The longest commonly positioned increasing subsequences problem

Abstract

Based on the well-known longest increasing subsequence problem and longest common increasing subsequence (LCIS) problem, we propose the longest commonly positioned increasing subsequences (LCPIS) problem. Let $A=\langle a_1,a_2,\ldots ,a_n\rangle$ and $B{=}\left\langle b_1,b_2,\ldots ,b_n\right\rangle$ be two input sequences. Let ${ Asub}=\left\langle a_{i_1},a_{i_2},\ldots ,a_{i_l}\right\rangle$ be a subsequence of A and ${ Bsub}=\left\langle b_{j_1},b_{j_2},\ldots ,b_{j_l}\right\rangle$ be a subsequence of B such that $a_{i_k}\le a_{i_{k+1}}, b_{j_k}\le b_{j_{k+1}}(1\le k<l)$, and $a_{i_k}$ and $b_{j_k}$ ($1\le k\le l$) are commonly positioned (have the same index $i_k=j_k$) in A and B respectively but these two elements do not need to be equal. The LCPIS problem aims at finding a pair of subsequences Asub and ${ Bsub}$ as long as possible. When all the elements of the two input sequences are positive integers, this paper presents an algorithm with $O(n\log n \log \log M)$ time to compute the LCPIS, where $M={ min}\{{ max}_{1\le i\le n}a_i,{ max}_{1\le j\le n}b_j\}$. And we also show a dual relationship between the LCPIS problem and the LCIS problem.