1, and V be the set of vertices connected by the edges added previous to e. After that one endpoint of e is in V and the other is not. As T1 is a spanning tree of G, there is a path in T1 joining the two endpoints. As one travels along with the path, one should encounter an edge f joining a vertex in V to one that is not in V. Now here, at the iteration while e was added to T, f could as well have been added and it would be added in place of e if its weight was less than e. As f was not added, we conclude that w(f) ≥ w(e).

Suppose T2 be the graph acquired by removing f and adding e from T1. It is simple to show that T2 is connected, has similar number of edges as T1, and the total weights of its edges is not larger as compared to that of T1, therefore it is as well a minimum spanning tree of G and it consists of e and all the edges added before it throughout the construction of V. Repeat the steps above and we will eventually acquired a minimum spanning tree of G that is similar to T. This depicts T is a minimum spanning tree.

 


Related Discussions:- Prove that prims algorithm produces a minimum spanning tree

The volume and surface area of this solid , The region bounded by y=e -x a...

The region bounded by y=e -x and the x-axis among x = 0 and x = 1 is revolved around the x-axis. Determine the volume and surface area of this solid of revolution.

Find out the radius of convergence, Example: Find out the radius of conver...

Example: Find out the radius of convergence for the following power series. Solution : Therefore, in this case we have, a n = ((-3) n )/(n7 n+1 )   a n+1 = (

Unbounded intervals, Intervals which extend indefinitely in both the ...

Intervals which extend indefinitely in both the directions are known as unbounded intervals. These are written with the aid of symbols +∞  and -  ∞  . The various types

Semi-infinite slab solution in fourier number, Consider the temperature dis...

Consider the temperature distribution in a 1D flat plate, insulated at x = L and exposed to convective heat transfer at x = 0. On the axes below, sketch what the distribution looks

Equation of the plane x + 4y 3z = 1, Find the equation of the plane thro...

Find the equation of the plane through (2, 1, 0) and parallel to x + 4y   3z = 1.

Function notation, Now we need to move onto something called function notat...

Now we need to move onto something called function notation.  Function notation will be utilized heavily throughout most of remaining section and so it is important to understand i

Find the probability, Q. Suppose Jessica has 10 pairs of shorts and 5 pair...

Q. Suppose Jessica has 10 pairs of shorts and 5 pairs of jeans in her drawer. How many ways could she pick out something to wear for the day? What is the probability that she pick

Determine the quotient and remainder , Let a = 5200 and b = 1320. (a) If...

Let a = 5200 and b = 1320. (a) If a is the dividend and b is the divisor, determine the quotient q and remainder r. (b) Use the Euclidean Algorithm to find gcd(a; b). (c)

Marvin helping teachers plan trip what is the minimum no, Marvin is helping...

Marvin is helping his teachers plan a ?eld trip. There are 125 people going on the ?eld trip and each school bus holds 48 people. What is the minimum number of school buses they wi

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd