Rearranging a sequence of points onto a line

초록

Given a sequence of n weighted points (p(1), p(2), ... , p(n)) in the plane, we consider the problem of finding a rearrangement of the points, q(i) for each p(i), onto a line such that any two consecutive points q(i) and q(i+1) are at distance no more than their weight difference, and the maximum distance between pi and qi over all i is minimized. We present efficient algorithms that compute optimal rearrangements for three variants of the problem either under the L-1 metric or under the Euclidean metric. When the line is fully specified or partially specified by only its orientation, our algorithms take near-linear time. When we have to choose a best target line over all lines in the plane, onto which the input sequence can be rearranged with the optimal rearrangement cost, we present an O (n(3)log(4 )n)-time algorithm.