# Resolution of the Burrows-Wheeler Transform Conjecture

Burrows-Wheeler Transform (BWT) is an invertible text transformation that permutes symbols of a text according to the lexicographical order of its suffixes. BWT is the main component of some of the most popular lossless compression methods as well as of compressed indexes, central in modern bioinformatics. The compression ratio of BWT-based compressors, such as bzip2, is quantified by the number r of maximal equal-letter runs in the BWT. This is also (up to polylog n factors, where n is the length of the text) the space used by the state-of-the-art BWT-based indexes, such as the recent r-index [Gagie et al., SODA 2018]. The output size of virtually every known compression method is known to be either within a polylog n factor from z, the size of Lempel-Ziv (LZ77) parsing of the text, or significantly larger (by a n^ϵ factor for ϵ > 0). The value of r has resisted, however, all attempts and until now, no non-trivial upper bounds on r were known. In this paper, we show that every text satisfies r=O(zlog^2 n). This result has a number of immediate implications: (1) it proves that a large body of work related to BWT automatically applies to the so-far disjoint field of Lempel–Ziv indexing and compression, e.g., it is possible to obtain full functionality of the suffix tree and the suffix array in O(z polylog n) space; (2) it lets us relate the number of runs in the BWT of the text and its reverse; (3) it shows that many fundamental text processing tasks can be solved in the optimal time assuming that the text is compressible by a sufficiently large polylog n factor using LZ77.

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