Efficiently Approximating Edit Distance Between Pseudorandom Strings
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We present an algorithm for approximating the edit distance ed(x, y) between two strings x and y in time parameterized by the degree to which one of the strings x satisfies a natural pseudorandomness property. The pseudorandomness model is asymmetric in that no requirements are placed on the second string y, which may be constructed by an adversary with full knowledge of x. We say that x is (p, B)-pseudorandom if all pairs a and b of disjoint B-letter substrings of x satisfy ed(a, b) > pB. Given parameters p and B, our algorithm computes the edit distance between a (p, B)-pseudorandom string x and an arbitrary string y within a factor of O(1/p) in time Õ(nB), with high probability. Our algorithm is robust in the sense that it can handle a small portion of x being adversarial (i.e., not satisfying the pseudorandomness property). In this case, the algorithm incurs an additive approximation error proportional to the fraction of x which behaves maliciously. The asymmetry of our pseudorandomness model has particular appeal for the case where x is a source string, meaning that ed(x, y) will be computed for many strings y. Suppose that one wishes to achieve an O(α)-approximation for each ed(x, y) computation, and that B is the smallest block-size for which the string x is (1/α, B)-pseudorandom. We show that without knowing B beforehand, x may be preprocessed in time Õ(n^1.5√(B)), so that all future computations of the form ed(x, y) may be O(α)-approximated in time Õ(nB). Furthermore, for the special case where only a single ed(x, y) computation will be performed, we show how to achieve an O(α)-approximation in time Õ(n^4/3B^2/3).
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