9 research outputs found
Succinct Data Structures for Families of Interval Graphs
We consider the problem of designing succinct data structures for interval
graphs with vertices while supporting degree, adjacency, neighborhood and
shortest path queries in optimal time in the -bit 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 bits
are necessary to represent any unlabeled interval graph with vertices,
answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017].
This is augmented by a data structure of size 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 Data Structures for Chordal Graphs
We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time.
We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries:
whether two vertices u,v are adjacent in time f(n) for any f(n) \in \omega(1).
the degree of a vertex in O(1) time.
the vertices adjacent to u in O(f(n)^2) time per neighbour
the length of the shortest path from u to v in O(n f(n)) tim
Succinct representation for (non)deterministic finite automata
International audienceNon)-Deterministic finite automata are one of the simplest models of computation studied in automata theory. Here we study them through the lens of succinct data structures. Towards this goal, we design a data structure for any deterministic automaton D having n states over a σ-letter alphabet using (σ − 1)n log n(1 + o(1)) bits, that determines, given a string x, whether D accepts x in optimal O (|x|) time. We also consider the case when there are N < σ n non-failure transitions, and obtain various time-space trade-offs. Here some of our results are better than the recent work of Cotumaccio and Prezza (SODA 2021). We also exhibit a data structure for non-deterministic automaton N using σ n 2 + n bits that takes O (n 2 |x|) time for string membership checking. Finally, we also provide time and space efficient algorithms for performing several standard operations on the languages accepted by finite automata