Parallel cleaning of a network with brushes

Theoretical Computer Science, 2008

Following the decontamination metaphor for searching a graph, we introduce a cleaning process, which is related to both the chip-firing game and edge searching. Brushes (instead of chips) are placed on some vertices and, initially, all the edges are dirty. When a vertex is 'fired', each dirty incident edge is traversed by only one brush, cleaning it, but a brush is not allowed to traverse an already cleaned edge; consequently, a vertex may not need degree-many brushes to fire. The model presented is one where the edges are continually recontaminated, say by algae, so that cleaning is regarded as an on-going process. Ideally, the final configuration of the brushes, after all the edges have been cleaned, should be a viable starting configuration to clean the graph again. We show that this is possible with the least number of brushes if the vertices are fired sequentially but not if fired in parallel. We also present bounds for the least number of brushes required to clean graphs in general and some specific families of graphs.

2014

Graphs and Combinatorics, 2011

A model for cleaning a graph with brushes was recently introduced. Most of the existing papers consider the minimum number of brushes needed to clean a given graph G in this model, the so-called brush number b(G). In this paper, we focus on the broom number, B(G), that is, the maximum number of brushes that can be used to clean a graph G in this model.

In this paper, we consider a contaminated network with an intruder. The task for the mobile agents is to decontaminate all hosts while preventing a recontamination and to do so as efficiently as possible.

Lecture Notes in Computer Science, 2007

In the recently introduced model for cleaning a graph with brushes, we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even). We then use a differential equations method to find the (asymptotic) number of brushes needed to clean a random d-regular graph using this algorithm. As well as the case for general d, interesting results for specific values of d are examined. We also state various open problems.

Networks, 2013

This paper considers the problem of using synchronous mobile agents to decontaminate the nodes of a graph given a spreading contamination. We begin by considering the problem of minimizing cleaning time, given initial agent and contamination locations. Then, we take as input a set of all possible locations in which a contamination can start, and examine problems in which we strategically pre-position agents. In one problem, we minimize the number of agents, and prescribe their initial locations, so that the graph can be cleaned within a time limit for any potential initial contamination. We also determine the best initial locations for some pre-determined number of agents to minimize expected cleaning time, given probability estimates of potential initial contamination locations. We analyze the complexity of each variant, and formulate the problems as mixed-integer programs. As an alternative method, we also provide a construction heuristic for the cleaning problem, and cutting-plane algorithms for the agent location problems. Computational results using these approaches demonstrate the efficacy of our procedures.

2013

Faults and viruses often spread in networked environments by propa-10 gating from site to neighboring site. We model this process of network contamina-11 tion by graphs. Consider a graph G = (V, E), whose vertex set is contaminated 12 and our goal is to decontaminate the set V (G) using mobile decontamination 13 agents that traverse along the edge set of G. Temporal immunity τ (G) ≥ 0 is 14 defined as the time that a decontaminated vertex of G can remain continuously 15 exposed to some contaminated neighbor without getting infected itself. The im-16 munity number of G, ι k (G), is the least τ that is required to decontaminate G 17 using k agents. We study immunity number for some classes of graphs corre-18 sponding to network topologies and present upper bounds on ι1(G), in some 19 cases with matching lower bounds. Variations of this problem have been exten-20 sively studied in literature, but proposed algorithms have been restricted to mono-21 tone strategies, where a vertex, once decontaminated, may not be recontaminated. 22 We exploit nonmonotonicity to give bounds which are strictly better than those 23 derived using monotone strategies. 24 arXiv:1307.7307v1 [math.CO] 27 Jul 2013 amount of time after which it becomes contaminated. Actual decontamination is per-41 formed by mobile cleaning agents which which move from host to host over network 42 connections. 43 1.1 Previous Work 44 Graph Search. The decontamination problem considered in this paper is a variation of 45 a problem extensively studied in the literature known as graph search. The graph search 46 problem was first introduced by Breish in [5], where an approach for the problem of 47 finding an explorer that is lost in a complicated system of dark caves is given. Parsons 48 ([20][21]) proposed and studied the pursuit-evasion problem on graphs. Members of 49 a team of searchers traverse the edges of a graph in pursuit of a fugitive, who moves 50 along the edges of the graph with complete knowledge of the locations of the pursuers. 51 The efficiency of a graph search solution is based on the size of the search team. Size of 52 smallest search team that can clear a graph G is called search number, and is denoted in 53 literature by s(G). In [19], Megiddo et al. approached the algorithmic question: Given 54 an arbitrary G, how should one calculate s(G)? They proved that for arbitrary graphs, 55 determining if the search number is less than or equal to an integer k is NP-Hard. They 56 also gave algorithms to compute s(G) where G is a special case of trees. For their 57 results, they use the fact that recontamination of a cleared vertex does not help reduce 58 s(G), which was proved by LaPaugh in [16]. A search plan for G that does not involve 59 recontamination of cleared vertices is referred to as a monotone plan.

Lecture Notes in Computer Science, 2015

We investigate the parallel traversal of a graph with multiple robots unaware of each other. All robots traverse the graph in parallel forever and the goal is to minimize the time needed until the last node is visited (first visit time) and the time between revisits of a node (revisit time). We also want to minimize the visit time, i.e. the maximum of the first visit time and the time between revisits of a node. We present randomized algorithms for uncoordinated robots, which can compete with the optimal coordinated traversal by a small factor, the so-called competitive ratio. For ring and path graph simple traversal strategies allow constant competitive factors even in the worst case. For grid and torus graphs with n nodes there is a O(log n)-competitive algorithm for both visit problems succeeding with high probability, i.e. with probability 1 − n −O(1). For general graphs we present an O(log 2 n)-competitive algorithm for the first visit problem, while for the visit problem we show an O(log 3 n)-competitive algorithm both succeeding with high probability.

Information Processing Letters, 2009

We prove a relationship between the Cleaning problem and the Balanced Vertex-Ordering problem, namely that the minimum total imbalance of a graph equals twice the brush number of a graph. This equality has consequences for both problems. On one hand, it allows us to prove the N P-completeness of the Cleaning problem, which was conjectured by Messinger et al. . On the other hand, it also enables us to design a faster algorithm for the Balanced Vertex-Ordering problem .

2015

We consider the problem of decontaminating an infected network using as few mobile cleaning agents as possible and avoiding recontamination. After a cleaning agent has left a vertex v, this vertex will become recontaminated if m or more of its neighbours are infected, where m ≥ 1 is a threshold parameter of the system indicating the local immunity level of the network. This network decontamination problem, also called monotone connected graph search and intruder capture, has been extensively studied in the literature when m = 1 (no immunity). In this paper, we extend these investigations and consider for the first time the network decontamination problem when the parameter m is an arbitrary integer value m ≥ 1. We direct our study to widely used interconnection networks, namely meshes, tori, and trees. For each of these classes of networks, we present decontamination algorithms with threshold m; these algorithms work even in asynchronous setting, either directly or with a simple mod...