66,427 research outputs found
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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Analysis and design of algorithms : double hashing and parallel graph searching
The following is in two parts, corresponding to the two separate topics in the dissertation.Probabilistic Analysis of Double HashingIn [GS78], a deep and elegant analysis shows that double hashing is asymptotically equivalent to the ideal uniform hashing up to a load factor of about 0.319. In this paper we show how a resampling technique can be used to develop a surprisingly simple proof of the result that this equivalence holds for load factors arbitrarily close to 1.Parallel Depth First Search of Planar Directed Acyclic GraphsIn 1988, Kao [Kao88] presented the first NC algorithm for the depth first search of a directed planar graph. Recently, Kao and Klein [KK90] reduced the number of processors required from O(n^4) to linear, but the time bound is O(log^8 n).We present an algorithm for the depth first search of a planar directed acyclic graph with k sources using O(n) processors and O(log k log n) time on a CRCW PRAM model. For planar dags with a single source and a single sink, we present a simple optimal algorithm which gives the depth first search in O(log n) time with O(n/log n) processors on an EREW PRAM. For a single-source multiple-sink planar dag, we have an O(log n) time O(n) processor EREW algorithm. The EREW algorithms assume that the embedding is given. A simplified variant of the depth first search of a multisource planar dag can be used to solve the single source reachability problem for a planar directed acyclic graph in O(log^2 n) time and O(n) processors on an CRCW PRAM. Since an O(log^4 n) algorithm for this problem is used as a subroutine by Kao and Klein in their depth first search for the general planar directed graph, this will lower their time bound by a factor of log^2 n. Our work uses the concept of a planar Euler tour depth first search, a depth first search in which the Euler tour around the tree is planar and crosses no tree edge. This concept may prove to be of use in other parallel algorithms for planar graphs
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
Optimal randomized incremental construction for guaranteed logarithmic planar point location
Given a planar map of segments in which we wish to efficiently locate
points, we present the first randomized incremental construction of the
well-known trapezoidal-map search-structure that only requires expected preprocessing time while deterministically guaranteeing worst-case
linear storage space and worst-case logarithmic query time. This settles a long
standing open problem; the best previously known construction time of such a
structure, which is based on a directed acyclic graph, so-called the history
DAG, and with the above worst-case space and query-time guarantees, was
expected . The result is based on a deeper understanding of the
structure of the history DAG, its depth in relation to the length of its
longest search path, as well as its correspondence to the trapezoidal search
tree. Our results immediately extend to planar maps induced by finite
collections of pairwise interior disjoint well-behaved curves.Comment: The article significantly extends the theoretical aspects of the work
presented in http://arxiv.org/abs/1205.543
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