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NAME: Ambrose Kipkoech

REG: CS/MG/1094/05/18

Digital Analysis of Algorithms Assignment

The traveling salesperson problem (TSP) can be solved via the minimum spanning tree (MST) heuristic, which is used to estimate the cost of completing a tour, given that a partial tour has already been constructed. The MST cost of a set of cities is the smallest sum of the link costs of any tree that connects all the cities.

(a) Show that this heuristic can be derived from a relaxed version of the TSP.                                                        we can get a solution for MST problem.  If the TSP constraints are relaxed such that each city can be visited more than once and the cost of repeated edges are not counted, then we can find a solution for MST problem.

(b) Show that MST heuristic dominates straight-line distance.                                                                                                      MST dominates straight-line distance because the shortest distance between any two points u, v is their straight-line distance. Moreover, let GMST (V, E¯) be the solution of MST problem given 1 the instance G. The cost from u to v for MST will be equal to the straight-line distance only if (u, v) ∈ E¯. Thus, MST heuristic dominates straight-line distance.

 (c) Write a problem generator for instances of the TSP where cities are represented by random points in the unit square.                                                                                                                                                                                                 Generator(Integer Nums)

{

P OINT S = nil

N = 0

 while(N < Nums)

{

 (X, Y ) = (Random(0, 1), Random(0, 1))

if ((X, Y ) ∈/ P OINT S)

{

P OINT S ← (X, Y )

N = N + 1

 }

}

return P OINT S }

b

Oval: b

c

Oval: c

s

Oval: s

a

Oval: a

d

Oval: d(d) Find an efficient algorithm in the literature for constructing the MST, and use it with an admissible search algorithm to solve instances of the TSP.                                                                                                                                  Any MST algorithm is acceptable.

 

You can use the MST algorithm as an oracle.

The detailed description of MST algorithm can be omitted in this problem. A TSP instance is shown below

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