41,893 research outputs found
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
Alignment based Network Coding for Two-Unicast-Z Networks
In this paper, we study the wireline two-unicast-Z communication network over
directed acyclic graphs. The two-unicast-Z network is a two-unicast network
where the destination intending to decode the second message has apriori side
information of the first message. We make three contributions in this paper:
1. We describe a new linear network coding algorithm for two-unicast-Z
networks over directed acyclic graphs. Our approach includes the idea of
interference alignment as one of its key ingredients. For graphs of a bounded
degree, our algorithm has linear complexity in terms of the number of vertices,
and polynomial complexity in terms of the number of edges.
2. We prove that our algorithm achieves the rate-pair (1, 1) whenever it is
feasible in the network. Our proof serves as an alternative, albeit restricted
to two-unicast-Z networks over directed acyclic graphs, to an earlier result of
Wang et al. which studied necessary and sufficient conditions for feasibility
of the rate pair (1, 1) in two-unicast networks.
3. We provide a new proof of the classical max-flow min-cut theorem for
directed acyclic graphs.Comment: The paper is an extended version of our earlier paper at ITW 201
Directed Acyclic Graphs
This code is copyright (2015) by the University of Hertfordshire and is
made available to third parties for research or private study, criticism
or review, and for the purpose of reporting the state of the art, under the
normal fair use/fair dealing exceptions in Sections 29 and 30 of the
Copyright, Designs and Patents Act 1988. Use of the code under this
provision is limited to non-commercial use: please contact us if you
wish to arrange a licence covering commercial use of the code.This source code implements a unified framework for pre-processing Directed Acyclic Graphs (DAGs) to lookup reachability between two vertices as well as compute the least upper bound of two vertices in constant time. Our framework builds on the adaptive pre-processing algorithm for constant time reachability lookups and extends this to compute the least upper bound of a vertex-pair in constant time.
The theoretical details of this work can be found in the research paper which is available at http://uhra.herts.ac.uk/handle/2299/1215
Complete Graphical Characterization and Construction of Adjustment Sets in Markov Equivalence Classes of Ancestral Graphs
We present a graphical criterion for covariate adjustment that is sound and
complete for four different classes of causal graphical models: directed
acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially
directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our
criterion unifies covariate adjustment for a large set of graph classes.
Moreover, we define an explicit set that satisfies our criterion, if there is
any set that satisfies our criterion. We also give efficient algorithms for
constructing all sets that fulfill our criterion, implemented in the R package
dagitty. Finally, we discuss the relationship between our criterion and other
criteria for adjustment, and we provide new soundness and completeness proofs
for the adjustment criterion for DAGs.Comment: 58 pages, 12 figures, to appear in JML
Covering Pairs in Directed Acyclic Graphs
The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a
classical problem that provides a clear and simple mathematical formulation for
several applications in different areas and that has an efficient algorithmic
solution. In this paper, we study the computational complexity of two
constrained variants of Minimum Path Cover motivated by the recent introduction
of next-generation sequencing technologies in bioinformatics. The first problem
(MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum
cardinality set of paths "covering" all the vertices such that both vertices of
each pair belong to the same path. For this problem, we show that, while it is
NP-hard to compute if there exists a solution consisting of at most three
paths, it is possible to decide in polynomial time whether a solution
consisting of at most two paths exists. The second problem (MaxRPSP), given a
DAG and a set of pairs of vertices, asks for a path containing the maximum
number of the given pairs of vertices. We show its NP-hardness and also its
W[1]-hardness when parametrized by the number of covered pairs. On the positive
side, we give a fixed-parameter algorithm when the parameter is the maximum
overlapping degree, a natural parameter in the bioinformatics applications of
the problem
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
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