- Saks M
- Seshadhri C

SODA '13 Proceedings of the Twenty-Forth Annual ACM-SIAM Symposium on Discrete Algorithms (2013) 1698-1709

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Approximating the length of the longest increasing sequence (LIS) of a data stream is a well-studied problem. There are many algorithms that estimate the size of the complement of the LIS, referred to as the \emph{distance to monotonicity}, both in the streaming and property testing setting. Let $n$ denote the size of an input array. Our aim is to develop a one-pass streaming algorithm that accurately approximates the distance to monotonicity, and only uses polylogarithmic storage. For any $\delta > 0$, our algorithm provides a $(1+\delta)$-multiplicative approximation for the distance, and uses only $O((\log^2 n)/\delta)$ space. The previous best known approximation using poly-logarithmic space was a multiplicative 2-factor. Our algorithm is simple and natural, being just 3 lines of pseudocode. It is essentially a polylogarithmic space implementation of a classic dynamic program that computes the LIS. Our technique is more general and is applicable to other problems that are exactly solvable by dynamic programs. We are able to get a streaming algorithm for the longest common subsequence problem (in the asymmetric setting introduced in \cite{AKO10}) whose space is small on instances where no symbol appears very many times. Consider two strings (of length $n$) $x$ and $y$. The string $y$ is known to us, and we only have streaming access to $x$. The size of the complement of the LCS is the edit distance between $x$ and $y$ with only insertions and deletions. If no symbol occurs more than $k$ times in $y$, we get a $O(k(\log^2 n)/\delta)$-space streaming algorithm that provides a $(1+\delta)$-multiplicative approximation for the LCS complement. In general, we also provide a deterministic 1-pass streaming algorithm that outputs a $(1+\delta)$-multiplicative approximation for the LCS complement and uses $O(\sqrt{(n\log n)/\delta})$ space.

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