210 research outputs found

    Space-Efficient Algorithms for Longest Increasing Subsequence

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    Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For sqrt(n) <= s <= n, we present algorithms that use O(s log n) bits and O(1/s n^2 log n) time for computing the length of a longest increasing subsequence, and O(1/s n^2 log^2 n) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space

    Quantum Meets Fine-Grained Complexity: Sublinear Time Quantum Algorithms for String Problems

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    Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence

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    A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem

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    The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Theta~(sqrt N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Omega(sqrt N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which the lower bound is based have very efficient randomized protocols. The purpose of this note is to guide and alert future researchers working on this very interesting problem
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