Abstract
This paper's main result is an [Formula: see text]-time algorithm for computing the kth smallest entry in each row of an m×n totally monotone array. (A two-dimensional array A={a[i, j]} is totally monotone if for all i1<i2 and j1<j2, a[i1, j1]<a[i1, j2] implies a[i2, j1]<a[i2, j2].) 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(n3/2 lg 1/2 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×n totally monotone array; this approximate median is an entry whose rank in its row lies between ⌊n/4⌋ and ⌈3n/4⌉ − 1.
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