We define and design succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. The main advantage of succinct indexes as opposed to succinct (integrated data/index) encodings is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this article, we present succinct indexes for various data types, namely strings, binary relations and multilabeled trees. Given the support for the interface of the ADTs of these data types, we can support various useful operations efficiently by constructing succinct indexes for them. When the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data.Using our techniques, we design a succinct encoding that represents a string of length n over an alphabet of size σ using nH k (S)+lg σ ·o(n)+O(nlg σ /lg lg lg σ ) bits to support access/rank/select operations in o((lg lg σ ) 1+ ) time, for any fixed constant > 0. We also design a succinct text index using nH 0 (S) + O(n lg σ /lg lg σ ) bits that supports finding all the occ occurrences of a given pattern of length m in O(mlg lg σ + occ lg n/ lg σ ) time, for any fixed constant 0 < < 1. Previous results on these two problems either have a lg σ factor instead of lg lg σ in the running time, or are not compressed. Finally, we present succinct encodings of binary relations and multi-labeled trees that are more compact than previous structures.

Abstract. We present a succinct representation of a set of n points on an n × n grid using n lg n + o(n lg n) bits 3 to support orthogonal range counting in O(lg n/ lg lg n) time, and range reporting in O(k lg n/ lg lg n) time, where k is the size of the output. This achieves an improvement on query time by a factor of lg lg n upon the previous result of Mäkinen and Navarro [15], while using essentially the information-theoretic minimum space. Our data structure not only can be used as a key component in solutions to the general orthogonal range search problem to save storage cost, but also has applications in text indexing. In particular, we apply it to improve two previous space-efficient text indexes that support substring search [7] and position-restricted substring search [15]. We also use it to extend previous results on succinct representations of sequences of small integers, and to design succinct data structures supporting certain types of orthogonal range query in the plane.

These results have been published in extended form in "Algorithmica 62(1-2): 224-257 (2012)"International audienceIn many applications, the properties of an object being modeled are stored as labels on vertices or edges of a graph. In this paper, we consider succinct representation of labeled graphs. Our main results are the succinct representations of labeled and multi-labeled graphs (we consider vertex labeled planar triangulations, as well as edge labeled planar graphs and the more general $k$-page graphs) to support various label queries efficiently. The additional space cost to store the labels is essentially the information-theoretic minimum. As far as we know, our representations are the first succinct representations of labeled graphs. We also have two preliminary results to achieve the main results. First, we design a succinct representation of unlabeled planar triangulations to support the rank/select of edges in ccw (counter clockwise) order in addition to the other operations supported in previous work. Second, we design a succinct representation for a $k$-page graph when $k$ is large to support various navigational operations more efficiently. In particular, we can test the adjacency of two vertices in $O(\lg k\lg\lg k)$ time, while previous work uses $O(k)$ time

We consider the problem of edge orientation, whose goal is to orient the edges of an undirected dynamic graph with n vertices such that vertex out-degrees are bounded, typically by a function of the graph's arboricity. Our main result is to show that an O(βα)-orientation can be maintained in O( lg(n/(βα)) β ) amortized edge insertion time and O(βα) worst-case edge deletion time, for any β ≥ 1, where α is the maximum arboricity of the graph during update. This is achieved by performing a new analysis of the algorithm of Brodal and Fagerberg [2]. Not only can it be shown that these bounds are comparable to the analysis in Brodal and Fagerberg [2] and that in Kowalik [7] by setting appropriate values of β, it also presents tradeoffs that can not be proved in previous work. Its main application is an approach that maintains a maximal matching of a graph in O(α + √ α lg n) amortized update time, which is currently the best result for graphs with low arboricity regarding this fundamental problem in graph algorithms. When α is a constant which is the case with planar graphs, for instance, our work shows that a maximal matching can be maintained in O( √ lg n) amortized time, while previously the best approach required O(lg n/ lg lg n) amortized time [13]. We further design an alternative solution with worst-case time bounds for edge orientation, and applied it to achieve new results on maximal matchings and adjacency queries.