Undirected Graph Connectivity Research Articles

Graph Connectivity in Log Steps Using Label Propagation

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

Strong subgraph k‐connectivity

AbstractGeneralized connectivity introduced by Hager [J. Combin. Theory Ser. B 38 (1985), pp. 179–189] has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized ‐connectivity of undirected graphs to directed graphs (we call it strong subgraph ‐connectivity) by replacing connectivity with strong connectivity. We prove NP‐completeness results and the existence of polynomial algorithms. We show that strong subgraph ‐connectivity is, in a sense, harder to compute than generalized ‐connectivity. However, strong subgraph ‐connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph ‐connectivity and pose some open questions.

Progress in Computational Complexity Theory

We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to average-case reductions; pseudorandomness and extractor constructions; and Valiant's new theory of holographic algorithms and reductions.

On Rayleigh–Ritz ratios of a generalized Laplacian matrix of directed graphs

In [C.W. Wu, Algebraic connectivity of directed graphs, Linear and Multilinear Algebra, in press] a generalization of Fiedler’s notion of algebraic connectivity to directed graphs was presented that inherits many properties of the algebraic connectivity of undirected graphs and has applications to the synchronization of coupled dynamical systems with both constant and time-varying coupling. However, it did not inherit nonnegativity and some of its bounds for combinatorial properties such as maximum directed cut and isoperimetric number are less strict that their undirected counterparts. In particular, these bounds do not have the corresponding bounds for undirected graphs as limiting cases. The purpose of this paper is to present a refinement of this algebraic connectivity which preserve nonnegativity for strongly connected graphs and have bounds which contain the undirected graphs as special cases. In particular, we study quantities related to a generalized Laplacian matrix of directed graphs and obtain bounds on combinatorial properties such as diameter, bandwidth, and bisection width for general directed graphs. Finally, we give an application to the synchronization of coupled dynamical systems with constant coupling. In particular, we show that strong enough cooperative coupling will synchronize a network of coupled systems if the underlying directed graph is strongly connected.

Consensus Problems in Networks of Agents With Switching Topology and Time-Delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

The “log rank” conjecture for modular communication complexity

The “log rank” conjecture involves the question of how precisely the deterministic communication complexity of a problem can be described in terms of algebraic invariants of the communication matrix of this problem. We answer this question in the context of modular communication complexity. We show that the modular communication complexity can be exactly characterised in terms of the logarithm of a certain rigidity function of the communication matrix. Thus, we are able to exactly determine the modular communication complexity of several problems, such as, e.g., set disjointness, comparability, and undirected graph connectivity. From the bounds obtained for the modular communication complexity we deduce exponential lower bounds on the size of depth-two circuits having arbitrary symmetric gates at the bottom level and a MOD m -gate at the top.

Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity

While it is straightforward to simulate a very general class of random processes space-efficiently by non-unitary quantum computations (e.g., quantum computations that allow intermediate measurements to occur), it is not currently known to what extent restricting quantum computations to be unitary affects the space required for such simulations. This paper presents a method by which a limited class of random processes—random walks on undirected graphs—can be simulated by unitary quantum computations in a space-efficient (and time-efficient) manner. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in a quantum analogue of randomized logspace that disallows intermediate measurements.

Bounds on the subdominant eigenvalue involving group inverse with applications to graphs

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ1〉λ2≥ ... ≥λn. In this paper we derive several lower and upper bounds, in particular on λ2 and λn, but also, indirectly, on \(\mu = \mathop {\max }\limits_{2 \leqslant i \leqslant n} |\lambda _i |\). The bounds are in terms of the diagonal entries of the group generalized inverse, Q#, of the singular and irreducible M-matrix Q = λ1I − A. Our starting point is a spectral resolution for Q#. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now Q becomes L, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of L# and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of L, by exploiting the diagonal entries of L#.

An Asynchronous Parallel Algorithm for Undirected Graph Connectivity

An algorithm for computing the components of an undirected graph in the (asynchronous) APRAM model is given; the algorithm uses O( n + e) processes and O(log n) rounds.

Problems complete for deterministic logarithmic space

We exhibit several problems complete for deterministic logarithmic space under NC 1 (i.e., log depth) reducibility. The list includes breadth-first search and depth-first search of an undirected tree, connectivity of undirected graphs known to be made up of one or more disjoint cycles, undirected graph acyclicity, and several problems related to representing and to operating with permutations of a finite set.

Partition semantics for relations

We use set-theoretic partitions to assign semantics to relation schemes, relations, and dependencies. This approach leads to a natural extension of functional dependencies, the most common database constraints, which is based on the duality between product and sum of partitions. These more general constraints, which we call partition dependencies (PDs), have the power to express both functional determination and connectivity in undirected graphs. We show that the implication problem for PDs is the uniform word problem for lattices, and we give a polynomial time algorithm for this implication problem. Our decision procedure for the uniform word problem for lattices improves upon previous exponential decision procedures. The algebraic techniques we use are new to database theory. We investigate the expressive power of PDs, and show that partition semantics justify a number of variants of an assumption commonly used in database theory, namely the weak instance assumption. Finally, we provide a polynomial time test for consistency of a set of relations with a set of PDs.