We study approximation algorithms for the following three string measures that are widely used in practice: edit distance (ED), longest common subsequence (LCS), and longest increasing sequence (LIS). All three problems can be solved exactly by standard algorithms that run in polynomial time with roughly $Θ(n)$ 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 $1+ε$ or $1-ε$ approximation for all three problems, where $ε>0$ can be any constant and even slightly sub constant. Our algorithms are flexible and can be adjusted to achieve the following two regimes of parameters: 1) space $n^δ$ for any constant $δ>0$ with running time essentially the same as or slightly more than the standard algorithms; and 2) space $\mathsf{polylog}(n)$ with (a larger) polynomial running time, which puts the approximation versions of the three problems in Steve's class (\textbf{SC}). Our algorithms significantly improve previous results in terms of space complexity, where all known results need to use space at least $Ω(\sqrt{n})$. Some of our algorithms can also be adapted to work in the asymmetric streaming model [SS13], and output the corresponding sequence. Furthermore, our results can be used to improve a recent result by Farhadi et. al. [FHRS20] about approximating ED in the asymmetric streaming model, reducing the running time from being exponential in [FHRS20] to a polynomial. Our algorithms are based on the idea of using recursion as in Savitch's theorem [Sav70], and a careful adaption 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.