Representing point sets on the plane as permutations

Abstract

Let Σ={σ1,…,σn} be a permutation of the set [n]={1,…,n}. A subsequence σi1,…,σim of Σ is increasing (respectively decreasing) if σi1<…<σim (respectively σi1>…>σim). It is well known that any permutation of [n] contains an increasing or a decreasing subsequence of at least n elements. A classical proof of this result is the following [6]: To every σi assign the coordinates f(σi)=(xi,yi) on the plane, where xi is the length of a longest increasing subsequence of Σ ending at σi, and yi is the length of a longest decreasing subsequence of Σ starting at σi. Then it is easy to see that at least one point in f(Σ)={f(σi):i=1,…,n} has its x or y-coordinate greater than or equal to n. In this paper, we characterize the point sets S in N2 such that S=f(Σ) for some permutation Σ. Furthermore, given a set S of n points in N2, we give an O(nlog⁡n)-time algorithm to obtain, if it exists, a permutation Σ such that f(Σ)=S. Additionally, we show that by slightly modifying S, the permutation Σ always exists and can be found in O(nlog⁡n) time. Our results can be used, for example, to encrypt keywords by generating permutations from which the keywords can be recovered.