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Original citation: Agarwala, R., Bafna, V., Farach, M., Paterson, Michael S. and Thorup, M. (1997) On the approximability of numerical taxonomy (fitting distances by tree metrics Copies of full items can be used for personal research or study, educational, or not-forprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. A note on versions:The version presented in WRAP is the published version or, version of record, and may be cited as it appears here.For more information, please contact the WRAP

The current explosion of stored information necessitates a new model of pattern matching, that of compressed matching. In this model one tries to find all occurrences of a pattern in a compressed text in time proportional to the compressed text size, i.e., without decompressing the text. The most effective general purpose compression algorithms are adaptive, in that the text represented by each compression symbol is determined dynamically by the data. As a result, the encoding of a substring depends on its location. Thus the same substring may``look different'' every time it appears in the compressed text. In this paper we consider pattern matching without decompression in the UNIX Z-compression. This is a variant of the Lempel Ziv adaptive compression scheme. If n is the length of the compressed text and m is the length of the pattern, our algorithms find the first pattern occurrence in time O(n+m 2 ) or O(n log m+m). We also introduce a new criterion to measure compressed matching algorithms, that of extra space. We show how to modify our algorithms to achieve a trade-off between the amount of extra space used and the algorithm's time complexity.

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String matching and compression are two widely studied areas of computer science. The theory of string matching has a long association with compression algorithms. Data structures from string matching can be used to derive fast implementations of many important compression schemes, most notably the Lempel-Ziv (LZ77) algorithm. Intuitively, once a string has been compressed-and therefore its repetitive nature has been elucidated-one might be tempted to exploit this knowledge to speed up string matching. The Compressed Matching Problem is that of performing string matching in a compressed text, without uncompressing it. More formally, let T be a text, let Z be the compressed string representing T , and let P be a pattern. The Compressed Matching Problem is that of deciding if P occurs in T , given only P and Z. Compressed matching algorithms have been given for several compression schemes such as LZW.In this paper we give the first nontrivial compressed matching algorithm for the classic adaptive compression scheme, the LZ77 algorithm. In practice, the LZ77 algorithm is known to compress more than other dictionary compression schemes, such as LZ78 and LZW, though for strings with constant per bit entropy, all these schemes compress optimally in the limit. However, for strings with o(1) per bit entropy, while it was recently shown that the LZ77 gives compression to within a constant factor of optimal, schemes such as LZ78 and LZW may deviate from optimality by an exponential factor. Asymptotically, compressed matching is only relevant if |Z| = o(|T |), i.e., if the compression ratio |T |/|Z| is more than a constant. These results show that LZ77 is the appropriate compression method in such settings.We present an LZ77 compressed matching algorithm which runs in time O(N log 2 U/N+P) where N = |Z|, U = |T |, and P = |P|. Compare with the naïve "decompresion" algorithm, which takes time (U + P) to decide if P occurs in T . Writing U + P as N · U/N + P, we see that we have improved the complexity, replacing the compression factor U/N by a factor log 2 U/N. Our algorithm is competitive in the sense that O(N log 2 U/N + P) = O(U + P), and opportunistic in the sense that O(N log 2 U/N + P) = o(U + P) if N = o(U) and P = o(U).

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