32 research outputs found
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs
While algorithms for planar graphs have received a lot of attention, few
papers have focused on the additional power that one gets from assuming an
embedding of the graph is available. While in the classic sequential setting,
this assumption gives no additional power (as a planar graph can be embedded in
linear time), we show that this is far from being the case in other settings.
We assume that the embedding is straight-line, but our methods also generalize
to non-straight-line embeddings. Specifically, we focus on sublinear-time
computation and massively parallel computation (MPC).
Our main technical contribution is a sublinear-time algorithm for computing a
relaxed version of an -division. We then show how this can be used to
estimate Lipschitz additive graph parameters. This includes, for example, the
maximum matching, maximum independent set, or the minimum dominating set. We
also show how this can be used to solve some property testing problems with
respect to the vertex edit distance.
In the second part of our paper, we show an MPC algorithm that computes an
-division of the input graph. We show how this can be used to solve various
classical graph problems with space per machine of for
some , and while performing rounds. This includes for
example approximate shortest paths or the minimum spanning tree. Our results
also imply an improved MPC algorithm for Euclidean minimum spanning tree
Approximate Distance Sensitivity Oracles in Subquadratic Space
An -edge fault-tolerant distance sensitive oracle (-DSO) with stretch
is a data structure that preprocesses a given undirected,
unweighted graph with vertices and edges, and a positive integer
. When queried with a pair of vertices and a set of at most
edges, it returns a -approximation of the --distance in . We
study -DSOs that take subquadratic space. Thorup and Zwick [JACM 2015]
showed that this is only possible for . We present, for any
constant and , and any , an -DSO with stretch that takes
space and has an query time.
We also give an improved construction for graphs with diameter at most . For
any constant , we devise an -DSO with stretch that takes
space and has query time,
with a preprocessing time of . Chechik, Cohen, Fiat,
and Kaplan [SODA 2017] presented an -DSO with stretch and
preprocessing time , albeit with a super-quadratic
space requirement. We show how to reduce their preprocessing time to
.Comment: accepted at STOC 202
Undirected Connectivity of Sparse Yao Graphs
Given a finite set S of points in the plane and a real value d > 0, the
d-radius disk graph G^d contains all edges connecting pairs of points in S that
are within distance d of each other. For a given graph G with vertex set S, the
Yao subgraph Y_k[G] with integer parameter k > 0 contains, for each point p in
S, a shortest edge pq from G (if any) in each of the k sectors defined by k
equally-spaced rays with origin p. Motivated by communication issues in mobile
networks with directional antennas, we study the connectivity properties of
Y_k[G^d], for small values of k and d. In particular, we derive lower and upper
bounds on the minimum radius d that renders Y_k[G^d] connected, relative to the
unit radius assumed to render G^d connected. We show that d=sqrt(2) is
necessary and sufficient for the connectivity of Y_4[G^d]. We also show that,
for d =
2/sqrt(3), Y_3[G^d] is always connected. Finally, we show that Y_2[G^d] can be
disconnected, for any d >= 1.Comment: 7 pages, 11 figure
The Complexity of Geodesic Spanners
A geometric t-spanner for a set S of n point sites is an edge-weighted graph for which the (weighted) distance between any two sites p, q ∈ S is at most t times the original distance between p and q. We study geometric t-spanners for point sets in a constrained two-dimensional environment P. In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let S be a set of n point sites in a simple polygon P with m vertices. We present an algorithm to construct, for any constant ε > 0 and fixed integer k ≥ 1, a (2k + ε)-spanner with complexity O(mn1/k + n log2 n) in O(n log2 n + m log n + K) time, where K denotes the output complexity. When we consider sites in a polygonal domain P with holes, we can construct such a (2k + ε)-spanner of similar complexity in O(n2 log m + nm log m + K) time. Additionally, for any constant ε ∈ (0, 1) and integer constant t ≥ 2, we show a lower bound for the complexity of any (t − ε)-spanner of (Equation presented)
On the Stretch Factor of Polygonal Chains
Let be a polygonal chain. The stretch factor of
is the ratio between the total length of and the distance of its
endpoints, . For a parameter , we call a -chain if , for
every triple , . The stretch factor is a global
property: it measures how close is to a straight line, and it involves all
the vertices of ; being a -chain, on the other hand, is a
fingerprint-property: it only depends on subsets of vertices of the
chain.
We investigate how the -chain property influences the stretch factor in
the plane: (i) we show that for every , there is a noncrossing
-chain that has stretch factor , for
sufficiently large constant ; (ii) on the other hand, the
stretch factor of a -chain is , for every
constant , regardless of whether is crossing or noncrossing; and
(iii) we give a randomized algorithm that can determine, for a polygonal chain
in with vertices, the minimum for which is
a -chain in expected time and
space.Comment: 16 pages, 11 figure