Publications Details

Improved selection in totally monotone arrays

Mansour, Y.; Park, J.K.; Schieber, B.; Sen, S.

This paper's main result is an O(({radical}{bar m}lgm)(n lg n) + mlg n)-time algorithm for computing the kth smallest entry in each row of an m {times} n totally monotone array. (A two-dimensional A = a(i,j) is totally monotone if for all i{sub 1} < i{sub 2} and j{sub 1} < j{sup 2}, < a(i{sub 1},j{sub 2}) implies a(i{sub 2},j{sub 1})). For large values of k (in particular, for k=(n/2)), this algorithm is significantly faster than the O(k(m+n))-time algorithm for the same problem due to Kravets and Park. An immediate consequence of this result is an O(n{sup 3/2} lg{sup 2}n)-time algorithm for computing the kth nearest neighbor of each vertex of a convex n-gon. In addition to the main result, we also give an O(n lg m)-time algorithm for computing an approximate median in each row of an m {times} n totally monotone array; this approximate median is an entry whose rank in its row lies between (n/4) and (3n/4) {minus} 1. 20 refs., 3 figs.