635 research outputs found
On Strong Diameter Padded Decompositions
Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles
Exploration of Graphs with Excluded Minors
We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ? 1 and recovers the known tight bound for the planar case (g = 0)
Exploration of graphs with excluded minors
We study the online graph exploration problem proposed by Kalyanasundaram and
Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This
result encompasses and significantly extends the graph classes that were
previously known to admit a constant competitive ratio. The main ingredient of
our proof is that we find a connection between the performance of the
particular exploration algorithm Blocking and the existence of light spanners.
Conversely, we exploit this connection to construct light spanners of bounded
genus graphs. In particular, we achieve a lightness that improves on the best
known upper bound for genus g>0 and recovers the known tight bound for the
planar case (g=0).Comment: to appear at ESA 202
A Unified and Fine-Grained Approach for Light Spanners
Seminal works on light spanners from recent years provide near-optimal
tradeoffs between the stretch and lightness of spanners in general graphs,
minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a
"truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these
papers employ inherently different techniques than others. Moreover, the
runtime of these constructions is rather high.
In this work, we present a unified and fine-grained approach for light
spanners. Besides the obvious theoretical importance of unification, we
demonstrate the power of our approach in obtaining (1) stronger lightness
bounds, and (2) faster construction times. Our results include:
_ -minor-free graphs: A truly optimal spanner construction and a fast
construction.
_ General graphs: A truly optimal spanner -- almost and a linear-time
construction with near-optimal lightness.
_ Low dimensional Euclidean spaces: We demonstrate that Steiner points help
in reducing the lightness of Euclidean -spanners almost
quadratically for .Comment: We split this paper into two papers: arXiv:2106.15596 and
arXiv:2111.1374
Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings
Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive
orderings (LSO) for Euclidean space. A -LSO is a collection
of orderings such that for every there is an
ordering , where all the points between and w.r.t.
are in the -neighborhood of either or . In essence, LSO
allow one to reduce problems to the -dimensional line. Later, Filtser and Le
[STOC 2022] developed LSO's for doubling metrics, general metric spaces, and
minor free graphs.
For Euclidean and doubling spaces, the number of orderings in the LSO is
exponential in the dimension, which made them mainly useful for the low
dimensional regime. In this paper, we develop new LSO's for Euclidean,
, and doubling spaces that allow us to trade larger stretch for a much
smaller number of orderings. We then use our new LSO's (as well as the previous
ones) to construct path reporting low hop spanners, fault tolerant spanners,
reliable spanners, and light spanners for different metric spaces.
While many nearest neighbor search (NNS) data structures were constructed for
metric spaces with implicit distance representations (where the distance
between two metric points can be computed using their names, e.g. Euclidean
space), for other spaces almost nothing is known. In this paper we initiate the
study of the labeled NNS problem, where one is allowed to artificially assign
labels (short names) to metric points. We use LSO's to construct efficient
labeled NNS data structures in this model
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