Some permutation routing algorithms for low-dimensional hypercubes

Oblivious permutation routing in binary d-cubes has been well studied in the literature. In a permutation routing, each node initially contains a packet with a destination such that all the 2^d destinations are distinct. Kaklamanis et al. (Math. Syst. Theory 24 (1991) 223–232) used the decomposability of hypercubes into Hamiltonian circuits to give an asymptotically optimal routing algorithm. The notion of “destination graph” was first introduced by Borodin and Hopcroft to derive lower bounds on routing algorithms. This idea was recently used by Grammatikakis et al. (Proceedings of the Advancement in Parallel Computing, Elsevier, Amsterdam, 1993) to construct many–one routing algorithms for the binary 2-cube and 3-cube. In the present paper, further theoretical development is made along this line. It is then applied to obtain algorithms for binary d-cubes with d up to 12, which compare favorably with the above-mentioned “Hamiltonian circuit” algorithm. Some results on t-nary cubes with t3 are also obtained.     

Routing algorithms for the shuffle-exchange permutation network

The Journal of Supercomputing - For a natural number n, let $$S_n$$ denote the symmetric group on n letters. Let $$textit{SEP}_n$$ be the Cayley graph $$text {Cay}(S_n,{sigma ,sigma ^{-1},tau...     

Fully dynamic algorithms for permutation graph coloring

We introduce a dynamic model for maintaining permutation graph coloring. Our motivation comes from the strait type river routing problem in VLSI. This paper presents fully dynamic algorithms for the permutation graph coloring problem. These algorithms are designed to handle Insert and Delete operations and answer some queries. The aim is to provide for running times that are asymptotically more efficient than recomputation (off-line algorithms that run in 0(n logw) time, are known [5,6,10,3]). First, the algorithm A^ that runs in 0(n) uniform running time per Insert/Delete operation is presented. Second, a more sophisticated data structure leads to the algorithm A2 that runs in (9(m logw) uniform running time per Insert I Delete, where m denotes the number of chains in the decomposition. It follows from [7,4] that the running time of A2 when the points from the dynamically changing set are drawn independently from a uniform distribution on the unit square is G(yfn logn) per Insert/Delete in probability. Third, we sketch a composite algorithm A3 that switches between A± and A2 guarantees an amortized running time of (min{n,m logw)) per Insert/Delete. Finally, we outline a number of applications     

Energy-efficient permutation routing in radio networks

A radio network (RN, for short) is a distributed system populated by small, hand-held commodity devices running on batteries. Since recharging batteries may not be possible while on mission, we are interested in designing protocols that are highly energy efficient. One of the most effective energy-saving strategies is to mandate that the stations go to sleep whenever they do not transmit or receive messages. It is well known that a station is expending power while its transceiver is active, that is, while transmitting or receiving a packet. It is perhaps surprising at first that a station is expending power even if it receives a packet that is not destined for it. Since, in single-hop radio networks, every station is within transmission range from every other station, the design of energy-efficient protocols is highly nontrivial. An instance of the permutation routing problem involves p stations of an RN, each storing n/p items. Each item has a unique destination which is the identity of the station to which the item must be routed. The goal is to route all the items to their destinations while expending as little energy as possible. Since, in the worst case, each item must be transmitted at least once, every permutation routing protocol must take n/k time slots. Similarly, each station must be awake for at least n/p time slots to transmit and/or receive packets. Our main contribution is to present an almost optimal energy-efficient permutation routing protocol for a k-channel, a p-station RN that routes n packets in at most (2d+2b+1)n/k+k time slots with no station being awake for more than (4d+7b-1)n/p time slots, where d=[(logp/k)/(logn/p)], b=[(log k)/(logn/p)] and k⩽√(p/2). Since, in most real-life situations, the number n of packets to route, the number p of stations in the RN, and the number k of channels available satisfy the relation k≪p≪n, it follows that d and b are very small     

An energy-efficient permutation routing protocol for single-hop radio networks

A radio network (RN) is a distributed system where each station or node is a small hand-held commodity device called a station. Typically, each station has access to a few channels for transmitting and receiving messages. By RN(p, k), we denote a radio network with p stations, where each station has access to k channels. In a single-hop RN, every station is within the transmission range of every other station. Each station consumes power while transmitting or receiving a message, even when it receives a message that is not destined for it. It is extremely important that the stations consume power only when it is necessary since it is not possible to recharge batteries when the stations are on a mission. We are interested in designing an energy-efficient protocol for permutation routing, which is one of the most fundamental problems in any distributed system. An instance of the permutation routing problem involves p stations of an RN, each storing n/p items. Each item has a unique destination address which is the identity of the destination station to which the item should be sent. The goal is to route all the items to their destinations while consuming as little energy as possible. We show that the permutation routing problem of n packets on an RN(p, k) can be solved in 2n/k+(p/k)/sup 2/+p+2k/sup 2/ slots and each station needs to be awake for at most 6n/p+2p/k+8k slots. When k/spl Lt/p/spl Lt/n, our protocol is more efficient, both in terms of total number of slots and the number of slots each station is awake compared to a previously published protocol by Nakano et al. (2001).     

Optimal algorithms for adjacent side routing

We consider the switchbox routing problem of two-terminal nets in the case when all thek nets lie on two adjacent sides of the rectangle. Our routing model is the standard two-layer model. We develop an optimal algorithm that routes all the nets whenever a routing exists. The routing obtained uses the fewest possible number of vias. A more general version of this problem (adjacent staircase) is also optimally solved.     

Two algorithms for three-layer channel routing

A channel router is an important design aid in the design automation of VLSI circuit layout. Many algorithms have been developed based on various wiring models with routing done on two layers. With the recent advances in VLSI process technology, it is possible to have three independent layers for interconnection. In this paper two algorithms are presented for three-layer channel routing. The first assumes a very simple wiring model. This enables the routing problem to be solved optimally in a time of $\text{O}\text{(}\text{n}\text{log}\text{n}\text{)}$. The second algorithm is for a different wiring model and has an upper bound of $\text{O(}\text{n}{}^2\text{)}$ for its execution time. It uses fewer horizontal tracks than the first algorithm. For the second model the channel width is not bounded by the channel density.     

无线多媒体传感器网络能量均衡的QoS路由算法

为保障能量受限的无线多媒体传感器网络(WMSNs)多服务质量(QoS)需求,提出了一种能量均衡的QoS路由(EBQR)算法。该算法通过蚁群优化将网络带宽、时延、丢包率和能量等因素作为目标函数,并根据函数值大小动态调整蚁群信息素的挥发系数和浓度增量,提供网络业务中满足不同QoS需求的最优路径。仿真结果表明:与AntWMSNs算法和ASAR算法相比,EBQR算法平均端到端时延降低了16%,丢包率减少22%,生命周期延长了近50%,有效实现了网络中节点能耗的均衡性。     

Work-efficient routing algorithms for rearrangeable symmetricalnetworks

The work performed by a parallel algorithm is the product of its running time and the number of processors it requires. This paper presents work-efficient (or cost-optimal) routing algorithms to determine the switch settings for realizing permutations on rearrangeable symmetrical networks such as Benes and the reduced Ω _NΩ_N^-1. These networks have 2n-1 stages with N=2ⁿ inputs/outputs, each stage consisting of N/2 crossbar switches of size (2×2). Previously known parallel routing algorithms for a rearrangeable network with N inputs determine the states of all switches recursively in O(n) iterations using N processors. Each iteration determines the switch settings of at most two stages of the network and requires at least O(n) time on a computer of N processors, regardless of the type of its interconnection network. Hence, the work of any previously known parallel routing algorithm equals at least O(Nn²) for setting up all the switches of a rearrangeable network. The new routing algorithms run on a computer of p processors, 1⩽p⩽N/n, and perform work O(Nn). Moreover, because the range of p is large, the new routing algorithms do not have to be changed in case some processors become faulty     

Optimal algorithms for adjacent side routing

We consider the switchbox routing problem of two-terminal nets in the case when all thek nets lie on two adjacent sides of the rectangle. Our routing model is the standard two-layer model. We develop an optimal algorithm that routes all the nets whenever a routing exists. The routing obtained uses the fewest possible number of vias. A more general version of this problem (adjacent staircase) is also optimally solved.This research was supported in part by NSA Contract No. MDA-904-85H-0015, NSF Grant No. DCR-86-00378, and by NSF Engineering Research Centers Program NSFD CDR 88003012.     

A unified framework for off-line permutation routing in parallel networks

In this paper we present a general strategy for finding efficient permutation routes in parallel networks. Among the popular parallel networks to which the strategy applies are mesh networks, hypercube networks, hypercube-derivative networks, ring networks, and star networks. The routes produced are generally congestion-free and take a number of routing steps that is within a small constant factor of the diameter of the network. Our basic strategy is derived from an algorithm that finds (in polynomial time) efficient permutation routes for aproduct network, G×H, given efficient permutation routes forG andH. We investigate the use of this algorithm for routingmultiple permutations and extend its applicability to a wide class of graphs, including several families ofCayley graphs. Finally, we show that our approach can be used to find efficient permutation routes among the remaining live nodes infaulty networks.This research was supported in part by a grant from the NSF, Grant No. CCR-88-12567.     

LSI channel routing algorithms

A new method is proposed and an algorithm is developed for two-wire channel routing in a two-layer board, with horizontal interconnections made in one layer and vertical interconnections in the other layer.Translated from Kibernetika, No. 3, pp. 32–35, May–June, 1991.     

Optimal routing algorithms for mesh-connected processor arrays

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than Θ(logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.     

Optimal routing algorithms for mesh-connected processor arrays

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than (logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.Preliminary reports of portions of the results contained in this paper have appeared in theProceedings of the 1988 Aegean Workshop on Computing [5], and in theProceedings of the 1987 Conference on Foundations of Software Technology and Theoretical Computer Science [18]. The work of the first author was supported in part by NSF Grant NSF-DCR-85-03251 and ONR contract N00014-80-C-0647. The work of the second author was supported in part by NSF Grant NSF-DCR-86-00379.     

Parallel algorithms for routing in nonblocking networks

We construct nonblocking networks that are efficient not only as regards their cost and delay, but also as regards the time and space required to control them. In this paper we present the first simultaneous weakly optimal solutions for the explicit construction of nonblocking networks, the design of algorithms and data-structures. Weakly optimal is in the sense that all measures of complexity (size and depth of the network, time for the algorithm, space for the data-structure, and number of processor-time product) are within one or more logarithmic factors of their smallest possible values. In fact, we construct a scheme in which networks withn inputs andn outputs have sizeO(n(logn)²) and depthO(logn), and we present deterministic and randomized on-line parallel algorithms to establish and abolish routes dynamically in these networks. In particular, the deterministic algorithm usesO((logn)⁵) steps to process any number of transactions in parallel (with one processor per transaction), maintaining a data structure that useO(n(logn)²) words.     

Fast algorithms for the dominating set problem on permutation graphs

AnO(n log logn) (resp.O(n² log² n)) algorithm is presented to solve the minimum cardinality (resp. weight) dominating set problem on permutation graphs, assuming the input is a permutation. The best-known previous algorithm was given by FÄrber and Keil, where they use dynamic programming to get anO(n² (resp.O(n³)) algorithm. Our improvement is based on the following three factors: (1) an observation on the order among the intermediate terms in the dynamic programming, (2) a new construction formula for the intermediate terms, and (3) efficient data structures for manipulating these terms.This research was supported in part by the National Science Foundation under Grant CCR-8905415 to Northwestern University.     

A heuristic to accelerate in-situ permutation algorithms

In-situ permutation algorithms permute an array of n items without using a second array. Little space is needed when one proceeds permuting along a cycle. Thus, those algorithms first search for one element called the leader on each cycle of the permutation. Then they move the items cycle by cycle. A great fraction of the runtime is spent with testing whether elements are leaders. We present a simple-to-implement heuristic to accelerate the search for leaders and present performance gains for randomly chosen permutations on three algorithms.     

Static and run-time algorithms for all-to-many personalizedcommunication on permutation networks

With the advent of new routing methods, the distance that a message is sent is becoming relatively less and less important. Thus, assuming no link contention, permutation seems to be an efficient collective communication primitive. In this paper, we present several algorithms for decomposing all-to-many personalized communication into a set of disjoint partial permutations. We discuss several algorithms and study their effectiveness from the view of static scheduling as well as run-time scheduling. An approximate analysis shows that with n processors, and assuming that every processor sends and receives d messages to random destinations, our algorithm can perform the scheduling in O(dn In d) time, on average, and can use an expected number of d+log d partial permutations to carry out the communication. We present experimental results of our algorithms on the CM-5     

Robust algorithms for packet routing in a mesh

This paper considers the problem of permutation packet routing on a n×n mesh-connected array of processors. Each node in the array is assumed to be independently faulty with a probability bounded above by a valuep. This paper gives a routing algorithm which, ifp 0.29, will with very high probability route every packet that can be routed inO(n logn) steps with queue lengths that areO(log² n). Extensions to higher-dimensional meshes are given.