Abstract
We consider the complexity of computing a longest increasing subsequence (LIS) parameterised by the length of the output. Namely, we show that the maximal length k of an increasing subsequence of a permutation of the set of integers { 1 , 2 , … , n } can be computed in time O ( n log log k ) in the RAM model, improving the previous 30-year bound of O ( n log k ) . The algorithm also improves on the previous O ( n log log n ) bound. The optimality of the new bound is an open question. Reducing the computation of a longest common subsequence (LCS) between two strings to an LIS computation leads to a simple O ( r log log k ) -time algorithm for two sequences having r pairs of matching symbols and an LCS of length k.
Published Version
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