There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
On sparse spanners of weighted graphs
Discrete & Computational Geometry
A fast algorithm for constructing sparse Euclidean spanners
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Parallel shortcutting of rooted trees
Journal of Algorithms
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A new way to weigh Malnourished Euclidean graphs
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Applications of Path Compression on Balanced Trees
Journal of the ACM (JACM)
Sparse communication networks and efficient routing in the plane (extended abstract)
Proceedings of the nineteenth annual ACM symposium on Principles of distributed computing
Approximate distance oracles for geometric graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Trade-offs in non-reversing diameter
Nordic Journal of Computing
Space-time tradeoff for answering range queries (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Tight bounds for the partial-sums problem
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Lower bound for sparse Euclidean spanners
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Small hop-diameter sparse spanners for doubling metrics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Geometric Spanner Networks
Approximate distance oracles for geometric spanners
ACM Transactions on Algorithms (TALG)
Shallow-Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
An optimal-time construction of sparse Euclidean spanners with tiny diameter
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Sparse Euclidean Spanners with Tiny Diameter
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Optimal euclidean spanners: really short, thin and lanky
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
New doubling spanners: better and simpler
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Hi-index | 0.00 |
In a seminal STOC'95 paper, Arya et al. [4] devised a construction that for any set S of n points in Rd and any ε 0, provides a (1 + ε)-spanner with diameter O(log n), weight O(log2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ε)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(log n) and weight O(log n)w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+ε)-spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n log n)w(MST(S)), and O(n) edges. For k = O(1) this gives rise to maximum degree O(1), diameter O(log n) and weight O(log2 n)w(MST(S)), which is one of the aforementioned results of [4]. For k = n1/a(n) this gives rise to diameter O(a(n)), weight O(n1/α(n)(log n)α(n))w(MST(S)) and maximum degree O(n1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log2 n, but the diameter is allowed to grow beyond log n.