Random Access to Grammar-Compressed Strings and Trees

Abstract

Abstract. Grammar based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures (sometimes with slight reduction in efficiency) many of the popular compression schemes, including the Lempel-Ziv family, Run-Length Encoding, Byte-Pair Encoding, Sequitur, and Re-Pair. In this paper, we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string.Let S be a string of length N compressed into a context-free grammar S of size n. We present two representations of S achieving O(log N ) random access time, and either O(n·α k (n)) construction time and space on the pointer machine model, or O(n) construction time and space on the RAM. Here, α k (n) is the inverse of the k th row of Ackermann's function. Our representations also efficiently support decompression of any substring in S: we can decompress any substring of length m in the same complexity as a single random access query and additional O(m) time. Combining these results with fast algorithms for uncompressed approximate string matching leads to several efficient algorithms for approximate string matching on grammar compressed strings without decompression. For instance, we can find all approximate occurrences of a pattern P with at most k errors in time O(n(min{|P |k, k 4 + |P |} + log N ) + occ), where occ is the number of occurrences of P in S. Finally, we generalize our results to navigation and other operations on grammar-compressed ordered trees.All of the above bounds significantly improve the currently best known results. To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two "biased" weighted ancestor data structures, and a compact representation of heavy paths in grammars.Key words. grammar-based compression, straight-line program, approximate string matching, tree compression AMS subject classifications. 68P05, 68P301. Introduction. Modern textual or semi-structured databases, e.g. for biological and WWW data, are huge, and are typically stored in compressed form. A query to such databases will typically retrieve only a small portion of the data. This presents several challenges: how to query the compressed data directly and efficiently, without the need for additional data structures (which can be many times larger than the compressed data), and how to retrieve the answers to the queries. In many practical cases, the naive approach of first decompressing the entire data and then processing it is completely unacceptable -for instance XML data compresses by an order of magnitude on disk [25] but expands by an order of magnitude when represented in-memory [22]; as we will shortly see, this approach is very problematic from an

Tree Compression Using String Grammars

“…We extend some of the results from [8] on querying SLP-compressed balanced parenthesis representations to our context. Specifically, we show that after a linear time preprocessing we can navigate (i.e., move to the parent node and the k th child), compute lowest common ancestors and subtree sizes in time O(log N ), where N is the size of the tree represented by the SLP (Theorem 10).…”

“…In [8] it is shown that for a given SLP A of size n that produces the balanced parenthesis representation of an unranked tree t of size N , one can produce in time O(n) a data structure of size O(n) that supports navigation as well as other important tree queries (e.g. lowest common ancestors queries) in time O(log N ).…”

“…, |t| in the string t. Proof. In [8], it is shown that for an SLP A of size n that produces a well-parenthesized string w ∈ {(, )} * of length N , one can produce in time O(n) a data structure of size O(n) that allows to do the following computations in time O(log N ) on a word RAM, where 1 ≤ k, j ≤ N are given in binary notation and b ∈ {(, )}:…”

“…In this paper we study the compression of ranked trees by SLPs for their preorder traversals. This approach is very similar to [8], where unranked unlabelled trees are compressed by SLPs for their balanced parenthesis representations. In [37] this idea is used together with the grammar-based compressor RePair to get a new compressed suffix tree implementation.…”

“…In Section 4 we compare the size of SLPs for preorder traversals with two other grammar-based compressed tree representations: the above mentioned SLPs for balanced parenthesis representations from [8] and (ii) tree straight-line programs (TSLPs) [10,22,26,33]. The latter directly generalize string SLPs to trees using context-free tree grammars that produce a single tree, see [31] for a survey.…”