Minimum max spannig tree slides

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Minimum Max Stretch Spanning Tree Problem

Ashok Naval

Under the Guidance of Dr Alok Singh

November 27, 2014

Ashok Naval Minimum Max Stretch Spanning Tree Problem

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Introduction

Given a graph G and a spanning tree T of G,we define the stretchof u and v in T as

str T ,G = dist T (u ,v )dist G (u ,v )

Denote the maximum stretch of T with respect to G by

maxstr (T , G ) = max

x ,y ∈V (G ){str T ,G (x , y )}

We denote the minimum max-stretch of G byminstr (G ) = min{maxstr (T , G )|T is a spanning tree of G }


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Application of MMST problem

Sequence alignment problems in computational biology.

Network design where the cost of an edge may represent thecost incurred in routing messages between its endpoints.


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Previous attempt on the problem

Most of the work on MMST is focused on designingapproximation algorithm for it.

The approximation algorithms provide solutions that areguaranteed to be within a certain factor of optimum

But on many problems, solution obtained with metaheuristicoutperform them.


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Difference between Heuristic and Meta-heuristics

Heuristics are problem-dependent techniques.They are oftentoo greedy, they usually get trapped in a local optimum andthus fail, in general, to obtain the global optimum solution.

Meta-heuristics, on the other hand, are problem-independenttechniques.They are not greedy. They may accept atemporary deterioration of the solution, which allows them toexplore more thoroughly the solution space and thus to get ahopefully better solution


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Overview of ABC algorithm

Based on metaheuristic approach motivated by intelligentforaging behavior of honeybee swarm.

Foraging bees are divided into three classesEmloyedOnlookersScouts


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The ABC algorithm follows an iterative process.Begins by associating all employed bees with randomlygenrated food sources(solution).

The probabiltiy p i of selecting a food source i is determined

using expressionp i =

F i mi =1

With this scheme, higher the fitness of a food source,higherwill be the probability of selection.


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Good food source will get more onlookers than the bad ones

After all onlookers have selected their food sources,each willdetermine food source in the neighborhood.

Among all the neighboring food sources i and food source iitself, the best food source is determined.

This best food source becomes the new location of the foodsource i in the next iteration.


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Pseudo code for ABC algorithm

1 Initilization

2 Move the employeed bees onto their food sources andevaluate their nectar amounts.

3 Place the onlookers depending upon the nectar amounts

obtained by employeed bees.

4 Send the scouts for exploring new food source.

5 Memorize the best food sources obtained so far.

6 If a termination criterion is not satisfied, go to step 2;

otherwise stop the procedure and display the best food sourceobtained so far.


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Pseudo Code for Improving the solution

1 S i =S 1,S 2,...S n

2 Choose the path which is causing the highest stretch factor.

3 Randomly delete an edge.

4 Find the candidate edge connecting the two partition resultingfrom step 3.

5 Recompute the stretch factor.

6 If stretch factor improves, go to step 2.

7 If solution doesn’t imrpove for required no. of time then pickanother solution.


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Results obtained so far


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My solution

Any known algorithm for all pair shortest path is order of O(n3).

Spanning tree have number of edges=number of vertices-1.

Breadth first traversal gives level-wise traversal of node a in atree.

If we do BFS on each node and represent the tree withadjacency list, we can reduce this time compelxity to O(n2).


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What we are planning to do in future

Further improving the all pair shortest algorithm for spanningtree.

To implement other metaheuristic technique for the sameproblem and compare it with ABC algorithm


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References

C. Liebchen, G. Wunsch The zoo of tree spanner problems ,Discrete Applied Mathematics - DAM , vol. 156, no. 5, pp.569-587, 2008.

A. Singh, S. Sundar An artificial bee colony algorithm for MMST problem, Springer-Verlag, 2011.

Y. Emek, D. Peleg Approximating MMST on unweighted

graphs , SIAM J. COMPUT., Vol. 38, No. 5, pp. 1761-1781,

2008.


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Thank You


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Minimum max spannig tree slides