On View Consistency in Multi-Server Distributed Virtual Environments Supplementary Material

where μ(Pi,Pj) = ∑ ck∈Pi,cl∈Pj σ(ck, cl). We now transform this simplified VCtoS problem into the quadratic assignment (QAP) problem, which is known to be NP-hard [SG76], as follows. Given n facilities denoted by set NF , n locations denoted by set NL, a flow matrix A, where each entry Aij represents the flow of materials moving from facility i to facility j, and a distance matrix D, where each entry Dij represents the distance between locations i to j, the objective of the QAP problem is to find a one-to-one function π:NF −→ NL, which assigns the facilities to locations such that the cost function, ∑ i,j AijDπ(i)π(j), is minimized. As π is a one-to-one function, we use a permutation matrix X to realize it, where π(i) = j iff Xij = 1. Then, the QAP problem can be rewritten as [KB57]: