Fault Tolerant Reachability for Directed Graphs

2021

In this paper, we consider reachability oracles and reachability preservers for directed graphs/networks prone to edge/node failures. Let G = (V,E) be a directed graph on n-nodes, and P ⊆ V × V be a set of vertex pairs in G. We present the first non-trivial constructions of single and dual fault-tolerant pairwise reachability oracle with constant query time. Furthermore, we provide extremal bounds for sparse faulttolerant reachability preservers, resilient to two or more failures. Prior to this work, such oracles and reachability preservers were widely studied for the special scenario of single-source and all-pairs settings. However, for the scenario of arbitrary pairs, no prior (non-trivial) results were known for dual (or more) failures, except those implied from the single-source setting. One of the main questions is whether it is possible to beat the O(n|P|) size bound (derived from the single-source setting) for reachability oracle and preserver for dual failures (or O(2n|P|) b...

Theory of Computing Systems, 2013

For a given collection G of directed graphs we define the joinreachability graph of G, denoted by J (G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J (G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest joinreachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.

Mathematics and Statistics, 2020

The problem of reachability on graphs with restriction is studied. Such restrictions mean that only those paths that satisfy certain conditions are valid paths on the graph. Because of this, for classical optimization problems one has to consider only a subset of feasible paths on the graph, which significantly complicates their solution. Reachability constraints arise naturally in various applied problems, for example, in the problem of navigation in telecommunication networks with areas of strong signal attenuation or when modeling technological processes in which there is a condition for the order of actions or the compatibility of operations. General concepts of a graph with non-standard reachability and a valid path on it are introduced. It is shown that the classical graphs, as well as graphs with restrictions on passing through the selected arcs subsets are special cases of graphs with non-standard reachability. General approach to solving the shortest path problem on a graph with non-standard achiev-ability is developed. This approach consists in constructing an auxiliary graph and reducing the shortest path problem on a graph with non-standard reachability to a similar problem on an auxiliary graph. The theorem on the correspondence of the paths of the original and auxiliary graphs is proved.

2008 IEEE 24th International Conference on Data Engineering, 2008

⎯ Given a directed graph G, to check whether a node v is reachable from another node u through a path is often required. In a database system, such an operation is called a recursion computation or reachability checking and not efficiently supported. The reason for this is that the space to store the whole transitive closure of G is prohibitively high. In this paper, we address this issue and propose an O(n 2 + bn) time algorithm to decompose a directed acyclic graph (DAG) into a minimized set of disjoint chains to facilitate reachability checking, where n is the number of the nodes and b is the DAG's width, defined to be the size of a largest node subset U of the DAG such that for every pair of nodes u, v ∈ U, there does not exist a path from u to v or from v to u. Using this algorithm, we are able to label a graph in O(be) time and store all the labels in O(bn) space with O(logb) reachability checking time, where e is the number of the edges of the DAG. The method can also be extended to handle cyclic directed graphs. Experiments have been performed, showing that our method is promising.

We present an algorithm for a fault tolerant Depth First Search (DFS) Tree in an undirected graph. This algorithm is drastically simpler than the current state-of-the-art algorithms for this problem, uses optimal space and optimal preprocessing time, and still achieves better time complexity. This algorithm also leads to a better time complexity for maintaining a DFS tree in a fully dynamic environment.

Scientia Iranica, 2018

Considering G as a weighted digraph, and s and t as two vertices of G, the Reachability Assurance (RA) problem is how to label the edges of G such that every path starting at s nally reaches t and the sum of the weights of the labeled edges, called the RA cost, is minimal. The common approach to the RA problem is path nding, in which a path is sought from s to t and, then, the edges of the path are labeled. This paper introduces a new approach, the Marking Problem (MP), to the RA problem. Compared to the common path nding approach, the proposed MP approach has a lower RA cost. It is shown that the MP is NP-complete, even when the underlying digraph is an unweighted Directed Acyclic Graph (DAG) or a weighted DAG with an out-degree of two. An appropriate heuristic algorithm to solve the MP in polynomial time is provided. To mitigate the RA problem as a serious challenge in this area, application of the MP in software testing is also presented. By evaluating the datasets from various program ow graphs, it is shown that the MP is superior to the path nding in the context of test case generation.

2020

In the past decade, the design of fault tolerant data structures for networks has become a central topic of research. Particular attention has been given to the construction of a subgraph H of a given digraph D with as fewest arcs/vertices as possible such that, after the failure of any set F of at most k ≥ 1 arcs, testing whether D − F has a certain property P is equivalent to testing whether H − F has that property. Here, reachability (or, more generally, distance preservation) is the most basic requirement to maintain to ensure that the network functions properly. Given a vertex s ∈ V (D), Baswana et al. [STOC’16] presented a construction of H with O(2n) arcs in time O(2nm) where n = |V (D)| and m = |E(D)| such that for any vertex v ∈ V (D): if there exists a path from s to v in D − F , then there also exists a path from s to v in H − F . Additionally, they gave a tight matching lower bound. While the question of the improvement of the dependency on k arises for special classes o...

Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes O(m + n) time to build a DFS tree for a given undirected graph G = (V, E) on n vertices and m edges. We address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem. 1. Fault tolerant DFS tree: There exists a data structure of sizeÕ(m) 1 such that given any set F of failed vertices or edges, a DFS tree of the graph G \ F can be reported inÕ(n|F|) time. 2. Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst caseÕ(√ mn) time per update for any arbitrary online sequence of updates. 3. Incremental DFS tree: There exists an incremental algorithm for maintaining a DFS tree that takes worst caseÕ(n) time per update for any arbitrary online sequence of edge insertion. These are the first o(m) worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with O(1) query time and o(m) worst case update time for connectivity, biconnectivity, and 2-edge connectivity in the dynamic subgraph model.

Mathematics

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

Proceedings of the 2009 C3S2E conference on - C3S2E '09, 2009

Graph reachability is fundamental to a wide range of applications, including CAD/CAM, CASE, office systems, software management, as well as geographical navigation and internet routing. Many applications involve huge graphs and requires fast answering of reachability queries. Several reachability labeling methods have been proposed for this purpose. They assign labels to the nodes, such that the reachability between any two nodes can be determined using their labels only. In this paper, we propose a new data structure, called a general spanning tree of a directed acyclic graph (DAG) to minimize label space. Different from a traditional spanning tree, an edge in a general spanning tree T of a DAG G may corresponds to a path in G. That is, for each edge u → v in T, we have a path from u to v in G. An algorithm is discussed to find such a tree with the least number of leaf nodes in O(bn) time, where n is the number of the nodes of G, and b is the number of the leaf nodes of T. It can be proven that b equals G's width, defined to be the size of a largest node subset U of G such that for every pair of nodes u, v ∈ U, there does not exist a path from u to v or from v to u. Based on T, we are able to reduce the label space to O(bn) with O(logb) reachability query time. Our method can also be extended for graphs containing cycles.