On random routing in wireless sensor grids: A mathematical model for rendezvous probability and performance optimization

Random routing protocols in Wireless Sensor Networks (WSNs) forward packets to randomly selected neighbors. These packets are ‘agents’ carrying information about events or ‘queries’ seeking such information. A novel mathematical framework is proposed for analyzing random routing protocols. Exact probability of a packet visiting a given node within a given hop count as well as the rendezvous probability of agents and queries meeting at a given node in a 2-D grid-based WSN are derived. The basic relationship needed for extending the models to a 3-D grid topology is provided. Exact probabilities of agents meeting queries are derived while ignoring physical boundary effects and packet losses, under two different strategies for forwarding the packet to a neighbor: (a) with equal probability, and (b) self-avoiding forwarding. We then extend the model to account for packet losses by considering the case where a packet is forwarded to a neighbor with equal probability. Also provided is the extension of the analysis for a network with rectangular boundaries. The exact solutions presented, unlike existing models relying on asymptotic behavior, are also applicable to small and medium scale networks. They can be used to set parameters and optimize performance of several classes of random routing protocols. All the models are validated using Monte Carlo simulations. Simulation results indicate that the model is also a good approximation for sparse arrays with 75% or higher node density. Finally, the utility of the model is demonstrated by determining the protocol parameters to optimize the performance of rumor routing protocol under a fixed energy budget.     

Time-Constrained Scheduling of Weighted Packets on Trees and Meshes

Abstract. The time-constrained packet routing problem is to schedule a set of packets to be transmitted through a multinode network, where every packet has a source and a destination (as in traditional packet routing problems) as well as a release time and a deadline. The objective is to schedule the maximum number of packets subject to deadline constraints. This problem is studied in [1], where it is shown that the problem is NP-Complete even when the underlying topology is a linear array. Approximation algorithms are also provided in [1] for the linear array and the unidirectional ring for both the case where packets may be buffered in transit and the case where they may not be. In this paper we extend the results of [1] in two directions. First, we consider the more general network topologies of trees and two-dimensional meshes. Second, we associate with each packet a measure of utility, called a weight, and study the problem of maximizing the total weight of the packets that are scheduled subject to their timing constraints. For the bufferless case, we provide constant factor approximation algorithms for the time-constrained scheduling problem with weighted packets on trees and meshes. We also provide logarithmic approximations for the same problems in the buffered case. These results are complemented by new lower bounds, which demonstrate that we cannot hope to achieve the same results for general network topologies. For example, we show that if k packets are required to follow prescribed paths in an arbitrary graph, then unless NP = ZPP, there is no polynomial-time k ^1-ε -approximation, for any ε > 0 , to the optimal set of packets that can be scheduled.     

Time-Constrained Scheduling of Weighted Packets on Trees and Meshes

The time-constrained packet routing problem is to schedule a set of packets to be transmitted through a multinode network, where every packet has a source and a destination (as in traditional packet routing problems) as well as a release time and a deadline. The objective is to schedule the maximum number of packets subject to deadline constraints. This problem is studied in [1], where it is shown that the problem is NP-Complete even when the underlying topology is a linear array. Approximation algorithms are also provided in [1] for the linear array and the unidirectional ring for both the case where packets may be buffered in transit and the case where they may not be. In this paper we extend the results of [1] in two directions. First, we consider the more general network topologies of trees and two-dimensional meshes. Second, we associate with each packet a measure of utility, called a weight, and study the problem of maximizing the total weight of the packets that are scheduled subject to their timing constraints. For the bufferless case, we provide constant factor approximation algorithms for the time-constrained scheduling problem with weighted packets on trees and meshes. We also provide logarithmic approximations for the same problems in the buffered case. These results are complemented by new lower bounds, which demonstrate that we cannot hope to achieve the same results for general network topologies. For example, we show that if k packets are required to follow prescribed paths in an arbitrary graph, then unless NP = ZPP, there is no polynomial-time k1-ε -approximation, for any ε > 0 , to the optimal set of packets that can be scheduled.     

Edge-Cut Bounds on Network Coding Rates

Active networks are network architectures with processors that are capable of executing code carried by the packets passing through them. A critical network management concern is the optimization of such networks and tight bounds on their performance serve as useful design benchmarks. A new bound on communication rates is developed that applies to network coding, which is a promising active network application that has processors transmit packets that are general functions, for example a bit-wise XOR, of selected received packets. The bound generalizes an edge-cut bound on routing rates by progressively removing edges from the network graph and checking whether certain strengthened d-separation conditions are satisfied. The bound improves on the cut-set bound and its efficacy is demonstrated by showing that routing is rate-optimal for some commonly cited examples in the networking literature.     

Permutation Routing on Reconfigurable Meshes

In this paper we present efficient algorithms for packet routing on the reconfigurable linear array and the reconfigurable two-dimensional mesh. We introduce algorithms that are efficient in the worst case and algorithms that are better on average. The time bounds presented are better than those achievable on the conventional mesh and previously known algorithms. We present two variants of the reconfigurable mesh. In the first model, M _r , the processors are attached to a reconfigurable bus, the individual edge connections being bidirectional. In the second model, M _mr , the processors are attached to two unidirectional buses. In this paper we present lower bounds and nearly matching upper bounds for packet routing on these two models. As a consequence, we solve two of the open problems mentioned in [9]. Received August 17, 1998; revised November 3, 1999.     

Potential Function Analysis of Greedy Hot-Potato Routing

We study the problem of packet routing in synchronous networks. We put forward a notion of greedy hot-potato routing algorithms and devise tech- niques for analyzing such algorithms. A greedy hot-potato routing algorithm is one where: • The processors have no buffer space for storing delayed packets. Therefore, each packet must leave any intermediate processor at the step following its arrival. • Packets always advance toward their destination if they can. Namely, a packet must leave its current intermediate node via a link which takes it closer to its destination, unless all these links are taken by other packets. Moreover, in this case all these other packets must advance toward their destinations. We use potential function analysis to obtain an upper bound of O(n k ^1/2 ) on the running time of a wide class of algorithms in the two-dimensional n × n mesh, for routing problems with a total of k packets. The same techniques can be generalized to obtain an upper bound of O(exp(d) n ^d-1 k ^1/d ) on the running time of a wide class of algorithms in the d -dimensional n ^d mesh, for routing problems with a total of k packets. Received December 1993, and in final form March 1997.     

Minimal Adaptive Routing on the Mesh with Bounded Queue Size

An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters. Such algorithms potentially avoid network bottlenecks by routing packets around “hot spots.” Minimal adaptive routing algorithms have the additional advantage that the path each packet takes is a shortest one. For a large class of minimal adaptive routing algorithms, we present an Ω(n²/k²) bound on the worst case time to route a static permutation of packets on ann×nmesh or torus with nodes that can hold up tok≥ 1 packets each. This is the first nontrivial lower bound on adaptive routing algorithms. The argument extends to more general routing problems, such as theh–hrouting problem. It also extends to a large class of dimension order routing algorithms, yielding an Ω(n²/k) time bound. To complement these lower bounds, we present two upper bounds. One is anO(n²/k+n) time dimension order routing algorithm that matches the lower bound. The other is the first instance of a minimal adaptive routing algorithm that achievesO(n) time with constant sized queues per node. We point out why the latter algorithm is outside the model of our lower bounds.     

An optimal time bound for oblivious routing

The problem of routing data packets in a constant-degree network is considered. A routing scheme is calledoblivious if the route taken by each packet is uniquely determined by its source and destination. The time required for the oblivious routing ofn packets onn processors is known to be Θ(√n). It is demonstrated that the presence of extra processors can expedite oblivious routing. More specifically, the time required for the oblivious routing ofn packets onp processors is Θ(n/√p + logn).     

Bounds on evacuation time for deflection routing

Summary Upper bounds are given on the timeT needed to evacuatek packets from a synchronous packet network using deflection routing, whereby all packets that arrive at a node during one time slot leave the node during the next slot. For example,Tn+2(k–1) for a binaryn-cube network when priority is given to packets closer to their destination, and for a single destination networkT is less than or equal to the network diameter plusk–1 times the network deflection index. Deflection routing for one pass through an omega network is also considered. Bruce Hajek received a B.S. in Mathematics and an M.S. in Electrical Engineering from the University of Illinois and a Ph.D. in Electrical Engineering from the University of California at Berkeley. He is a Professor in the Coordinated Science Laboratory and the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign, where he has been since August, 1979. His research interests include communication and computer network algorithms, information theory, random processes, and combinatorial optimization.Supported by the National Science Foundation under contract NSF ECS 83 52030 with matching funds provided by AT&T. Postal address: CSL, 1101 W. Springfield, Urbana IL 61801, USA.     

Routing on Meshes with Buses

We consider the problem of routing packets on an MIMD mesh-connected array of processors augmented with row and column buses. We give lower bounds and randomized algorithms for the problem of routing k-permutations (where each processor is the source and destination of exactly k packets) on a d-dimensional mesh with buses, which we call the (k,d)-routing problem. We give a general class of ``hard' permutations which we use to prove lower bounds for the (k,d)-routing problem, for all k,d≥ 1. For the (1,1)- and (1,2)-routing problems the worst-case permutations from this class are identical to ones published by other authors, as are the resulting lower bounds. However, we further show that the (1,d)-routing problem requires 0.72 ... n steps for d=3, 0.76 ... n steps for d=4, and slightly more than steps for all d≥ 5. We also obtain new lower bounds for the (k,d)-routing problem for k,d > 1, which improve on the bisection lower bound in some cases. These lower bounds hold for off-line routing as well. We develop efficient algorithms for the (k,1)-routing problem and for the problem of routing k-randomizations (where each processor has k packets initially and each packet is routed to a random destination) on the one-dimensional mesh and use them in a general (k,d)-routing algorithm which improves considerably on previous results. In particular, the routing time for the (1,d)-routing problem is bounded by steps with high probability (whp), whenever for some constant ε > 0, and the routing time for the (k,d)-routing problem is steps whp whenever for some constant ε > 0 and k≥ 3.6 ... d, matching the bisection lower bound. We then present a simple algorithm for the (2,2)-routing problem running in 1.39 ... n+o(n) steps whp. Finally, for the important special case of routing permutations on two-dimensional meshes with buses, the (1,2)-routing problem, we give a more sophisticated algorithm that runs in 0.78 ... n+o(n) steps whp. Received May 18, 1994; revised June 23, 1995.     

An optimal time bound for oblivious routing

The problem of routing data packets in a constant-degree network is considered. A routing scheme is calledoblivious if the route taken by each packet is uniquely determined by its source and destination. The time required for the oblivious routing ofn packets onn processors is known to be (n). It is demonstrated that the presence of extra processors can expedite oblivious routing. More specifically, the time required for the oblivious routing ofn packets onp processors is (n/p + logn).     

Universal wormhole routing

We examine the wormhole routing problem in terms of the “congestion” c and “dilation” d for a set of packet paths. We show, with mild restrictions, that there is a simple randomized algorithm for routing any set of P packets in O(cdη+dη log P) time with high probability, where L is the number of flits in a packet, and η=min {d, L}; only a constant number of flits are stored in each queue at any time. Using this result, we show that a fat tree network of area ⊖(A) can simulate wormhole routing on any network of comparable area with O(log³ A) slowdown, when all worms have the same length. Variable length worms are also considered. We run some simulations on the fat tree which show that not only does wormhole routing tend to perform better than the more heavily studied store and forward routing in this context, but that performance superior to our provable bound is attainable in practice     

Bounds on maximum delay in networks with deflection routing

We consider the problem of finding the worst case packet transit delay in networks using deflection routing. Several classes of networks are studied, including many topologies for which deflection routing has been proposed or implemented (e.g., hypercube, Manhattan Street Network, shuffle-exchange network). We derive new upper bounds on the evacuation time of batch admissions, and present simple proofs for several existing bounds. Also derived are bounds on worst case transit delay for certain networks admitting packets continuously. To demonstrate the practical utility of our results, we compare a new delay bound to the maximum delay observed in simulations. The results have application in both protocol design and the determination of the required capacity of packet resequencing buffers     

Multi-core implementation of decomposition-based packet classification algorithms

Multi-field packet classification is a network kernel function where packets are classified based on a set of predefined rules. Decomposition-based classification approaches are of major interest to the research community because of the parallel search in each packet header field. This paper presents four decomposition-based approaches on multi-core processors. We search in parallel for all the fields using linear search or range-tree search; we store the partial results in a linked list or a bit vector. The partial results are merged to produce the final packet header match. We evaluate the performance with respect to latency and throughput varying the rule set size (1–64 K). Experimental results show that our approaches can achieve 128 ns latency per packet and 11.5 Gbps throughput on state-of-the-art 16-core platforms.     

Fast Generation of Random Permutations Via Networks Simulation

We consider the problem of generating random permutations with uniform distribution. That is, we require that for an arbitrary permutation π of n elements, with probability 1/n! the machine halts with the i th output cell containing π(i) , for 1 ≤ i ≤ n . We study this problem on two models of parallel computations: the CREW PRAM and the EREW PRAM. The main result of the paper is an algorithm for generating random permutations that runs in O(log log n) time and uses O(n ^1+o(1) ) processors on the CREW PRAM. This is the first o(log n) -time CREW PRAM algorithm for this problem. On the EREW PRAM we present a simple algorithm that generates a random permutation in time O(log n) using n processors and O(n) space. This algorithm outperforms each of the previously known algorithms for the exclusive write PRAMs. The common and novel feature of both our algorithms is first to design a suitable random switching network generating a permutation and then to simulate this network on the PRAM model in a fast way. Received November 1996; revised March 1997.     

Networks on which hot-potato routing does not livelock

Summary. Hot-potato routing is a form of synchronous routing which makes no use of buffers at intermediate nodes. Packets must move at every time step, until they reach their destination. If contention prevents a packet from taking its preferred outgoing edge, it is deflected on a different edge. Two simple design principles for hot potato routing algorithms are minimum advance, that advances at least one packet towards its destination from every nonempty node (and possibly deflects all other packets), and maximum advance, that advances the maximum possible number of packets. Livelock is a situation in which packets keep moving indefinitely in the network without any packet ever reaching its destination. It is known that even maximum advance algorithms might livelock on some networks. We show that minimum advance algorithms never livelock on tree networks, and that maximum advance algorithms never livelock on triangulated networks. Received: March 1999 / Accepted: August 1999     

A Lower Bound for Nearly Minimal Adaptive and Hot Potato Algorithms

Recently, Chinn et al. [10] presented lower bounds for store-and-forward permutation routing algorithms on the n \times n mesh with bounded buffer size and where a packet must take a shortest (or minimal ) path to its destination. We extend their analysis to algorithms that are nearly minimal. We also apply this technique to the domain of hot potato algorithms, where there is no storage of packets and the shortest path to a destination is not assumed (and is in general impossible). We show that ``natural' variants and ``improvements' of several algorithms in the literature perform poorly in the worst case. As a result, we identify algorithmic features that are undesirable for worst-case hot potato permutation routing. Recent works in hot potato routing have tried to define simple and greedy classes of algorithms. We show that when an algorithm is too simple and too greedy, its performance in routing permutations is poor in the worst case. Specifically, the technique of [10] is also applicable to algorithms that do not necessarily send packets in minimal or even nearly minimal paths: it may be enough that they naively attempt to do so when possible. In particular, our results show that a certain class of greedy algorithms that was suggested recently by Ben-Dor et al. [6] contains algorithms that have poor performance in routing worst-case permutations. Received August 24, 1995; revised May 27, 1997.     

Õ(Congestion + Dilation) Hot-Potato Routing on Leveled Networks

We study packet routing problems, in which we are given a set of N packets which will be sent on preselected paths with congestion C and dilation D. For store-and-forward routing, in which nodes have buffers for packets in transit, there are routing algorithms with a performance that matches the lower bound Ω(C+D). Motivated from optical networks, we study hot-potato routing in which the nodes are bufferless. Due to the lack of buffers, in hot-potato routing the packets may be delayed more than in store-and-forward routing. An interesting question is how much is the performance of routing algorithms affected by the absence of buffers. Here, we answer this question for the class of leveled networks, in which the nodes are partitioned into L+1 distinct levels. We present a randomized hot-potato routing algorithm for leveled networks, which routes the packets in O((C + L) ln⁹ (LN)) time with high probability. For routing problems with dilation Ω(L), and where N is a polynonial in L, this bound is within polylogarithmic factors of the lower bound Ω(C+L). Our algorithm demonstrates that for such routing problems the benefit from using buffers is no more than polylogarithmic; thus, hot-potato routing is an efficient way to route packets in leveled networks. In hot-potato routing, due to the lack of buffers, the packets may not be able to remain on their preselected paths during the course of routing (while in store-and-forward routing the packets remain on their preselected paths). However, in our algorithm the actual path that each packet follows contains its original preselected path; thus the lower bound Ω(C+L) is also a lower bound for the new paths. Our algorithm is distributed, that is, routing decisions are taken locally at each node while packets are routed in the network. To our knowledge, this is the first hot-potato algorithm designed and analyzed, in terms of congestion and dilation, for leveled networks.     

基于拥塞控制的非结构化P2P网络的路由负载均衡策略

非结构化P2P(unstructured peer-to-peer network)对等网络中的节点资源定位的路由查询是对等网络研究中的一个主要难题,特别是当网络中客户端节点由于其频繁加入、离开导致网络结构动态变化所带来的资源查询难题.提出了一种新的基于拥塞控制的路由查询方法来实现动态网络下的资源查询.该方法分两部分实现:首先是网络资源的分组与节点重连策略.该策略使得具有同等资源的节点相互连接,并周期性地调整节点上的节点连接数量以减少同组资源节点上的负载.通过以上策略,使得网络的拓扑结构自动地从随机网络结构进化到以资源组为单位的聚类网络,从而使得网络中形成网络资源组间的查询负载均衡.另一方面,组内的节点之间的路由负载均衡是通过节点间协同学习实现的.采用协同Q-学习方法,所研究的方法不仅从节点上学习其处理能力、连接数和资源的个数等参数,还将节点的拥塞状态作为协同Q-学习的重要参数,并建立模型.通过这种技术,同一组节点上的资源查询被有目的地引导,以避开那些组内拥塞的节点,从而最终实现资源组内节点之间的查询均衡.仿真实验结果表明,相比常用的random walk资源查找方法,该研究所实现的资源定位方法能够更迅速地实现网络的资源查询.仿真结果还表明,相比random walk方法,所提出的方法在网络高强度查询和网络节点动态加入和退出的情况下进行查询具有更高的鲁棒性和适应性.     

Performance of ad hoc networks with two-hop relay routing and limited packet lifetime (extended version)

We consider a mobile ad hoc network consisting of three types of nodes (source, destination and relay nodes) and using the two-hop relay routing. This type of routing takes advantage of the mobility and the storage capacity of the nodes, called the relay nodes, in order to route packets between a source and a destination. Packets at relay nodes are assumed to have a limited lifetime in the network. Nodes are moving inside a bounded region according to some random mobility model. Closed-form expressions and asymptotic results when the number of nodes is large are provided for the packet delivery delay and for the energy needed to transmit a packet from the source to its destination. We also introduce and evaluate a variant of the two-hop relay protocol that limits the number of generated copies in the network. Our model is validated through simulations for two mobility models (random waypoint and random direction mobility models), and the performance of the two-hop routing and of the epidemic routing protocols are compared.