This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AMS/MBA 546 (Spring, 2008) Estie Arkin Network Flows  Final, sketch of solutions Average 85. 1). Consider a set of n numbers a 1 , a 2 , . . . , a n arranged in nondecreasing order of their values ( a i a i +1 ). We wish to partition these numbers into clusters so that (1) each cluster contains at least p numbers; (2) each cluster contains consecutive numbers from the list a 1 , a 2 , . . . , a n ; and (3) the sum of the squared deviation of the numbers from their cluster mean is as small as possible. Let a ( S ) = ( i S a i ) /  S  denote the mean of a set S of numbers forming a cluster. For a number a k in the cluster, its squared deviation from the cluster mean is ( a k a ( S )) 2 . Describe how to formulate this problem as a shortest path problem. Make sure to clearly define the nodes and arcs of your graph, the costs on the arcs, and what are s and t . Construct nodes 0 , 1 , 2 , . . . , n where s = 0 and t = n . We build arcs ( i, j ) if j i + p . The cost of an arc ( i, j ) is the cost of a cluster i +1 , i +2 , . . . j which is j k = i +1 ( a k a ( S )) 2 , where a ( S ) = ( j k = i +1 a k ) / ( j i ). 2). (a). Given a tree on n > 2 nodes, with exactly 2 nodes of degree equal to 1, show that all other nodes of this tree must have degree equal to 2. Let d i be the degree of node i . We have 1 + 1 + d 3 + d 4 + d n = 2( n 1), since we know that the sum of the node degrees is 2 times the number of edges in any graph, and the number of edges in a tree with n nodes is n 1. So n 3 d i = 2 n 4, and d i 2 for i = 3 , ..., n . So 2( n 2) 2 n 4. Since we have equality, this implies that d i = 2 for i = 3 , ..., n ....
View Full
Document
 Fall '08
 Arkin,E

Click to edit the document details