3,016 research outputs found
An Efficient Representation for Filtrations of Simplicial Complexes
A filtration over a simplicial complex is an ordering of the simplices of
such that all prefixes in the ordering are subcomplexes of . Filtrations
are at the core of Persistent Homology, a major tool in Topological Data
Analysis. In order to represent the filtration of a simplicial complex, the
entire filtration can be appended to any data structure that explicitly stores
all the simplices of the complex such as the Hasse diagram or the recently
introduced Simplex Tree [Algorithmica '14]. However, with the popularity of
various computational methods that need to handle simplicial complexes, and
with the rapidly increasing size of the complexes, the task of finding a
compact data structure that can still support efficient queries is of great
interest.
In this paper, we propose a new data structure called the Critical Simplex
Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica
'17]. Our data structure allows one to store in a compact way the filtration of
a simplicial complex, and allows for the efficient implementation of a large
range of basic operations. Moreover, we prove that our data structure is
essentially optimal with respect to the requisite storage space. Finally, we
show that the CSD representation admits fast construction algorithms for Flag
complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set
An independent dominating set D of a graph G = (V,E) is a subset of vertices
such that every vertex in V \ D has at least one neighbor in D and D is an
independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum
independent dominating set in a graph is an NP-hard problem. Whereas it is hard
to cope with this problem using parameterized and approximation algorithms,
there is a simple exact O(1.4423^n)-time algorithm solving the problem by
enumerating all maximal independent sets. In this paper we improve the latter
result, providing the first non trivial algorithm computing a minimum
independent dominating set of a graph in time O(1.3569^n). Furthermore, we give
a lower bound of \Omega(1.3247^n) on the worst-case running time of this
algorithm, showing that the running time analysis is almost tight.Comment: Full version. A preliminary version appeared in the proceedings of WG
200
Self-stabilizing algorithms for Connected Vertex Cover and Clique decomposition problems
In many wireless networks, there is no fixed physical backbone nor
centralized network management. The nodes of such a network have to
self-organize in order to maintain a virtual backbone used to route messages.
Moreover, any node of the network can be a priori at the origin of a malicious
attack. Thus, in one hand the backbone must be fault-tolerant and in other hand
it can be useful to monitor all network communications to identify an attack as
soon as possible. We are interested in the minimum \emph{Connected Vertex
Cover} problem, a generalization of the classical minimum Vertex Cover problem,
which allows to obtain a connected backbone. Recently, Delbot et
al.~\cite{DelbotLP13} proposed a new centralized algorithm with a constant
approximation ratio of for this problem. In this paper, we propose a
distributed and self-stabilizing version of their algorithm with the same
approximation guarantee. To the best knowledge of the authors, it is the first
distributed and fault-tolerant algorithm for this problem. The approach
followed to solve the considered problem is based on the construction of a
connected minimal clique partition. Therefore, we also design the first
distributed self-stabilizing algorithm for this problem, which is of
independent interest
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