We present an index to search a two-dimensional pattern of size m × m in a twodimensional text of size n × n, even when the pattern appears rotated in the text. The index is based on (path compressed) tries. By using O(n 2 ) (i.e. linear) space the index can search the pattern in O((log oe n) 5=2 ) time on average, where oe is the alphabet size. We also consider various schemes for approximate matching, for which we obtain either O(polylog oe n) or O(n 2 ) search time, where ! 1 in most useful cases. A larger index of size O(n 2 (log oe n) 3=2 ) yields an average time of O(log oe n) for the simplest matching model. The algorithms have applications e.g. in content based information retrieval from image databases.

We focus on how to compute the edit distance (or similarity) between two images and the problem of approximate string matching in two dimensions, that is, to find a pattern of size mm in a text of size n n with at most k errors (character substitutions, insertions and deletions). Pattern and text are matrices over an alphabet of size . We present new models and give the first sublinear time search algorithms for the new and the existing models.

Abstract. We give fast filtering algorithms to search for a 2- dimensional pattern in a 2-dimensional text allowing any rotation of the pattern. We consider the cases of exact and approximate matching under several matching models, improving the previous results. For a text of size n \Theta n characters and a pattern of size m \Theta m characters, the exact matching takes average time O(n2 log m=m2), which is optimal. If we allow k mismatches of characters, then our best algorithm achieves O(n2k log m=m2) average time, for reasonable k values. For large k, we obtain an O(n2k3=2 p log m=m) average time algorithm. We generalize

We present new and faster algorithms to search for a 2-dimensional pattern in a 2-dimensional text allowing any rotation of the pattern. This has applications such as image databases and computational biology. We consider the cases of exact and approximate matching under several matching models, using a combinatorial approach that generalizes string matching techniques. We focus on sequential algorithms, where only the pattern can be preprocessed, as well as on indexed algorithms, where the text is preprocessed and an index built on it. On sequential searching we derive average-case lower bounds and then obtain optimal average-case algorithms for all the matching models. At the same time, these algorithms are worst-case optimal. On indexed searching we obtain search time polylogarithmic on the text size, as well as sublinear time in general for approximate searching.

. We address the problem of approximate string matching in d dimensions, that is, to find a pattern of size m d in a text of size n d with at most k ! m d errors (substitutions, insertions and deletions along any dimension). We use a novel and very flexible error model, for which there exists only an algorithm to evaluate the similarity between two elements in two dimensions at O(m 4 ) time. We extend the algorithm to d dimensions, at O(d!m 2d ) time and O(d!m 2d\Gamma1 ) space. We also give the first search algorithm for such model, which is O(d!m d n d ) time and O(d!m d n d\Gamma1 ) space. We show how to reduce the space cost to O(d!3 d m 2d\Gamma1 ) with little time penalty. Finally, we present the first sublinear-time (on average) searching algorithm (i.e. not all text cells are inspected), which is O(kn d =m d\Gamma1 ) for k ! (m=(d(log oe m \Gamma log oe d))) d\Gamma1 , where oe is the alphabet size. After that error level the filter still remains ...