The longest commonly positioned increasing subsequences problem

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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.