Average Sensitivity of Graph Algorithms

In modern applications of graphs algorithms, where the graphs of interest are large and dynamic, it is unrealistic to assume that an input representation contains the full information of a graph being studied. Hence, it is desirable to use algorithms that, even when only a (large) subgraph is available, output solutions that are close to the solutions output when the whole graph is available. We formalize this idea by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover's distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity. After deriving basic properties of average sensitivity such as composition, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the minimum $s$-$t$ cut problem, and the maximum matching problem. In addition, we prove that the average sensitivity of our global minimum cut algorithm is almost optimal, by showing a nearly matching lower bound. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. One of the main ideas involved in designing our algorithms with low average sensitivity is the following fact; if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms and local computation algorithms (LCAs). Using this connection, we show that every LCA for 2-coloring has linear query complexity, thereby answering an open question.Comment: 39 pages, 1 figur

Engineering DFS-Based Graph Algorithms

Depth-first search (DFS) is the basis for many efficient graph algorithms. We introduce general techniques for the efficient implementation of DFS-based graph algorithms and exemplify them on three algorithms for computing strongly connected components. The techniques lead to speed-ups by a factor of two to three compared to the implementations provided by LEDA and BOOST. We have obtained similar speed-ups for biconnected components algorithms. We also compare the graph data types of LEDA and BOOST

Applications of graph algorithms in GIS

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Modular design of data-parallel graph algorithms

Amorphous Data Parallelism has proven to be a suitable vehicle for implementing concurrent graph algorithms effectively on multi-core architectures. In view of the growing complexity of graph algorithms for information analysis, there is a need to facilitate modular design techniques in the context of Amorphous Data Parallelism. In this paper, we investigate what it takes to formulate algorithms possessing Amorphous Data Parallelism in a modular fashion enabling a large degree of code re-use. Using the betweenness centrality algorithm, a widely popular algorithm in the analysis of social networks, we demonstrate that a single optimisation technique can suffice to enable a modular programming style without loosing the efficiency of a tailor-made monolithic implementation

Fast Dynamic Graph Algorithms for Parameterized Problems

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are $o(|G|)$, that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT). We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size $k$. These dynamic graphs achieve almost the best possible update time $O(\mathrm{poly}(k)\log n)$ and the query time $O(f(\mathrm{poly}(k),k))$, where $f(n,k)$ is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem. We also give a dynamic graph for Chromatic Number parameterized by the solution size of Cluster Vertex Deletion, and a dynamic graph for bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming the parameter is a constant, each dynamic graph can be updated in $O(\log n)$ time and can compute a solution in $O(1)$ time. These results are obtained by another approach.Comment: SWAT 2014 to appea

GraphTool : a tool for interactive design and manipulation of graphs and graph algorithms

GraphTool is an interactive tool for editing graphs and visualizing the execution and results of graph algorithms. It runs under both the SunView and X Windows environments and has a full window/mouse interface which is as similar as possible for the two windowing systems. In addition, there is a standalone program called the Wrapper which simulates the Graph-Tool interface without graphics for batch processing of graph algorithms. While the primary purpose of GraphTool is to provide a means for experimentally investigating the performance of graph algorithms, it has other useful features as well. It provides features for printing graphs in a visually appealing format, which makes it easier to prepare papers for publication. It also provides a facility for "animating" algorithms, which means that it can be used in computer assisted instruction (CAI) and for preparing video presentations of algorithms