Permutation Routing Problem Research Articles

CIBORG: CIrcuit-Based and ORiented Graph theory permutation routing protocol for single-hop IoT networks

The Internet of Things (IoT) has emerged as a promising paradigm which facilitates the seamless integration of physical devices and digital systems, thereby transforming multiple sectors such as healthcare, transportation, and urban planning. This paradigm is also known as ad-hoc networks. IoT is characterized by several pieces of equipment called objects. These objects have different and limited capacities such as battery, memory, and computing power. These limited capabilities make it difficult to design routing protocols for IoT networks because of the high number of objects in a network. In IoT, objects often have data which does not belong to them and which should be sent to other objects, then leading to a problem known as permutation routing problems. The solution to that problem is found when each object receives its items. In this paper, we propose a new approach to addressing the permutation routing problem in single-hop IoT networks. To this end, we start by representing an IoT network as an oriented graph, and then, based on a reservation channel protocol, we first define a permutation routing protocol for an IoT in a single channel. Secondly, we generalize the previous protocol to make it work in multiple channels. Routing is done using graph theory approaches. The obtained results show that the wake-up times and activities of IoT objects are greatly reduced, thus optimizing network lifetime. This is an effective solution for the permutation routing problem in IoT networks. The proposed approach considerably reduces energy consumption and computation time. It saves 5.2 to 32.04% residual energy depending on the number of items and channels used. Low energy and low computational cost demonstrate that the performance of circuit-based and oriented graph theory is better than the state-of-the-art protocol and therefore is a better candidate for the resolution of the permutation routing problem in single-hop environment.

NESEPRIN: A new scheme for energy-efficient permutation routing in IoT networks

Internet of Things (IoT) consists of a variety of heterogeneous interconnected devices called objects or things. These objects are generally equipped with sensing, processing and wireless communication capabilities. Unfortunately, these capabilities are not enough compared to those of devices in traditional networks. For example, objects have low battery power, limited memory storage, and less processing power. There are some cases where objects have to communicate with each other. In an Autonomous Vehicular Network for example, when a vehicle needs to change its direction, it has to alert other vehicles in the same network. This is known as the permutation routing problem. More precisely, the permutation routing problem in IoT occurs when some things of the network possess items that belong to others. The goal is to send items to their respective owners. A number of solutions to the problem have been proposed in literature which focus mainly on Wireless Sensor Networks (where the memory size of objects are the same). In this paper, we propose an efficient permutation routing scheme for a single-hop IoT network (the memory size differs from one object to another). The proposed NESEPRIN protocol consists of two phases: In the first phase, we solve the permutation routing problem in a single-hop environment with a single channel, and secondly we generalize the previous solution to a network with multiple channels. Our solution makes use of the wake and sleep technique to improve the energy conservation of objects. The simulation results show that our protocol outperforms the existing protocols designed to solve the permutation routing problem in terms of energy saving when the volume of data to route is large. NESEPRIN is the better candidate to solve the permutation routing problem in a single-hop multi-channels IoT environment where the volume of data to route is huge.

An Efficient Permutation Routing Protocol in Multi-Hop Wireless Sensor Networks

A wireless sensors network (WSN for short) can be represented as a graph, tree, or other structures. The structure is more or less important depending on the problem that we will deal with and the respect of constraints on WSNs. WSN (p, n) is a wireless sensor network of p stations with n items saved on it. In this paper we are interested in the permutation routing problem on multi-hop WSN(p, n). We derive a new protocol that performs with less number of broadcast rounds compared to the one in [3]. We present some simulation results comparing our protocols to the one in [3].

A Deterministic Protocol for Permutation Routing in Dense Multi-Hop Sensor Networks

A large variety of permutation routing protocols in a single-hop Network are known in the literature. Since they are single hop, there is always a wireless link connecting two nodes. One way to solve this problem in a multiple hop environment is to partition nodes into clusters, where a node in each cluster called clusterhead is responsible for the routing service. In this paper, we propose a hybrid clustering mechanism to perform permutation routing in multi-hop ad hoc Networks. We first propose to partition the network in single-hop clusters also named cliques. Secondly, we run a local permutation routing to broadcast items to their local destinations in each clique. Next we partition the clusterheads of cliques with the hierarchical clustering technique. We show how the outgoing items can be routed to their destination cliques. We give an estimation of the number of broadcast rounds in the worse case. More precisely, we show that solving the permutation routing problem on a multi-hop sensor network need 2 (1 ) ( ) 2 max max (/ ) kO k HUB HUB np    k in the worse case. Where n is the number of the data items stored in the network, p is the number of sensors, |HUBmax| is the number of sensors in the clique of maximum size and k is the number of cliques after the first clustering. Finally, simulation results show that our algorithm performs better than the naive multiple gossiping. To the best of our knowledge, it is the first algorithm for permutation routing in multi-hop radio networks.

An optimal permutation routing algorithm on full-duplex hexagonal networks

Distributed Computing and Networking In the permutation routing problem, each processor is the origin of at most one packet and the destination of no more than one packet. The goal is to minimize the number of time steps required to route all packets to their respective destinations, under the constraint that each link can be crossed simultaneously by no more than one packet. We study this problem in a hexagonal network, i.e. a finite subgraph of a triangular grid, which is a widely used network in practical applications. We present an optimal distributed permutation routing algorithm on full-duplex hexagonal networks, using the addressing scheme described by F.G. Nocetti, I. Stojmenovic and J. Zhang (IEEE TPDS 13(9): 962-971, 2002). Furthermore, we prove that this algorithm is oblivious and translation invariant.

CROSSTALK-FREE REARRANGEABLE MULTISTAGE INTERCONNECTION NETWORKS

In this paper, the notion of crosstalk-free rearrangeability (CF-rearrangeability) of multistage interconnection networks (MINs) is formally defined. Using the concept of line digraphs from graph theory, we show that the problem of crosstalk-free routing on any bit permutation network (BPN) is always equivalent to the classical permutation routing problem on a BPN of smaller size and with fewer stages. We also show the CF-rearrangeability and minimality (in stage number) of three families of BPNs, including the dilated Benes network. Some necessary conditions for a BPN to be CF-rearrangeable are given, and a brief discussion of CF-rearrangeable networks with dilation in time or space is included.

Hypercube computations on partitioned optical passive stars networks

This paper shows that an n=2k processor partitioned optical passive stars (POPS) network with g groups and d processors per group can simulate every bidirectional move of an n processor hypercube using one slot when d g. Moreover, the same POPS network can simulate every monodirectional move of a processor hypercube using one slot when d=g. All these results are shown to be optimal. Our simulations improve on the literature whenever dneg and directly yield several important consequences. For example, as a direct consequence of our simulations, a POPS network, n=dg and d<g, can compute the prefix sums of n data values in log2n slots. This is faster than the best previously known ad hoc algorithm and is actually optimal. Similarly, we improve on the best POPS network algorithms for both the prefix sums problem on general POPS networks and the fundamental online permutation routing problem, among others

A Fault-Tolerant Protocol for Energy-Efficient Permutation Routing in Wireless Networks

A wireless network (WN) is a distributed system where each node is a small hand-held commodity device called a station. Wireless sensor networks have received increasing interest in recent years due to their usage in monitoring and data collection in a wide variety of environments like remote geographic locations, industrial plants, toxic locations, or even office buildings. Two of the most important issues related to a WN are their energy constraints and their potential for developing faults. A station is usually powered by a battery which cannot be recharged while on a mission. Hence, any protocol run by a WN should be energy-efficient. Moreover, it is possible that all stations deployed as part of a WN may not work perfectly. Hence, any protocol designed for a WN should work well even when some of the stations are faulty. The permutation routing problem is an abstraction of many routing problems in a wireless network. In an instance of the permutation routing problem, each of the p-stations in the network is the sender and recipient of n/p packets. The task is to route the packets to their correct destinations. We consider the permutation routing problem in a single-hop wireless network, where each station is within the transmission range of all other stations. We design a protocol for permutation routing on a WN which is both energy efficient and fault tolerant. We present both theoretical estimates and extensive simulation results to show that our protocol is efficient in terms of energy expenditure at each node even when some of the nodes are faulty. Moreover, we show that our protocol is also efficient for the unbalanced permutation routing problem when each station is the sender and recipient of an unequal number of packets.

Concentration, load balancing, partial permutation routing, and superconcentration on cube-connected cycles parallel computers

The cube-connected cycles (CCC) was proposed by Preparata and Vuillemin as an efficient general-purpose parallel system for its fixed-degree, and compact and regular layout. In this paper, a few of the basic algorithms on CCC ( n , 2 n ) interconnection networks are addressed and then applied to concentration, superconcentration, partial permutation routing, and load-balancing problems. The results show that both concentration and superconcentration problems can be solved in O ( n ) time and the on-line partial permutation routing problem in O ( n 2 ) time with O ( 1 ) buffers for each node, where n is the dimension of CCC ( n , 2 n ) interconnection networks. The load-balancing problem based on superconcentration can be solved in O ( Mn ) time, where M is the maximum number of tasks in each node.

An Application of an Initialization Protocol to Permutation Routing in a Single-Hop Mobile Ad Hoc Networks

In 1999 Nakano, Olariu, and Schwing in [20], they showed that the permutation routing of n items pretitled on a mobile ad hoc network (MANET for short) of p stations (p known) and k channels (MANET{(n, p, k)) with k < p, can be carried out in $\frac{2n}{k} + k - 1$ broadcast rounds if k ≤ √p and if each station has a $O(\frac{n}{k})$ -memory locations. And if k ≤ $\smash{\sqrt{\frac {p} {2}}}$ and if each station has a $O(\frac{n}{p})$ -memory locations, the permutations of these n pretitled items can be done also in $\smash{\frac{2n}{k} + k - 1}$ broadcast rounds. They used two assumptions: first they suppose that each station of the mobile ad hoc network has an identifier beforehand. Secondly, the stations are partitioned into k groups such that each group has $\frac{p}{k}$ stations, but it was not shown how this partition can be obtained. In this paper, the stations have not identifiers beforehand and p is unknown. We develop a protocol which first names the stations, secondly gives the value of p, and partitions stations in groups of $\frac{p}{k}$ stations. Finally we show that the permutation routing problem can be solved on it in $O (\frac{p}{\ln2}) + (\frac{2}{k} + 1)n + k - 1$ broadcast rounds in the worst case. It can be solved in $O(\frac{p}{\ln2})+(\frac{2}{k})n+k-1$ broadcast rounds in the better case. Note that our approach does not impose any restriction on k.

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).

Randomized routing and wavelength requirements in wavelength-routed WDM multistage, hypercube, and de Bruijn networks

This paper considers randomized routing in wavelength-routed wavelength division multiplexed (WDM) networks. Randomized routing is advantageous when compared to deterministic routing, as it is simple, oblivious, and suitable for centralized as well as distributed implementation. Recently, there has been a considerable interest in studying regular graphs such as shuffle networks and de Bruijn networks as a possible physical topology for wavelength-routed WDM networks (IEEE/ACM Trans. Network 3(3) (June 1995) 269; IEEE/ACM Trans. Network 2(1) (February 1994) 70; Comput. Networks ISDN Systems 30(24) (December 1998) 2349). In Pankaj and Gallager (1995), permutation routing has been taken as a benchmark problem to accept a network and a deterministic algorithm has been proposed for routing permutation traffic without any blocking in an N-node wrapped-around multi-stage shuffle network using O( log 3 N) wavelengths. In this paper, we look at the possible application of probabilistic techniques on a class of wrapped-around multi-stage networks called 2-multinets and study the wavelength requirements for routing permutation traffic, allowing a negligible blocking of sessions. We adopt the routing scheme proposed in (J. ACM 31 (1984) 507) for fast packet routing to find routes in the circuit-switched wavelength-routed WDM networks and analyze the wavelength requirements for different models. The models differ in their routing node architecture and wavelength selection policy. Our analysis shows that the permutation routing problem can be solved on these models with an overwhelming probability using O( log 2 N) wavelengths. We also study the routing problem on 2-multinets for a class of 1-expectedly bounded connection requests. For this class also, we propose two models and obtain similar bounds on the number of wavelengths required. We confirm the theoretical results by extensive simulation experiments. The theoretical as well as experimental results show that the number of wavelengths required is reduced significantly when randomized routing techniques are used. Finally, we discuss how the results obtained for 2-multinets can be extended to hypercubes and de Bruijn networks.

Fast permutation routing in a class of interconnection networks

AbstractThis paper considers the following permutation routing problem: Given an N × N augmented data manipulator (ADM) network and a permutation π between its N inputs and outputs, can all the traffic connections of π be routed through the network in one pass? A number of backtrack search algorithms have been devised for recognizing ADM admissible permutations. None of the published results, however, appears to settle the time complexity of the problem. The goal of this paper was to answer the question positively by showing the first polynomial time bound for solving the problem. The devised algorithm requires O(N1.695) time to decide whether a given permutation π is admissible and compute a setting of the switches whenever π is admissible. For many practical applications, the obtained bound compares favorably with the O(N lg N) size of an N‐input ADM network. © 2002 Wiley Periodicals, Inc.

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/spl les//spl radic/(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/spl Lt/p/spl Lt/n, it follows that d and b are very small.

Optimal Bounds for Matching Routing on Trees

The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (so-called) caterpillar tree partition, we prove that any permutation on an n-node tree (and thus graph) can be routed in $\frac{3}{2}n + O(\log n)$ steps. This answers an open problem of Alon, Chung, and Graham [ SIAM J. Discrete Math., 7 (1994), pp. 516--530].

Permutation routing in wavelength-routed wrapped-around shuffle networks using fewer wavelengths

Optical networks based on wavelength division multiplexing (WDM) and wavelength routing are considered to be potential candidates for the next generation of wide area networks. One of the main issues in these networks is the development of efficient routing algorithms which require minimum number of wavelengths. In this paper, we focus on the permutation routing problem in WDM networks with the shuffle network topology. In Pankaj and Gallager [Wavelength Requirements of All-Optical Networks, IEEE/ACM Transactions on Networking 3, No. 3 (June 1995) 269–280], an algorithm has been proposed for permutation routing in wrapped-around multi-stage shuffle networks using O(log 3 N) wavelengths, where N is the number of nodes. Here, the messages are routed using one (light) hop. We present a routing algorithm in these networks with only O(log 2 N) wavelengths, which routes the messages using O(loglog N) hops. We also discuss how our approach can be extended to the generalized routing. We study the worst case buffer requirement and bandwidth reduction due the use of multiple hops. We evaluate the performance of the proposed algorithm through extensive simulation by measuring the average number of hops required and delay incurred by a message. We also study the average number of messages queued at a node and average number of messages that share a lightpath at various time intervals. We also evaluate the system throughput achieved by our multi-hop routing algorithm.

Routing Permutations on Graphs via Factors

We deal with the permutation routing problem on graphs modeling interconnection networks. In our model, calledrouting via factors, at each routing step, the communication pattern is a directed 1-factor in a symmetric digraph. This adds a new feature, that of continuous packet movement, to preciously studied routing types, where the routing of a permutation is reduced to a sequence of permutations from a given class. We especially focus on bipartite graphs and we give sufficient conditions for a graph to be rearrangeable in our model. We propose a general technic for routing via factors that we apply to the 2D mesh and the hypercube.

PERMUTATION ROUTING AND SORTING ON THE RECONFIGURABLE MESH

In this paper we demonstrate the power of reconfiguration by presenting efficient randomized algorithms for both packet routing and sorting on a reconfigurable mesh connected computer. The run times of these algorithms are better than the best achievable time bounds on a conventional mesh. Many variations of the reconfigurable mesh can be found in the literature. We define yet another variation which we call as Mr. We also make use of the standard PARBUS model. We show that permutation routing problem can be solved on a linear array Mr of size n in [Formula: see text] steps, whereas n-1 is the best possible run time without reconfiguration. A trivial lower bound for routing on Mr will be [Formula: see text]. On the PARBUS linear array, n is a lower bound and hence any standard n-step routing algorithm will be optimal. We also show that permutation routing on an n×n reconfigurable mesh Mr can be done in time n+o(n) using a randomized algorithm or in time 1.25n+o(n) deterministically. In contrast, 2n-2 is the diameter of a conventional mesh and hence routing and sorting will need at least 2n-2 steps on a conventional mesh. A lower bound of [Formula: see text] is in effect for routing on the 2D mesh Mr as well. On the other hand, n is a lower bound for routing on the PARBUS and our algorithms have the same time bounds on the PARBUS as well. Thus our randomized routing algorithm is optimal upto a lower order term. In addition we show that the problem of sorting can be solved in randomized time n+o(n) on Mr as well as on PARBUS. Clearly, this sorting algorithm will be optimal on the PARBUS model. The time bounds of our randomized algorithms hold with high probability.

Routing Permutations on a 2D Grid with One-Way Edges

We deal with the permutation routing problem on the m × n grid with one-register vertices and one-way edges. We give a routing algrithm of O(m + n) steps which is local, that is, the routing decision of every vertex depends on the states of vertices O(1) apart. Previous work investigated either local routings on two-way edges or non local routings on one-way edges.

Flit-serial packet routing on meshes and tori

In this paper we consider theflit-serial packet-routing problem, where each packet consists of a sequence ofk flits and is, thus, called asnake. Based on the properties of the snake during the routing, we give a formal definition of three different packet-routing models, namely, thestore-and-forward model, thecut-through model, and thecut-through with partial cuts model. Surprisingly, all previous work has focused on the store-and-forward model. We also introduce therestricted cut-through model, which is unrealistic, but is proved to be a very powerful tool in the effort to bound the time required by a routing problem. We study the cut-through with partial cuts model which is most commonly used in practice. We present the first algorithms, deterministic and probabilistic, based on this model for the permutation routing problem on a chain, on a square mesh, and on a square torus.