273 research outputs found
Streaming and Query Once Space Complexity of Longest Increasing Subsequence
Longest Increasing Subsequence (LIS) is a fundamental problem in
combinatorics and computer science. Previously, there have been numerous works
on both upper bounds and lower bounds of the time complexity of computing and
approximating LIS, yet only a few on the equally important space complexity.
In this paper, we further study the space complexity of computing and
approximating LIS in various models. Specifically, we prove non-trivial space
lower bounds in the following two models: (1) the adaptive query-once model or
read-once branching programs, and (2) the streaming model where the order of
streaming is different from the natural order.
As far as we know, there are no previous works on the space complexity of LIS
in these models. Besides the bounds, our work also leaves many intriguing open
problems.Comment: This paper has been accepted to COCOON 202
Space-Efficient Algorithms for Longest Increasing Subsequence
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
A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem
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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