Space efficient streaming algorithms for the distance to monotonicity and asymmetric edit distance

Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem. We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used. We present an algorithm which, for any $\delta > 0$, given streaming access to an array of length $n$ provides a $(1+\delta)$-multiplicative approximation to the \emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((\log² n)/\delta)$ space. The previous best known approximation using polylogarithmic space was a multiplicative 2-factor. Our algorithm can be used to estimate the length of the LIS to within an additive $\delta n$ for any $\delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$. Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time. It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS. We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$. Our algorithm works in the asymmetric setting (inspired by \cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$. More precisely, it gives an additive-$\delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(\log² n)/\delta)$, where $k$ is the maximum number of times any one symbol appears in $y$.