1,834 research outputs found
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
Prioritized Metric Structures and Embedding
Metric data structures (distance oracles, distance labeling schemes, routing
schemes) and low-distortion embeddings provide a powerful algorithmic
methodology, which has been successfully applied for approximation algorithms
\cite{llr}, online algorithms \cite{BBMN11}, distributed algorithms
\cite{KKMPT12} and for computing sparsifiers \cite{ST04}. However, this
methodology appears to have a limitation: the worst-case performance inherently
depends on the cardinality of the metric, and one could not specify in advance
which vertices/points should enjoy a better service (i.e., stretch/distortion,
label size/dimension) than that given by the worst-case guarantee.
In this paper we alleviate this limitation by devising a suit of {\em
prioritized} metric data structures and embeddings. We show that given a
priority ranking of the graph vertices (respectively,
metric points) one can devise a metric data structure (respectively, embedding)
in which the stretch (resp., distortion) incurred by any pair containing a
vertex will depend on the rank of the vertex. We also show that other
important parameters, such as the label size and (in some sense) the dimension,
may depend only on . In some of our metric data structures (resp.,
embeddings) we achieve both prioritized stretch (resp., distortion) and label
size (resp., dimension) {\em simultaneously}. The worst-case performance of our
metric data structures and embeddings is typically asymptotically no worse than
of their non-prioritized counterparts.Comment: To appear at STOC 201
Fast Shortest Path Distance Estimation in Large Networks
We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications.
In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks.
We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random.
Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.Yahoo! Research (internship
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford
A \emph{metric tree embedding} of expected \emph{stretch~}
maps a weighted -node graph to a weighted tree with such that, for all ,
and
. Such embeddings are highly useful for designing
fast approximation algorithms, as many hard problems are easy to solve on tree
instances. However, to date the best parallel -depth algorithm that achieves an asymptotically optimal expected stretch of
requires
work and a metric as input.
In this paper, we show how to achieve the same guarantees using
depth and
work, where and is an arbitrarily small constant.
Moreover, one may further reduce the work to at the expense of increasing the expected stretch to
.
Our main tool in deriving these parallel algorithms is an algebraic
characterization of a generalization of the classic Moore-Bellman-Ford
algorithm. We consider this framework, which subsumes a variety of previous
"Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss
it in depth. In our tree embedding algorithm, we leverage it for providing
efficient query access to an approximate metric that allows sampling the tree
using depth and work.
We illustrate the generality and versatility of our techniques by various
examples and a number of additional results
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
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