11 research outputs found
Efficient Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs
Graphs are an important tool to model data in different domains, including
social networks, bioinformatics and the world wide web. Most of the networks
formed in these domains are directed graphs, where all the edges have a
direction and they are not symmetric. Betweenness centrality is an important
index widely used to analyze networks. In this paper, first given a directed
network and a vertex , we propose a new exact algorithm to
compute betweenness score of . Our algorithm pre-computes a set
, which is used to prune a huge amount of computations that do
not contribute in the betweenness score of . Time complexity of our exact
algorithm depends on and it is respectively
and
for unweighted graphs and weighted graphs with positive weights.
is bounded from above by and in most cases, it
is a small constant. Then, for the cases where is large, we
present a simple randomized algorithm that samples from and
performs computations for only the sampled elements. We show that this
algorithm provides an -approximation of the betweenness
score of . Finally, we perform extensive experiments over several real-world
datasets from different domains for several randomly chosen vertices as well as
for the vertices with the highest betweenness scores. Our experiments reveal
that in most cases, our algorithm significantly outperforms the most efficient
existing randomized algorithms, in terms of both running time and accuracy. Our
experiments also show that our proposed algorithm computes betweenness scores
of all vertices in the sets of sizes 5, 10 and 15, much faster and more
accurate than the most efficient existing algorithms.Comment: arXiv admin note: text overlap with arXiv:1704.0735
Deterministic Subgraph Detection in Broadcast CONGEST
We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation:
- For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds.
- For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n)
rounds.
- On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d + log n) rounds, and
5-cycles in O(d2 + log n) rounds.
In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/logn) and O(d2/logn), respect- ively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique
Deterministic subgraph detection in broadcast CONGEST
We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d+log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/ log n) and O(d2/log n), respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique. © 2017 Janne H. Korhonen and Joel Rybicki.Peer reviewe
Algorithms and Hardness for Multidimensional Range Updates and Queries
Traditional orthogonal range problems allow queries over a static set of points, each with some value. Dynamic variants allow points to be added or removed, one at a time. To support more powerful updates, we introduce the Grid Range class of data structure problems over arbitrarily large integer arrays in one or more dimensions. These problems allow range updates (such as filling all points in a range with a constant) and queries (such as finding the sum or maximum of values in a range). In this work, we consider these operations along with updates that replace each point in a range with the minimum, maximum, or sum of its existing value, and a constant. In one dimension, it is known that segment trees can be leveraged to facilitate any n of these operations in O?(n) time overall. Other than a few specific cases, until now, higher dimensional variants have been largely unexplored.
Despite their tightly-knit complexity in one dimension, we show that variants induced by subsets of these operations exhibit polynomial separation in two dimensions. In particular, no truly subquadratic time algorithm can support certain pairs of these updates simultaneously without falsifying several popular conjectures. On the positive side, we show that truly subquadratic algorithms can be obtained for variants induced by other subsets.
We provide two general approaches to designing such algorithms that can be generalised to online and higher dimensional settings. First, we give almost-tight O?(n^{3/2}) time algorithms for single-update variants where the update and query operations meet a set of natural conditions. Second, for other variants, we provide a general framework for reducing to instances with a special geometry. Using this, we show that O(m^{3/2-?}) time algorithms for counting paths and walks of length 2 and 3 between vertex pairs in sparse graphs imply truly subquadratic data structures for certain variants; to this end, we give an O?(m^{(4?-1)/(2?+1)}) = O(m^1.478) time algorithm for counting simple 3-paths between vertex pairs
Boosting Local Search for the Maximum Independent Set Problem
An independent set of a graph G = (V, E) with vertices V and edges E is a subset S ⊆ V, such that the subgraph induced by S does not contain any edges. The goal of the maximum independent set problem (MIS problem) is to find an independent set of maximum size. It is equivalent to the well-known vertex cover problem (VC problem) and maximum clique problem. This thesis consists of two main parts. In the first one we compare the currently best algorithms for finding near-optimal independent sets and vertex covers in large, sparse graphs. They are Iterated Local Search (ILS) by Andrade et al. [2], a heuristic that uses local search for the MIS problem and NuMVC by Cai et al. [6], a local search algorithm for the VC problem. As of now, there are no methods to solve these large instances exactly in any reasonable time. Therefore these heuristic algorithms are the best option. In the second part we analyze a series of techniques, some of which lead to a significant speed up of the ILS algorithm. This is done by removing specific ver
Recent Advances in Fully Dynamic Graph Algorithms
In recent years, significant advances have been made in the design and
analysis of fully dynamic algorithms. However, these theoretical results have
received very little attention from the practical perspective. Few of the
algorithms are implemented and tested on real datasets, and their practical
potential is far from understood. Here, we present a quick reference guide to
recent engineering and theory results in the area of fully dynamic graph
algorithms
Algorithms and Software for the Analysis of Large Complex Networks
The work presented intersects three main areas, namely graph algorithmics, network science and applied software engineering. Each computational method discussed relates to one of the main tasks of data analysis: to extract structural features from network data, such as methods for community detection; or to transform network data, such as methods to sparsify a network and reduce its size while keeping essential properties; or to realistically model networks through generative models