95 research outputs found
Speeding up shortest path algorithms
Given an arbitrary, non-negatively weighted, directed graph we
present an algorithm that computes all pairs shortest paths in time
, where is the number of
different edges contained in shortest paths and is a running
time of an algorithm to solve a single-source shortest path problem (SSSP).
This is a substantial improvement over a trivial times application of
that runs in . In our algorithm we use
as a black box and hence any improvement on results also in improvement
of our algorithm.
Furthermore, a combination of our method, Johnson's reweighting technique and
topological sorting results in an all-pairs
shortest path algorithm for arbitrarily-weighted directed acyclic graphs.
In addition, we also point out a connection between the complexity of a
certain sorting problem defined on shortest paths and SSSP.Comment: 10 page
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of
order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n))
(Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and
lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor,
1989). Our first result is an improvement of the upper-bound technique of
Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for
even s up to lower-order terms in the exponent. More importantly, we also
present a new technique for deriving upper bounds for lambda_s(n). With this
new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) +
O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new
upper bounds for general s; and (3) obtain improved upper bounds for the
generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and
Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) -
O(n), and therefore, the coefficient 2 is tight. We also present a simpler
version of the construction of Agarwal, Sharir, and Shor that achieves the
known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure
Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem
In this paper, we study linear programming based approaches to the maximum
matching problem in the semi-streaming model. The semi-streaming model has
gained attention as a model for processing massive graphs as the importance of
such graphs has increased. This is a model where edges are streamed-in in an
adversarial order and we are allowed a space proportional to the number of
vertices in a graph.
In recent years, there has been several new results in this semi-streaming
model. However broad techniques such as linear programming have not been
adapted to this model. We present several techniques to adapt and optimize
linear programming based approaches in the semi-streaming model with an
application to the maximum matching problem. As a consequence, we improve
(almost) all previous results on this problem, and also prove new results on
interesting variants
Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges
Computing \emph{all best swap edges} (ABSE) of a spanning tree of a given
-vertex and -edge undirected and weighted graph means to select, for
each edge of , a corresponding non-tree edge , in such a way that the
tree obtained by replacing with enjoys some optimality criterion (which
is naturally defined according to some objective function originally addressed
by ). Solving efficiently an ABSE problem is by now a classic algorithmic
issue, since it conveys a very successful way of coping with a (transient)
\emph{edge failure} in tree-based communication networks: just replace the
failing edge with its respective swap edge, so as that the connectivity is
promptly reestablished by minimizing the rerouting and set-up costs. In this
paper, we solve the ABSE problem for the case in which is a
\emph{single-source shortest-path tree} of , and our two selected swap
criteria aim to minimize either the \emph{maximum} or the \emph{average
stretch} in the swap tree of all the paths emanating from the source. Having
these criteria in mind, the obtained structures can then be reviewed as
\emph{edge-fault-tolerant single-source spanners}. For them, we propose two
efficient algorithms running in and time, respectively, and we show that the guaranteed (either
maximum or average, respectively) stretch factor is equal to 3, and this is
tight. Moreover, for the maximum stretch, we also propose an almost linear time algorithm computing a set of \emph{good} swap edges,
each of which will guarantee a relative approximation factor on the maximum
stretch of (tight) as opposed to that provided by the corresponding BSE.
Surprisingly, no previous results were known for these two very natural swap
problems.Comment: 15 pages, 4 figures, SIROCCO 201
Almost optimal exact distance oracles for planar graphs
We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q, and since the mid-1990s all results had polynomial time-space tradeoffs, e.g., Q = ~ Î(n/â S) or Q = ~Î(n5/2/S3/2).
In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n1+o(1) and almost optimal query time no(1). More precisely, we achieve the following space-time tradeoffs:
n1+o(1) space and log2+o(1) n query time,
n log2+o(1) n space and no(1) query time,
n4/3+o(1) space and log1+o(1) n query time.
We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures
Recommendations to encourage participation of individuals from diverse backgrounds in psychiatric genetic studies
We present innovative research practices in psychiatric genetic studies to ensure representation of individuals from diverse ancestry, sex assigned at birth, gender identity, age, body shape and size, and socioeconomic backgrounds. Due to histories of inappropriate and harmful practices against marginalized groups in both psychiatry and genetics, people of certain identities may be hesitant to participate in research studies. Yet their participation is essential to ensure diverse representation, as it is incorrect to assume that the same genetic and environmental factors influence the risk for various psychiatric disorders across all demographic groups. We present approaches developed as part of the Eating Disorders Genetics Initiative (EDGI), a study that required tailored approaches to recruit diverse populations across many countries. Considerations include research priorities and design, recruitment and study branding, transparency, and community investment and ownership. Ensuring representation in participants is costly and funders need to provide adequate support to achieve diversity in recruitment in prime awards, not just as supplemental afterthoughts. The need for diverse samples in genetic studies is critical to minimize the risk of perpetuating health disparities in psychiatry and other health research. Although the EDGI strategies were designed specifically to attract and enroll individuals with eating disorders, our approach is broadly applicable across psychiatry and other fields
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
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