Storing a Sparse Table with 0(1) Worst Case Access Time
Journal of the ACM (JACM)
Implicit representation of graphs
SIAM Journal on Discrete Mathematics
Short encodings of planar graphs and maps
Discrete Applied Mathematics
Membership in Constant Time and Almost-Minimum Space
SIAM Journal on Computing
Low Redundancy in Static Dictionaries with Constant Query Time
SIAM Journal on Computing
Optimal bounds for the predecessor problem and related problems
Journal of Computer and System Sciences - STOC 1999
Compact representations of separable graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Efficient Minimal Perfect Hashing in Nearly Minimal Space
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Succinct representation of balanced parentheses, static trees and planar graphs
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Rank/select operations on large alphabets: a tool for text indexing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Optimal succinct representations of planar maps
Proceedings of the twenty-second annual symposium on Computational geometry
Succinct indexes for strings, binary relations and multi-labeled trees
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets
ACM Transactions on Algorithms (TALG)
Succinct Representations of Arbitrary Graphs
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
An Introduction to Kolmogorov Complexity and Its Applications
An Introduction to Kolmogorov Complexity and Its Applications
Succinct representations of permutations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Optimal trade-offs for succinct string indexes
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Adaptive searching in succinctly encoded binary relations and tree-structured documents
CPM'06 Proceedings of the 17th Annual conference on Combinatorial Pattern Matching
New lower and upper bounds for representing sequences
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
| Hi-index | 5.23 |
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 @d0) or too dense (namely m=@w(n^2^-^@d) for any constant @d0). 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 @e0. 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 @d0), or very dense (mn^2/lg^1^-^@dn for an arbitrarily small constant @d0).