最長共同遞增子序列 (LCIS) 問題是兩個經典演算法問題的結合: 最長遞增子序列 (LIS) 問題以及最長共同子序列 (LCS) 問題. 在這邊論文中, 我們提出一個枚舉兩個長度為 n 的字串 a, b 之最長共同遞增子序列的演算法, 其時間及空間複雜度為 O(n+σ+Ia), 其中 σ 為字元集大小, Ia 為 a 的遞增子序列數量. 本演算法僅使用字串樹 (trie) 及少數簡單的資料結構, 有實作複雜性低的優勢, 以及在 σ=O(logn) 時, 可以被證明在時間及空間複雜度上是最佳的.

The longest common increasing subsequence (LCIS) problem is the combination of two classic problems in algorithms: the longest increasing subsequence (LIS) problem and the longest common subsequence (LCS) problem. In this paper, we propose an algorithm that finds every LCIS of two sequences a, b of length n in O(n+σ+Ia) time and space, where σ denotes the size of the alphabet set and Ia the total number of increasing subsequences contained in a. Our algorithm employs the trie and some simple data structures, and thus is implementation-wise simple. In addition, it can be proved that our algorithm is optimal in both time and space complexity when σ is O(logn).

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