New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

In this paper, we consider the question of computing sparse subgraphs for any input directed graph $G=(V,E)$ on $n$ vertices and $m$ edges, that preserves reachability and/or strong connectivity structures. We show $O(n+\min\{|{\cal P}|\sqrt{n},n\sqrt{|{\cal P}|}\})$ bound on a subgraph that is an $1$-fault-tolerant reachability preserver for a given vertex-pair set ${\cal P}\subseteq V\times V$, i.e., it preserves reachability between any pair of vertices in ${\cal P}$ under single edge (or vertex) failure. Our result is a significant improvement over the previous best $O(n |{\cal P}|)$ bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set ${\cal P}\subseteq V\times V$ of size $\Omega(n^\epsilon)$, for even any arbitrarily small $\epsilon$, requires at least $\Omega(n^{1+\epsilon/8})$ edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. We also present the first sub-quadratic bound of at most $\tilde{O}(k 2^k n^{2-1/k})$ size, for strong-connectivity preservers of directed graphs under $k$ failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general $k$. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95]

Rapid Recovery for Systems with Scarce Faults

Our goal is to achieve a high degree of fault tolerance through the control of a safety critical systems. This reduces to solving a game between a malicious environment that injects failures and a controller who tries to establish a correct behavior. We suggest a new control objective for such systems that offers a better balance between complexity and precision: we seek systems that are k-resilient. In order to be k-resilient, a system needs to be able to rapidly recover from a small number, up to k, of local faults infinitely many times, provided that blocks of up to k faults are separated by short recovery periods in which no fault occurs. k-resilience is a simple but powerful abstraction from the precise distribution of local faults, but much more refined than the traditional objective to maximize the number of local faults. We argue why we believe this to be the right level of abstraction for safety critical systems when local faults are few and far between. We show that the computational complexity of constructing optimal control with respect to resilience is low and demonstrate the feasibility through an implementation and experimental results.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Reachability Preservers: New Extremal Bounds and Approximation Algorithms

We abstract and study \emph{reachability preservers}, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph $G = (V, E)$ and a set of \emph{demand pairs} $P \subseteq V \times V$, a reachability preserver is a sparse subgraph $H$ that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an $n$-node graph and demand pairs of the form $P \subseteq S \times V$ for a small node subset $S$, there is always a reachability preserver on $O(n+\sqrt{n |P| |S|})$ edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which $O(n)$ size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the $O(n^{0.6+\varepsilon})$ of Chlamatac, Dinitz, Kortsarz, and Laekhanukit (SODA'17) to $O(n^{4/7+\varepsilon})$.Comment: SODA '1

\~Optimal Fault-Tolerant Reachability Labeling in Planar Graphs

We show how to assign labels of size $\tilde O(1)$ to the vertices of a directed planar graph $G$, such that from the labels of any three vertices $s,t,f$ we can deduce in $\tilde O(1)$ time whether $t$ is reachable from $s$ in the graph $G\setminus \{f\}$. Previously it was only known how to achieve $\tilde O(1)$ queries using a centralized $\tilde O(n)$ size oracle [SODA'21]

An Optimal Dual Fault Tolerant Reachability Oracle

Let G=(V,E) be an n-vertices m-edges directed graph. Let s inV be any designated source vertex. We address the problem of reporting the reachability information from s under two vertex failures. We show that it is possible to compute in polynomial time an O(n) size data structure that for any query vertex v, and any pair of failed vertices f_1, f_2, answers in O(1) time whether or not there exists a path from s to v in G{f_1,f_2}. For the simpler case of single vertex failure such a data structure can be obtained using the dominator-tree from the celebrated work of Lengauer and Tarjan [TOPLAS 1979, Vol. 1]. However, no efficient data structure was known in the past for handling more than one failures. We, in addition, also present a labeling scheme with O(log^3(n))-bit size labels such that for any f_1, f_2, v in Vit is possible to determine in poly-logarithmic time if v is reachable from s in G{f_1,f_2} using only the labels of f1, f_2 and v. Our data structure can also be seen as an efficient mechanism for verifying double-dominators. For any given x, y, v in V we can determine in O(1) time if the pair (x,y) is a double-dominator of v. Earlier the best known method for this problem was using dominator chain from which verification of double-dominators of only a single vertex was possible

An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model

In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G=(V,E) with n=|V| and m=|E|, and an integer value kgeq 1, there is an algorithm that computes in O(2^{k}n log^2 n) time for any set F of size at most k the strongly connected components of the graph GF. The running time of our algorithm is almost optimal since the time for outputting the SCCs of GF is at least Omega(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2^{k} n^2). Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest