134,369 research outputs found
Measures on the square as sparse graph limits
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on [0, 1](2). This leads to a generalization of dense graph limit theory to sparse graph sequences. (C) 2019 Elsevier Inc. All rights reserved
Extended Learning Graphs for Triangle Finding
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS\u2714]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC\u2715] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution.
Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP\u2711]
Cover Time and Broadcast Time
We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows:
begin{itemize}
item For any graph of size and minimum degree , we have , where denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs.
item For any -regular (or almost -regular) graph it holds that . Together with our upper bound on , this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree , since then the cover time equals the broadcast time multiplied by (neglecting logarithmic factors).
item Conversely, for any we construct almost -regular graphs that satisfy . Since any regular expander satisfies , the strong relationship given above does not hold if is polynomially smaller than .
end{itemize}
Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Exact Distance Oracles for Planar Graphs
We present new and improved data structures that answer exact node-to-node
distance queries in planar graphs. Such data structures are also known as
distance oracles. For any directed planar graph on n nodes with non-negative
lengths we obtain the following:
* Given a desired space allocation , we show how to
construct in time a data structure of size that answers
distance queries in time per query.
As a consequence, we obtain an improvement over the fastest algorithm for
k-many distances in planar graphs whenever .
* We provide a linear-space exact distance oracle for planar graphs with
query time for any constant eps>0. This is the first such data
structure with provable sublinear query time.
* For edge lengths at least one, we provide an exact distance oracle of space
such that for any pair of nodes at distance D the query time is
. Comparable query performance had been observed
experimentally but has never been explained theoretically.
Our data structures are based on the following new tool: given a
non-self-crossing cycle C with nodes, we can preprocess G in
time to produce a data structure of size that can
answer the following queries in time: for a query node u, output
the distance from u to all the nodes of C. This data structure builds on and
extends a related data structure of Klein (SODA'05), which reports distances to
the boundary of a face, rather than a cycle.
The best distance oracles for planar graphs until the current work are due to
Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For
and space , we essentially improve the query
time from to .Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on
Discrete Algorithms, SODA 201
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