Space Efficient Deterministic Approximation of String Measures

We study approximation algorithms for the following three string measures that are widely used in practice: edit distance, longest common subsequence, and longest increasing sequence.\ All three problems can be solved exactly by standard algorithms that run in polynomial time with roughly $O(n)$ space, where $n$ is the input length, and our goal is to design deterministic approximation algorithms that run in polynomial time with significantly smaller space. Towards this, we design several algorithms that achieve $1+\epsilon$ or $1-\epsilon$ approximation for all three problems, where $\epsilon>0$ can be any constant. Our algorithms use space $n^{\delta}$ for any constant $\delta>0$ and have running time essentially the same as or slightly more than the standard algorithms. Our algorithms significantly improve previous results in terms of space complexity, where all known results need to use space at least $\Omega(\sqrt{n})$. Some of our algorithms can also be adapted to work in the asymmetric streaming model \cite{saks2013space}, and output the corresponding sequence. Our algorithms are based on the idea of using recursion as in Savitch's theorem \cite{Savitch70}, and a careful modification of previous techniques to make the recursion work. Along the way we also give a new logspace reduction from longest common subsequence to longest increasing sequence, which may be of independent interest.