arXiv subjects

Roberto Grossi

Publications and source records attributed to Roberto Grossi.

28 records · Page 2Linked to original sources

arXiv2013

Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing

This paper presents a general technique for optimally transforming any dynamic data structure that operates on atomic and indivisible keys by constant-time comparisons, into a data structure that handles unbounded-length keys whose comparison cost is not a constant. Examples of these keys are strings, multi-dimensional points, multiple-precision numbers, multi-key data (e.g.~records), XML paths, URL addresses, etc. The technique is more general than what has been done in previous work as no particular exploitation of the underlying structure of is required. The only requirement is that the insertion of a key must identify its predecessor or its successor. Using the proposed technique, online suffix tree can be constructed in worst case time $O(\log n)$ per input symbol (as opposed to amortized $O(\log n)$ time per symbol, achieved by previously known algorithms). To our knowledge, our algorithm is the first that achieves $O(\log n)$ worst case time per input symbol. Searching for a pattern of length $m$ in the resulting suffix tree takes $O(\min(m\log |Σ|, m + \log n) + tocc)$ time, where $tocc$ is the number of occurrences of the pattern. The paper also describes more applications and show how to obtain alternative methods for dealing with suffix sorting, dynamic lowest common ancestors and order maintenance.

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arXiv2012

Optimal Listing of Cycles and st-Paths in Undirected Graphs

We present the first optimal algorithm for the classical problem of listing all the cycles in an undirected graph. We exploit their properties so that the total cost is the time taken to read the input graph plus the time to list the output, namely, the edges in each of the cycles. The algorithm uses a reduction to the problem of listing all the paths from a vertex s to a vertex t which we also solve optimally.

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arXiv2012

The Wavelet Trie: Maintaining an Indexed Sequence of Strings in Compressed Space

An indexed sequence of strings is a data structure for storing a string sequence that supports random access, searching, range counting and analytics operations, both for exact matches and prefix search. String sequences lie at the core of column-oriented databases, log processing, and other storage and query tasks. In these applications each string can appear several times and the order of the strings in the sequence is relevant. The prefix structure of the strings is relevant as well: common prefixes are sought in strings to extract interesting features from the sequence. Moreover, space-efficiency is highly desirable as it translates directly into higher performance, since more data can fit in fast memory. We introduce and study the problem of compressed indexed sequence of strings, representing indexed sequences of strings in nearly-optimal compressed space, both in the static and dynamic settings, while preserving provably good performance for the supported operations. We present a new data structure for this problem, the Wavelet Trie, which combines the classical Patricia Trie with the Wavelet Tree, a succinct data structure for storing a compressed sequence. The resulting Wavelet Trie smoothly adapts to a sequence of strings that changes over time. It improves on the state-of-the-art compressed data structures by supporting a dynamic alphabet (i.e. the set of distinct strings) and prefix queries, both crucial requirements in the aforementioned applications, and on traditional indexes by reducing space occupancy to close to the entropy of the sequence.

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arXiv2011

Fast Compressed Tries through Path Decompositions

Tries are popular data structures for storing a set of strings, where common prefixes are represented by common root-to-node paths. Over fifty years of usage have produced many variants and implementations to overcome some of their limitations. We explore new succinct representations of path-decomposed tries and experimentally evaluate the corresponding reduction in space usage and memory latency, comparing with the state of the art. We study two cases of applications: (1) a compressed dictionary for (compressed) strings, and (2) a monotone minimal perfect hash for strings that preserves their lexicographic order. For (1), we obtain data structures that outperform other state-of-the-art compressed dictionaries in space efficiency, while obtaining predictable query times that are competitive with data structures preferred by the practitioners. In (2), our tries perform several times faster than other trie-based monotone perfect hash functions, while occupying nearly the same space.

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arXiv2011

Partial Data Compression and Text Indexing via Optimal Suffix Multi-Selection

Consider an input text string T[1,N] drawn from an unbounded alphabet. We study partial computation in suffix-based problems for Data Compression and Text Indexing such as (I) retrieve any segment of K<=N consecutive symbols from the Burrows-Wheeler transform of T, and (II) retrieve any chunk of K<=N consecutive entries of the Suffix Array or the Suffix Tree. Prior literature would take O(N log N) comparisons (and time) to solve these problems by solving the total problem of building the entire Burrows-Wheeler transform or Text Index for T, and performing a post-processing to single out the wanted portion. We introduce a novel adaptive approach to partial computational problems above, and solve both the partial problems in O(K log K + N) comparisons and time, improving the best known running times of O(N log N) for K=o(N). These partial-computation problems are intimately related since they share a common bottleneck: the suffix multi-selection problem, which is to output the suffixes of rank r_1,r_2,...,r_K under the lexicographic order, where r_1<r_2<...<r_K, r_i in [1,N]. Special cases of this problem are well known: K=N is the suffix sorting problem that is the workhorse in Stringology with hundreds of applications, and K=1 is the recently studied suffix selection. We show that suffix multi-selection can be solved in Theta(N log N - sum_{j=0}^K Delta_j log Delta_j+N) time and comparisons, where r_0=0, r_{K+1}=N+1, and Delta_j=r_{j+1}-r_j for 0<=j<=K. This is asymptotically optimal, and also matches the bound in [Dobkin, Munro, JACM 28(3)] for multi-selection on atomic elements (not suffixes). Matching the bound known for atomic elements for strings is a long running theme and challenge from 70's, which we achieve for the suffix multi-selection problem. The partial suffix problems as well as the suffix multi-selection problem have many applications.

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arXiv2011

Consecutive Ones Property and PQ-Trees for Multisets: Hardness of Counting Their Orderings

A binary matrix satisfies the consecutive ones property (COP) if its columns can be permuted such that the ones in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q_1,..,Q_m}, where Q_i is subset of R for some universe R, satisfies the COP if the symbols in R can be permuted such that the elements of each set Q_i occur consecutively, as a contiguous segment of the permutation of R's symbols. We consider the COP version on multisets and prove that counting its solutions is difficult (#P-complete). We prove completeness results also for counting the frontiers of PQ-trees, which are typically used for testing the COP on sets, thus showing that a polynomial algorithm is unlikely to exist when dealing with multisets. We use a combinatorial approach based on parsimonious reductions from the Hamiltonian path problem, showing that the decisional version of our problems is therefore NP-complete.

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arXiv2010

Optimal Trade-Off for Succinct String Indexes

Let s be a string whose symbols are solely available through access(i), a read-only operation that probes s and returns the symbol at position i in s. Many compressed data structures for strings, trees, and graphs, require two kinds of queries on s: select(c, j), returning the position in s containing the jth occurrence of c, and rank(c, p), counting how many occurrences of c are found in the first p positions of s. We give matching upper and lower bounds for this problem, improving the lower bounds given by Golynski [Theor. Comput. Sci. 387 (2007)] [PhD thesis] and the upper bounds of Barbay et al. [SODA 2007]. We also present new results in another model, improving on Barbay et al. [SODA 2007] and matching a lower bound of Golynski [SODA 2009]. The main contribution of this paper is to introduce a general technique for proving lower bounds on succinct data structures, that is based on the access patterns of the supported operations, abstracting from the particular operations at hand. For this, it may find application to other interesting problems on succinct data structures.

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arXiv2010

MADMX: A Novel Strategy for Maximal Dense Motif Extraction

We develop, analyze and experiment with a new tool, called MADMX, which extracts frequent motifs, possibly including don't care characters, from biological sequences. We introduce density, a simple and flexible measure for bounding the number of don't cares in a motif, defined as the ratio of solid (i.e., different from don't care) characters to the total length of the motif. By extracting only maximal dense motifs, MADMX reduces the output size and improves performance, while enhancing the quality of the discoveries. The efficiency of our approach relies on a newly defined combining operation, dubbed fusion, which allows for the construction of maximal dense motifs in a bottom-up fashion, while avoiding the generation of nonmaximal ones. We provide experimental evidence of the efficiency and the quality of the motifs returned by MADMX

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arXiv2009

More Haste, Less Waste: Lowering the Redundancy in Fully Indexable Dictionaries

We consider the problem of representing, in a compressed format, a bit-vector $S$ of $m$ bits with $n$ 1s, supporting the following operations, where $b \in \{0, 1 \}$: $rank_b(S,i)$ returns the number of occurrences of bit $b$ in the prefix $S[1..i]$; $select_b(S,i)$ returns the position of the $i$th occurrence of bit $b$ in $S$. Such a data structure is called \emph{fully indexable dictionary (FID)} [Raman et al.,2007], and is at least as powerful as predecessor data structures. Our focus is on space-efficient FIDs on the \textsc{ram} model with word size $Θ(\lg m)$ and constant time for all operations, so that the time cost is independent of the input size. Given the bitstring $S$ to be encoded, having length $m$ and containing $n$ ones, the minimal amount of information that needs to be stored is $B(n,m) = \lceil \log {{m}\choose{n}} \rceil$. The state of the art in building a FID for $S$ is given in [Patrascu,2008] using $B(m,n)+O(m / ((\log m/ t) ^t)) + O(m^{3/4}) $ bits, to support the operations in $O(t)$ time. Here, we propose a parametric data structure exhibiting a time/space trade-off such that, for any real constants $0 < δ\leq 1/2$, $0 < \eps \leq 1$, and integer $s > 0$, it uses \[ B(n,m) + O(n^{1+δ} + n (\frac{m}{n^s})^\eps) \] bits and performs all the operations in time $O(sδ^{-1} + \eps^{-1})$. The improvement is twofold: our redundancy can be lowered parametrically and, fixing $s = O(1)$, we get a constant-time FID whose space is $B(n,m) + O(m^\eps/\poly{n})$ bits, for sufficiently large $m$. This is a significant improvement compared to the previous bounds for the general case.

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arXiv2009

Optimal cache-aware suffix selection

Given string $S[1..N]$ and integer $k$, the {\em suffix selection} problem is to determine the $k$th lexicographically smallest amongst the suffixes $S[i... N]$, $1 \leq i \leq N$. We study the suffix selection problem in the cache-aware model that captures two-level memory inherent in computing systems, for a \emph{cache} of limited size $M$ and block size $B$. The complexity of interest is the number of block transfers. We present an optimal suffix selection algorithm in the cache-aware model, requiring $\Thetah{N/B}$ block transfers, for any string $S$ over an unbounded alphabet (where characters can only be compared), under the common tall-cache assumption (i.e. $M=\Omegah{B^{1+ε}}$, where $ε<1$). Our algorithm beats the bottleneck bound for permuting an input array to the desired output array, which holds for nearly any nontrivial problem in hierarchical memory models.

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