Arash Farzan

A Uniform Approach Towards Succinct Representation of Trees

We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown fruitful in succinct representation of ordinal trees [7,10] and dynamic binary trees [16,21]. We take an approach that simplifies the existing representation of ordinal trees while allowing the full set of navigational operations. The approach applied to cardinal (i.e. k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g. determining child labeled i) as well as the full set of ordinal-type operations (e.g. reporting the number of siblings to the left of a node). Existing space-optimal succinct representations had not been able to support both types of operations [2,19]. We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. We show that our approach matches the degree-distribution entropy suggested by Jansson etal. [13]. We discuss that our approach can be made adaptive to various other entropy measures.

Succinct Representations of Arbitrary Graphs

We consider the problem of encoding a graph with nvertices and medges compactly supporting adjacency, neighborhood and degree queries in constant time in the logn-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex. We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of nand m, to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information-theory lower bound within lower order terms unless the graph is too sparse (namely m= o(ni¾?) for any constant i¾?> 0) or too dense (namely m= i¾?(n2 i¾? i¾?) for any constant i¾?> 0). Furthermore, we present a succinct encoding for graphs for all values of n,msupporting queries in constant time. The space requirement of the representation is always within a multiplicative 1 + i¾?factor of the information-theory lower bound for any arbitrarily small constant i¾?> 0. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly within lower order terms when the graph is sparse (m= o(ni¾?) for any constant i¾?> 0).

Entropy-Bounded Representation of Point Grids

We give the first fully compressed representation of a set of m points on an n ×n grid, taking H + o(H) bits of space, where \(H=\lg {n^2 \choose m}\) is the entropy of the set. This representation supports range counting, range reporting, and point selection queries, with a performance that is comparable to that of uncompressed structures and that improves upon the only previous compressed structure. Operating within entropy-bounded space opens a new line of research on an otherwise well-studied area, and is becoming extremely important for handling large datasets.

Succinct representations of separable graphs

We consider the problem of highly space-efficient representation of separable graphs while supporting queries in constant time in the RAM with logarithmic word size. In particular, we show constant-time support for adjacency, degree and neighborhood queries. For any monotone class of separable graphs, the storage requirement of the representation is optimal to within lower order terms. Separable graphs are those that admit a O(nc)-separator theorem where c < 1. Many graphs that arise in practice are indeed separable. For instance, graphs with a bounded genus are separable. In particular, planar graphs (genus 0) are separable and our scheme gives the first succinct representation of planar graphs with a storage requirement that matches the information-theory minimum to within lower order terms with constant time support for the queries. We, furthers, show that we can also modify the scheme to succinctly represent the combinatorial planar embedding of planar graphs (and hence encode planar maps).

Worst case optimal union-intersection expression evaluation

We consider the problem of evaluating an expression consisting of unions and intersections of some sorted sets. Given the expression and the sizes of the sets, we are interested in the worst-case complexity of evaluating the expression in terms of the sizes of the sets. We assume no set is repeated in the expression. We show a lower bound on this problem and present an algorithm that matches the lower bound asymptotically.

Succinct encoding of arbitrary graphs

We consider the problem of encoding graphs with n vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time in the @Q(logn)-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex. We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of n and m, to within lower order terms. We prove a lower bound in the cell probe model indicating it is impossible to achieve the information-theory lower bound up to lower order terms unless the graph is either too sparse (namely, m=o(n^@d) for any constant @d>0) or too dense (namely m=@w(n^2^-^@d) for any constant @d>0). Furthermore, we present a succinct encoding of graphs supporting aforementioned queries in constant time. The space requirement of the encoding is within a multiplicative 1+@e factor of the information-theory lower bound for any arbitrarily small constant @e>0. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly within lower order terms where the graph is very sparse (m=o(n^@d) for any constant @d>0), or very dense (m>n^2/lg^1^-^@dn for an arbitrarily small constant @d>0).

Compact Navigation and Distance Oracles for Graphs with Small Treewidth

Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build space-efficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model where the size of a word is Ω(logn) bits.The first oracle, we present, is the navigation oracle which facilitates primitive navigation operations of adjacency, neighborhood, and degree queries. By way of an enumeration argument, which is of interest in its own right, we show the space requirement of the oracle is optimal to within lower order terms for all graphs with n vertices and treewidth k. The oracle supports the mentioned queries all in constant worst-case time. The second oracle, we present, is an exact distance oracle which facilitates distance queries between any pair of vertices (i.e., an all-pairs shortest-path oracle). The space requirement of the oracle is also optimal to within lower order terms. Moreover, the distance queries perform in O(k3log3k) time. Particularly, for the class of graphs of popular interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worst-case time.

Compact representation of posets

We give a data structure for storing an n-element poset of width w in essentially minimal space. We then show how this data structure supports the most interesting queries on posets in either constant time, or in time that depends only on w and the size of the in-/output, but not on n. Our results also have direct applicability to DAGs of low width.

Succinct indices for range queries with applications to orthogonal range maxima

We consider the problem of preprocessing N points in 2D, each endowed with a priority, to answer the following queries: given a axis-parallel rectangle, determine the point with the largest priority in the rectangle. Using the ideas of the effective entropy of range maxima queries and succinct indices for range maxima queries, we obtain a structure that uses O(N) words and answers the above query in O(logN loglogN) time. This a direct improvement of Chazelles result from 1985 [10] for this problem --- Chazelle required O(N/e) words to answer queries in O((logN)1+e) time for any constant e>0.

Compact navigation and distance oracles for graphs with small treewidth

Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build space-efficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model where the size of a word is Ω (log n) bits. The first oracle, we present, is the navigation oracle which facilitates primitive navigation operations of adjacency, neighborhood, and degree queries. By way of an enumerate argument, which is of independent interest, we show the space requirement of the oracle is optimal to within lower order terms for all treewidths. The oracle supports the mentioned queries all in constant worst-case time. The second oracle, we present, is an exact distance oracle which facilitates distance queries between any pair of vertices (i.e., an all-pair shortest-path oracle). The space requirement of the oracle is also optimal to within lower order terms. Moreover, the distance queries perform in O(k2 log3 k) time. Particularly, for the class of graphs of our interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worst-case time.