In this problem, we explain why the MST algorithmworks. DefineS minimum spanning treeCt nodes connected after iteration t of MSTalgorithm has been completedCt nodes not connected after iteration t of MSTalgorithm has been completedAt set of arcs in minimum spanning tree after titerations of MST algorithm have beencompleted Suppose the MST algorithm does not yield a minimumspanning tree. Then, for some t, it must be the case that allarcs in At1 are in S, but the arc chosen at iteration t (call itat) of the MST algorithm is not in S. Then S must containsome arc at that leads from a node in Ct1 to a node in Ct1). Show that by replacing arc at with arc at, we can obtain ashorter spanning tree than S. This contradiction proves that allarcs chosen by the MST algorithm must be in S. Thus, theMST algorithm does indeed find a minimum spanning tree.

In this problem, we explain why the MST algorithmworks. DefineS minimum spanning treeCt nodes connected after iteration t of MSTalgorithm has been completedCt nodes not connected after iteration t of MSTalgorithm has been completedAt set of arcs in minimum spanning tree after titerations of MST algorithm have beencompleted Suppose the MST algorithm does not yield a minimumspanning tree. Then, for some t, it must be the case that allarcs in At1 are in S, but the arc chosen at iteration t (call itat) of the MST algorithm is not in S. Then S must containsome arc at that leads from a node in Ct1 to a node in Ct1). Show that by replacing arc at with arc at, we can obtain ashorter spanning tree than S. This contradiction proves that allarcs chosen by the MST algorithm must be in S. Thus, theMST algorithm does indeed find a minimum spanning tree.

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter6: Optimization Models With Integer Variables

Section: Chapter Questions

Problem 47P

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