The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we consider these problems in the asymmetric streaming model introduced by Andoni et al. (FOCS'10) and Saks and Seshadhri (SODA'13). In this model we have random access to one string and streaming access the other string. Our main contribution is a constant factor approximation algorithm for ED with the memory of $\tilde O(n^δ)$ for any constant $δ> 0$. In addition to this, we present an upper bound of $\tilde O_ε(\sqrt{n})$ on the memory needed to approximate ED or LCS within a factor $1+ε$. All our algorithms are deterministic and run in a single pass. For approximating ED within a constant factor, we discover yet another application of triangle inequality, this time in the context of streaming algorithms. Triangle inequality has been previously used to obtain subquadratic time approximation algorithms for ED. Our technique is novel and elegantly utilizes triangle inequality to save memory at the expense of an exponential increase in the runtime.