11 research outputs found
Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts
We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Omega(n^{1/6}), and that any O(m)-size shortcut set cannot bring it below Omega(n^{1/11}). These improve Hesse\u27s [Hesse, 2003] lower bound of Omega(n^{1/17}). By combining these constructions with Abboud and Bodwin\u27s [Abboud and Bodwin, 2017] edge-splitting technique, we get additive stretch lower bounds of +Omega(n^{1/13}) for O(n)-size spanners and +Omega(n^{1/18}) for O(n)-size emulators. These improve Abboud and Bodwin\u27s +Omega(n^{1/22}) lower bounds
Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond
For fixed h >= 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest h-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact h-hopset and may be applied in processing distance queries on graph G. In particular, a 2-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3-hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that 3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle, and also offer a speedup in query time when compared to simple oracles based on a direct application of 2-hopsets. Finally, we consider the problem of computing minimum-size h-hopset (for any h >= 2) for a given graph G, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When h=3, for a given bound on the space used by the distance oracle, we provide a construction of hopset achieving polylog approximation both for space and query time compared to the optimal 3-hopset oracle given the space bound
Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product
We obtain improved lower bounds for additive spanners, additive emulators,
and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs
that approximately preserve the distances of a given graph. A shortcut set is a
set of edges that when added to a directed graph, decreases its diameter. The
previous best known lower bounds for these three structures are given by Huang
and Pettie [SWAT 2018]. For -sized spanners, we improve the lower bound
on the additive stretch from to . For
-sized emulators, we improve the lower bound on the additive stretch from
to . For -sized shortcut sets, we
improve the lower bound on the graph diameter from to
. Our key technical contribution, which is the basis of all of
our bounds, is an improvement of a graph product known as an alternation
product.Comment: To appear in SODA 202
Balancing Degree, Diameter and Weight in Euclidean Spanners
In this paper we devise a novel \emph{unified} construction of Euclidean
spanners that trades between the maximum degree, diameter and weight
gracefully. For a positive integer k, our construction provides a
(1+eps)-spanner with maximum degree O(k), diameter O(log_k n + alpha(k)),
weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note
that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight
O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree
O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al.
\cite{ADMSS95}; in the corresponding result from \cite{ADMSS95} the spanner has
the same number of edges and diameter, but its weight and degree may be
arbitrarily large. Also, for k = O(1) this gives rise to maximum degree O(1),
diameter O(log n) and weight O(log^2 n) \cdot w(MST(S)), which reproves another
classical result of Arya et al. \cite{ADMSS95}. Our bound of O(log_k n +
alpha(k)) on the diameter is optimal under the constraints that the maximum
degree is O(k) and the number of edges is O(n). Our bound on the weight is
optimal up to a factor of log n. Our construction also provides a similar
tradeoff in the complementary range of parameters, i.e., when the weight should
be smaller than log^2 n, but the diameter is allowed to grow beyond log n.
For random point sets in the d-dimensional unit cube, we "shave" a factor of
log n from the weight bound. Specifically, in this case our construction
achieves maximum degree O(k), diameter O(log_k n + alpha(k)) and weight that is
with high probability O(k \cdot log_k n) \cdot w(MST(S)).
Finally, en route to these results we devise optimal constructions of
1-spanners for general tree metrics, which are of independent interest.Comment: 27 pages, 7 figures; a preliminary version of this paper appeared in
ESA'1
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum