Deterministic Length Reduction: Fast Convolution in Sparse Data and Applications
نویسندگان
چکیده
In this paper a deterministic algorithm for the length reduction problem is presented. This algorithm enables a new tool for performing fast convolution in sparse data. While the regular fast convolution of vectors V1, V2 whose sizes are N1, N2 respectively, takes O(N1 log N2) using FFT, the proposed algorithm performs the convolution in O(n1 log n1), where n1 is the number of non-zero values in V1. This algorithm assumes that V1 is given in advance, and the V2 is given in running time. This running time is achieved using a preprocessing phase on V1, which takes O(n1) if N1 is polynomial in n1, and O(n 4 1) if N1 is exponential in n1 (which is rarely the case in practical applications). This tool is used to obtain faster results for several well known problems, such as the dDimensional Point Set Matching and Searching in Music Archives.
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