We present a sparse dynamic programming algorithm that, given two strings s and t, a gap penalty λ, and an integer p, computes the value of the gap-weighted length-p subsequences kernel. The algorithm works in time O(p|M|log|t|), where M = {(i, j)|si = t j} is the set of matches of characters in the two sequences. The algorithm is easily adapted to handle bounded length subsequences and different gap-penalty schemes, including penalizing by the total length of gaps and the number of gaps as well as incorporating character-specific match/gap penalties. The new algorithm is empirically evaluated against a full dynamic programming approach and a trie-based algorithm both on synthetic and newswire article data. Based on the experiments, the full dynamic programming approach is the fastest on short strings, and on long strings if the alphabet is small. On large alphabets, the new sparse dynamic programming algorithm is the most efficient. On medium-sized alphabets the trie-based approach is best if the maximum number of allowed gaps is strongly restricted.

Min-convolution and more generally, rank-convolutions play an important role in many application areas, including nonlinear signal processing, pattern recognition, computer vision, and combinatorial optimization. In computer vision they have been used for object recognition, depth estimation, and image restoration. Rank-convolutions such as the median-convolution have the advantage of being more robust against noise. We define a “mask ” as a function taking only values zero and infinity. Rank-convolutions with masks are of special interest; they correspond to a type of order-statistic filtering used in image processing. We show how to compute a rank-convolution of any function with a mask in O(n 3/2 (log n) 1/2). No improvement better than a logarithmic factor over the O(n 2) brute force bound appears to have been previously known even for the special case of min-convolutions. Our result generalizes to the d-dimensional case. Our algorithm is also quite practical. We describe an implementation, illustrating its application to a problem in image processing. 1