Succinct Dynamic One-Dimensional Point Reporting

In this paper we present a succinct data structure for the dynamic one-dimensional range reporting problem. Given an interval [a,b] for some a,b in [m], the range reporting query on an integer set S subseteq [m] asks for all points in S cap [a,b]. We describe a data structure that answers reporting queries in optimal O(k+1) time, where k is the number of points in the answer, and supports updates in O(lg^epsilon m) expected time. Our data structure uses B(n,m) + o(B(n,m)) bits where B(n,m) is the minimum number of bits required to represent a set of size n from a universe of m elements. This is the first dynamic data structure for this problem that uses succinct space and achieves optimal query time

Succinct Color Searching in One Dimension

In this paper we study succinct data structures for one-dimensional color reporting and color counting problems. We are given a set of n points with integer coordinates in the range [1,m] and every point is assigned a color from the set {1,...sigma}. A color reporting query asks for the list of distinct colors that occur in a query interval [a,b] and a color counting query asks for the number of distinct colors in [a,b]. We describe a succinct data structure that answers approximate color counting queries in O(1) time and uses mathcal{B}(n,m) + O(n) + o(mathcal{B}(n,m)) bits, where mathcal{B}(n,m) is the minimum number of bits required to represent an arbitrary set of size n from a universe of m elements. Thus we show, somewhat counterintuitively, that it is not necessary to store colors of points in order to answer approximate color counting queries. In the special case when points are in the rank space (i.e., when n=m), our data structure needs only O(n) bits. Also, we show that Omega(n) bits are necessary in that case. Then we turn to succinct data structures for color reporting. We describe a data structure that uses mathcal{B}(n,m) + nH_d(S) + o(mathcal{B}(n,m)) + o(nlgsigma) bits and answers queries in O(k+1) time, where k is the number of colors in the answer, and nH_d(S) (d=log_sigma n) is the d-th order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates. Our dynamic data structure uses nH_d(S)+o(nlgsigma) bits and supports queries in O(k+1) time

Space-Efficient Data-Analysis Queries on Grids

We consider various data-analysis queries on two-dimensional points. We give new space/time tradeoffs over previous work on geometric queries such as dominance and rectangle visibility, and on semigroup and group queries such as sum, average, variance, minimum and maximum. We also introduce new solutions to queries less frequently considered in the literature such as two-dimensional quantiles, majorities, successor/predecessor, mode, and various top-$k$ queries, considering static and dynamic scenarios.Comment: 20 pages, 2 figures, submittin

Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ bits on the side. We then complement our data structure with a lower bound showing that any data structure storing $r$ bits on the side, with $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over $\mathbb{F}_2$

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 $O(\min(m\log |\Sigma|, m + \log n) + tocc)$ 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