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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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Abstract. Jumbled indexing is the problem of indexing a text T for queries that ask whether there is a substring of T matching a pattern represented as a Parikh vector, i.e., the vector of frequency counts for each character. Jumbled indexing has garnered a lot of interest in the last four years; for a partial list see [2,6,13,16,17,20,22,24,26,30,35,36]. There is a naive algorithm that preprocesses all answers in O(n 2 |Σ|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(n log |Σ|) query time. Despite a tremendous amount of effort there has been little improvement over these running times. In this paper we provide good reason for this. We show that, under a 3SUM-hardness assumption, jumbled indexing for alphabets of size ω(1) requires Ω(n 2−ǫ ) preprocessing time or Ω(n 1−δ ) query time for any ǫ, δ > 0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r ≥ 3 there exist describable fixed constant ǫr and δr such that jumbled indexing requires Ω(n 2−ǫr ) preprocessing time or Ω(n 1−δr ) query time.

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In this article, we address a new version of dynamic pattern matching. The dynamic text and static pattern matching problem is the problem of finding a static pattern in a text that is continuously being updated. The goal is to report all new occurrences of the pattern in the text after each text update. We present an algorithm for solving the problem where the text update operation is changing the symbol value of a text location. Given a text of length n and a pattern of length m, our algorithm preprocesses the text in time O(n log log m), and the pattern in time O(m log m). The extra space used is O(n + m log m). Following each text update, the algorithm deletes all prior occurrences of the pattern that no longer match, and reports all new occurrences of the pattern in the text in O(log log m) time. We note that the complexity is not proportional to the number of pattern occurrences, since all new occurrences can be reported in a succinct form.

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