The Random Adversary: A Lower-Bound Technique for Randomized Parallel Algorithms

We prove that the number of Hamilton cycles in the random graph $G(n,p)$ is $n!p^n(1+o(1))^n$ asymptotically almost surely (a.a.s.), provided that $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree $2$ creates $(\frac{\ln n}{e})^n(1+o(1))^n$ Hamilton cycles a.a.s.

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the G(n, p) random graph model, with number of nodes n and p = c ln n/n^δ (for any constant 0 0), that finds a Hamiltonian cycle with high probability in O(n^δ) rounds. Our algorithm works in the (synchronous) CONGEST model (i.e., only O(log n)-sized messages are communicated per edge per round) and its computational cost per node is sublinear (in n) per round and is fully-distributed (each node uses only o(n) memory and all nodes' computations are essentially balanced). Our algorithm improves over the previous best known result in terms of both the running time as well as the edge sparsity of the graphs where it can succeed; in particular, the denser the random graph, the smaller is the running time.

ABSTRACTWe discuss the existence of Hamilton cycles in the random graph where there are restrictions caused by (i) coloring sequences, (ii) a subset of vertices must occur in a specific order, and (iii) there is a bound on the number of inversions in the associated permutation.

We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$ -cycles.

Let H be an edge-colored hypergraph. We say that H contains a rainbow copy of a hypergraph S if it contains an isomorphic copy of S with all edges of distinct colors. We consider the following setting. A randomly edge-colored random hypergraph \(H \sim \mathcal{H}_{c}^{k}(n,p)\) is obtained by adding each k-subset of [n] with probability p, and assigning it a color from [c] uniformly, independently at random. As a first result we show that a typical \(H \sim \mathcal{H}_{c}^{2}(n,p)\) (that is, a random edge-colored graph) contains a rainbow Hamilton cycle, provided that \(c = (1 + o(1))n\) and \(p = \frac{\log n+\log \log n+\omega (1)} {n}\). This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh. Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding “nicely edge-colored” structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical \(H \sim \mathcal{H}_{c}^{k}(n,p)\) contains a rainbow copy of a hypergraph S, provided that \(c = (1 + o(1))\vert E(S)\vert\) and p is (up to a multiplicative constant) a threshold function for the property of containment of a copy of S. In the second application we show that a typical \(G \sim \mathcal{H}_{c}^{2}(n,p)\) contains \((1 - o(1))np/2\) edge-disjoint Hamilton cycles, each of which is rainbow, provided that c = ω(n) and \(p =\omega (\log n/n)\).

On Matchings and Hamiltonian Cycles in Random Graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős‐Rényi random graph Gn,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs.In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014

We consider the problem of diffusion-based load balancing on a distributed network with n processors. If the load is arbitrarily divisible, then the convergence is fairly well captured in terms of the second largest eigenvalue of the diffusion matrix. As for many applications load can not be arbitrarily divided, we consider a model where load consists of indivisible, unit-size tokens. Quantifying by how much this integrality assumption worsens the efficiency of load balancing algorithms is a natural question which has been posed by many authors [9, 15, 16, 6, 19, 17]. In this paper we show essentially that discrete load balancing is almost as easy as continuous load balancing. More precisely, we present a fully distributed, randomized algorithm for discrete load balancing that balances the load up to an additive constant error on any graph in time O(log (Kn)/(1-λ2)), where K is the initial imbalance and λ2 is the second largest eigenvalue of the diffusion matrix. This improves and tightens a result of Elsasser, Monien, Schamberger (2006) who proved a runtime bound of O((log (K) + (log n)2) / (1-λ2)). We also develop a load balancing algorithm based on routing that achieves a runtime of O(D ⋅ log n), where D is the diameter of the graph.

This paper addresses the problem of static load balancing in heterogeneous distributed computing systems taking into account both memory and communication capacity constraints. It first models the load balancing problem as an optimization problem. It then presents a modified genetic algorithm, called Adaptive Genetic Algorithm (AGA), to solve the problem. The performance of the proposed algorithm is evaluated by simulation studies on randomly generated instances and the results are compared with that obtained by applying both the Genetic Algorithm (GA) and the Simulated Annealing (SA). Also, the qualities of the results are compared with the optimal solutions that obtained by applying the Brach-and-Bound (BB) algorithm.

With the development of stream processing technology, achieving load balance of resource accessing becomes one of key problems. However, the existing technologies are difficult to balance the system and maintain the data integrity requirements when large data stream arrive. In this paper, combined with the dynamic load balancing algorithm, we propose a load balancing algorithm of stream data based on correlation analysis (SDCA-LBA). The algorithm analyses correlation of stream data through the window feature statistics. On the basis of ensuring the load balancing of the stream data system, we consider the data correlation. The experiment results show that this method can effectively solve the problem of stream data load balancing, maintain the data integrity, and improve the system data processing capabilities.

Packet switching remains one of the bottlenecks in building fast Internet routers. Load balancing and switch scheduling are two important algorithms in the effort to maximize the throughput and minimize the latency of these packet switches. A load balancing algorithm regulates the traffic to conform to the service rates while a switch scheduling algorithm allocates the service rates adaptive to the arrival patterns. Many existing load balancing and switch scheduling algorithms are very similar. We show that load balancing and switch scheduling systems are dual systems based on the linear queue dynamics approximation. This allows us to cast a load balancing problem as a scheduling problem, and vice versa. We further show an example of designing a new algorithm for load balancing using an existing scheduling algorithm based on the duality. The duality perspective also allows us to solve unknown problems. We find the entropy rate of the randomized bandwidth allocation system with linear queue dynamics based on the knowledge of the entropy rate of the randomized load balancing system. For the general case, we find both an upper and a lower bound on the entropy rate. The joint use of dual load balancing and switch scheduling algorithms leads to performance gains as we show using mean field analysis.

QoS-priority based load balancing algorithm for LTE systems with mixed users

Cloud-based railway interlocking system is a novel solution for interlocking systems with the development of cloud computing. The load balancing problem caused by imbalanced resources in computing platforms is an essential problem in realizing cloud-based railway interlocking system. Conventional load balancing algorithms focus on makespan, throughput and costs. However, a load balancing algorithm for a cloud-based railway interlocking system should focus on optimizing resource utilization and execution time. To this end, we propose a hybrid meta-heurisite algorithm called Improved Genetic Annealing Algorithm (IGAA). The proposed algorithm can guarantee global optimization ability by using the Metropolis criterion and can converge rapidly due to the parallel searching characteristic of the Genetic Algorithm. Finally, the feasibility and effectiveness of the proposed algorithm are verified by experiments.

This paper studies the load balancing problem for distributed computer systems. A load balancing algorithm is proposed to solve the problem. The algorithm is modeled with Petri nets and using Lyapunov methods a formal proof of stability and convergence is presented. The formulation is shown to be simple and straightforward.

Promoting cooperation in selfish computational grids