7,266 research outputs found
Improved Distributed Algorithms for Exact Shortest Paths
Computing shortest paths is one of the central problems in the theory of
distributed computing. For the last few years, substantial progress has been
made on the approximate single source shortest paths problem, culminating in an
algorithm of Becker et al. [DISC'17] which deterministically computes
-approximate shortest paths in time, where
is the hop-diameter of the graph. Up to logarithmic factors, this time
complexity is optimal, matching the lower bound of Elkin [STOC'04].
The question of exact shortest paths however saw no algorithmic progress for
decades, until the recent breakthrough of Elkin [STOC'17], which established a
sublinear-time algorithm for exact single source shortest paths on undirected
graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for
exact all pairs shortest paths problem on directed graphs.
In this paper, we present a new single-source shortest path algorithm with
complexity . For polylogarithmic , this improves
on Elkin's bound and gets closer to the
lower bound of Elkin [STOC'04]. For larger values of
, we present an improved variant of our algorithm which achieves complexity
, and
thus compares favorably with Elkin's bound of in essentially the entire range of parameters. This
algorithm provides also a qualitative improvement, because it works for the
more challenging case of directed graphs (i.e., graphs where the two directions
of an edge can have different weights), constituting the first sublinear-time
algorithm for directed graphs. Our algorithm also extends to the case of exact
-source shortest paths...Comment: 26 page
Enumeration of idempotents in planar diagram monoids
We classify and enumerate the idempotents in several planar diagram monoids:
namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The
classification is in terms of certain vertex- and edge-coloured graphs
associated to Motzkin diagrams. The enumeration is necessarily algorithmic in
nature, and is based on parameters associated to cycle components of these
graphs. We compare our algorithms to existing algorithms for enumerating
idempotents in arbitrary (regular *-) semigroups, and give several tables of
calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24
pages, 6 figures, 8 tables, 5 algorithm
Spanning embeddings of arrangeable graphs with sublinear bandwidth
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen
343 (1), 175-205] gives minimum degree conditions for the containment of
spanning graphs H with small bandwidth and bounded maximum degree. We
generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n),
where n is the number of vertices of H.
Our result implies that sufficiently large n-vertex graphs G with minimum
degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as
subgraphs. Using techniques developed by Allen, Brightwell and Skokan
[Combinatorica, to appear] we can also apply our methods to show that almost
all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding
results for graphs embeddable on different orientable surfaces.Comment: 20 page
Geodesic Distance in Planar Graphs
We derive the exact generating function for planar maps (genus zero
fatgraphs) with vertices of arbitrary even valence and with two marked points
at a fixed geodesic distance. This is done in a purely combinatorial way based
on a bijection with decorated trees, leading to a recursion relation on the
geodesic distance. The latter is solved exactly in terms of discrete
soliton-like expressions, suggesting an underlying integrable structure. We
extract from this solution the fractal dimensions at the various
(multi)-critical points, as well as the precise scaling forms of the continuum
two-point functions and the probability distributions for the geodesic distance
in (multi)-critical random surfaces. The two-point functions are shown to obey
differential equations involving the residues of the KdV hierarchy.Comment: 38 pages, 8 figures, tex, harvmac, eps
Distributed coloring in sparse graphs with fewer colors
This paper is concerned with efficiently coloring sparse graphs in the
distributed setting with as few colors as possible. According to the celebrated
Four Color Theorem, planar graphs can be colored with at most 4 colors, and the
proof gives a (sequential) quadratic algorithm finding such a coloring. A
natural problem is to improve this complexity in the distributed setting. Using
the fact that planar graphs contain linearly many vertices of degree at most 6,
Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm
coloring -vertex planar graphs with 7 colors in rounds. Here, we
show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds.
Our algorithm indeed works more generally in the list-coloring setting and for
sparse graphs (for such graphs we improve by at least one the number of colors
resulting from an efficient algorithm of Barenboim and Elkin, at the expense of
a slightly worst complexity). Our bounds on the number of colors turn out to be
quite sharp in general. Among other results, we show that no distributed
algorithm can color every -vertex planar graph with 4 colors in
rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented
at PODC'18 (ACM Symposium on Principles of Distributed Computing
New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification
We discover new P-time computable six-vertex models on planar graphs beyond
Kasteleyn's algorithm for counting planar perfect matchings. We further prove
that there are no more: Together, they exhaust all P-time computable six-vertex
models on planar graphs, assuming #P is not P. This leads to the following
exact complexity classification: For every parameter setting in
for the six-vertex model, the partition function is either (1) computable in
P-time for every graph, or (2) #P-hard for general graphs but computable in
P-time for planar graphs, or (3) #P-hard even for planar graphs. The
classification has an explicit criterion. The new P-time cases in (2) provably
cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local
connection to #CSP, defined in terms of a "loop space".
This is the first substantive advance toward a planar Holant classification
with not necessarily symmetric constraints. We introduce M\"obius
transformation on as a powerful new tool in hardness proofs for
counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202
Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling
We present a bijection between some quadrangular dissections of an hexagon
and unrooted binary trees, with interesting consequences for enumeration, mesh
compression and graph sampling. Our bijection yields an efficient uniform
random sampler for 3-connected planar graphs, which turns out to be determinant
for the quadratic complexity of the current best known uniform random sampler
for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also
provides an encoding for the set of -edge 3-connected
planar graphs that matches the entropy bound
bits per edge (bpe). This solves a
theoretical problem recently raised in mesh compression, as these graphs
abstract the combinatorial part of meshes with spherical topology. We also
achieve the {optimal parametric rate} bpe
for graphs of with vertices and faces, matching in
particular the optimal rate for triangulations. Our encoding relies on a linear
time algorithm to compute an orientation associated to the minimal Schnyder
wood of a 3-connected planar map. This algorithm is of independent interest,
and it is for instance a key ingredient in a recent straight line drawing
algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing
2005]
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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