Succinct Data Structures for Families of Interval Graphs

Abstract

We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the Θ(log n)bit 1 word RAM model. The degree query reports the number of incident edges to a given vertex in constant time, the adjacency query returns true if there is an edge between two vertices in constant time, the neighborhood query reports the set of all adjacent vertices in time proportional to the degree of the queried vertex, and the shortest path query returns a shortest path in time proportional to its length, thus the running times of these queries are optimal. Towards showing succinctness, we first show that at least n log n − 2n log log n − O(n) bits are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size n log n + O(n) bits while supporting not only the aforementioned queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on the input interval graph efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-proper and k-improper interval graphs, and circular-arc graphs, and design succinct/compact data structures for these graph classes as well along with supporting queries on them efficiently.

Succinct Representations for (Non)Deterministic Finite Automata

“…To illustrate this further, there already exists a large body of work on representing various combinatorial objects succinctly. A partial list of such combinatorial objects would be trees [18,21], various special graph classes like planar graphs [2], chordal graphs [19], partial k-trees [11], interval graphs [1] along with arbitrary general graphs [12], permutations [17], functions [17], bitvectors [22] among many others. We refer the reader to the recent book by Navarro [20] for a comprehensive treatment of this field.…”

Succinct Data Structure for Path Graphs

“…Recently such results have appeared in literature for intersection graphs like interval graphs [2] and chordal graphs [8]. For interval graphs, [2] gives an n log n + O(n) bit succinct data structure that supports degree and adjacency queries in O(1) time while neighborhood query for vertex a in O(d) time where d is the degree of the vertex a. In the case of chordal graphs, [8] gives an n 2 /4 + o(n) bit succinct data structure that supports adjacency query in f (n) time where f (n) ∈ ω(1), degree of a vertex in O(1) time and neighborhood in (f (n)) 2 time per neighbour.…”

“…Formally, given a set T consisting of combinatorial objects with certain property, our goal is to store any arbitrary member x ∈ T using the information theoretic minimum of log(|T |) + o(log(|T |)) bits (throughout this paper, log denotes the logarithm to the base 2) while still being able to support the queries efficiently on x. Recently Acan et al [2] showed that the information-theoretic lower bound for representing interval graphs with n vertices is at least n log n bits, and as path graphs are a proper superclass of interval graphs, this lower bound also holds true for path graphs. Surprisingly, we manage to construct an n log n + o(n log n)-bit data structure for representing path graphs matching this information-theoretic lower bound, thus, obtaining succinct data structure for path graphs for the first time in literature.…”

“…A partial list of such special graph classes would be trees [3,4], planar graphs [5], partial k-tree [6], and arbitrary graphs [7]. Recently such results have appeared in literature for intersection graphs like interval graphs [2] and chordal graphs [8]. For interval graphs, [2] gives an n log n + O(n) bit succinct data structure that supports degree and adjacency queries in O(1) time while neighborhood query for vertex a in O(d) time where d is the degree of the vertex a.…”

“…Central to our data structures is the usage of the classical heavy path decomposition, followed by a careful bookkeeping using an orthogonal range search data structure among others, which maybe of independent interest for designing succinct data structures for other graphs. It is the use of the results of Acan et al [2] in the second data structure that permits a simple and efficient implementation at the expense of more space.…”