# 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.

### Keywords

Longest increasing subsequence Common positions Algorithms Dual relationship Longest common increasing subsequence## Notes

### Acknowledgements

This research was supported by the National Natural Science Foundation of China under Grant 71371129.

### References

- Bergroth L, Hakonen H, Raita T (2000) A survey of longest common subsequence algorithms. In: String processing and information retrieval, SPIRE 2000. Proceedings. Seventh international symposium on, IEEE. pp 39–48Google Scholar
- Chan WT, Zhang Y, Fung SP, Ye D, Zhu H (2007) Efficient algorithms for finding a longest common increasing subsequence. J Comb Optim 13(3):277–288MathSciNetCrossRefMATHGoogle Scholar
- Crochemore M, Porat E (2010) Fast computation of a longest increasing subsequence and application. Inf Comput 208(9):1054–1059MathSciNetCrossRefMATHGoogle Scholar
- Fredman ML (1975) On computing the length of longest increasing subsequences. Discrete Math 11(1):29–35MathSciNetCrossRefMATHGoogle Scholar
- Kutz M, Brodal GS, Kaligosi K, Katriel I (2011) Faster algorithms for computing longest common increasing subsequences. J Discrete Algorithms 9(4):314–325MathSciNetCrossRefMATHGoogle Scholar
- Masek WJ, Paterson MS (1980) A faster algorithm computing string edit distances. J Comput Syst Sci 20(1):18–31MathSciNetCrossRefMATHGoogle Scholar
- van Emde Boas P (1977) Preserving order in a forest in less than logarithmic time and linear space. Inf Process Lett 6(3):80–82CrossRefMATHGoogle Scholar
- Yang IH, Huang CP, Chao KM (2005) A fast algorithm for computing a longest common increasing subsequence. Inf Process Lett 93(5):249–253MathSciNetCrossRefMATHGoogle Scholar