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Direct routing: Algorithms and Complexity
 In Proceedings of the 12th Annual European Symposium on Algorithms (ESA
, 2004
"... Direct routing is the special case of bufferless routing where N packets, once injected into the network, must be delivered to their destinations without collisions. We give a general treatment of three facets of direct routing: (i) Algorithms. We present a polynomial time greedy direct algorithm wh ..."
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Direct routing is the special case of bufferless routing where N packets, once injected into the network, must be delivered to their destinations without collisions. We give a general treatment of three facets of direct routing: (i) Algorithms. We present a polynomial time greedy direct algorithm which is worstcase optimal. We improve the bound of the greedy algorithm for special cases, by applying variants of the this algorithm to commonly used network topologies. In particular, we obtain nearoptimal routing time for the tree, mesh, butterfly and hypercube. (ii) Complexity. By a reduction from Vertex Coloring, we show that optimal Direct Routing is inapproximable, unless P=NP. (iii) Lower Bounds for Buffering. We show that certain direct routing problems cannot be solved efficiently; in order to solve these problems, any routing algorithm needs buffers. We give nontrivial lower bounds on such buffering requirements for general routing algorithms.
Dynamic Tree Routing under the "Matching with Consumption" Model
, 1996
"... . In this paper we consider dynamic routing on trees under the "matching with consumption" routing model, a natural extension of the matching routing model introduced by Alon, Chung and Graham [1, 2], which allows for the consumption of packets when they reach their destination. We present ..."
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. In this paper we consider dynamic routing on trees under the "matching with consumption" routing model, a natural extension of the matching routing model introduced by Alon, Chung and Graham [1, 2], which allows for the consumption of packets when they reach their destination. We present an asymptotically optimal online algorithm that routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps where d is the degree of tree T on which the routing takes place and dist is the maximum distance some packet has to travel. We present an offline algorithm that solves the same problem within 2(k \Gamma 1) + dist steps. Versions of both the online and the offline algorithms which avoid livelock situations are also provided. We establish a close relation between the "matching with consumption" and the hotpotato routing models, and we exploit it in the analysis of our routing algorithms. 1 Introduction In a packet routing problem on a connected undirected gra...
Permutation Routing via Matchings
, 1996
"... The following routing problem on an undirected graph is considered: Initially, each node of the graph contains exactly one packet. Each node is the destination node of exactly one packet, so the initial state can be considered a permutation of the packets. The packets are routed to their destination ..."
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Cited by 4 (1 self)
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The following routing problem on an undirected graph is considered: Initially, each node of the graph contains exactly one packet. Each node is the destination node of exactly one packet, so the initial state can be considered a permutation of the packets. The packets are routed to their destination nodes by a sequence of steps. In one step, each packet can either remain at its current location, or it can be swapped with a neighbor, i.e., a step is determined by a matching of the participating nodes. The time complexity of the previously best algorithm for routing all packets to their destination nodes given any initial permutation in a graph with n nodes was bounded by 13 5 n. We present an algorithm running in at most 2n \Gamma 3 steps, where n 2, at the same time simplifying the analysis of the time complexity.
Efficient Bufferless Packet Switching on Trees and Leveled Networks ∗
"... In bufferless networks the packets cannot be buffered while they are in transit; thus, once injected, the packets have to move constantly. Bufferless networks are interesting because they model optical networks. We consider the tree and leveled network topologies, which represent a wide class of net ..."
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In bufferless networks the packets cannot be buffered while they are in transit; thus, once injected, the packets have to move constantly. Bufferless networks are interesting because they model optical networks. We consider the tree and leveled network topologies, which represent a wide class of network configurations. On these networks, we study manytoone batch problems where each node is the source of at most one packet, and the destination of an arbitrary number of packets. Each packet is to follow a preselected path from the source to the destination. Let T ∗ be the optimal delivery time for the packets. We have the following results: • For trees, we present two bufferless algorithms: (i) a deterministic algorithm with delivery time O(δ · T ∗ · log n), and (ii) a randomized algorithm with delivery time O(T ∗ · log 2 n); where, δ is the maximum node degree, and n is the number of nodes. Both algorithms are distributed in the sense that packet forwarding decisions are made locally at the nodes. • For leveled networks, we present two algorithms: (i) a centralized algorithm with
UNIVERSAL BUFFERLESS PACKET SWITCHING ∗
"... Abstract. A packetswitching algorithm specifies the actions of the nodes in order to deliver packets in the network. A packetswitching algorithm is universal if it applies to any network topology and for any batch communication problem on the network. A long standing open problem has concerned the ..."
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Abstract. A packetswitching algorithm specifies the actions of the nodes in order to deliver packets in the network. A packetswitching algorithm is universal if it applies to any network topology and for any batch communication problem on the network. A long standing open problem has concerned the existence of a universal packetswitching algorithm with near optimal performance guarantees for the class of bufferless networks where the buffer size for packets in transit is zero. We give a positive answer to this question. In particular, we give a universal bufferless algorithm which is within a polylogarithmic factor from optimal for arbitrary batch problems: T = O ` T ∗ · log 3 (n + N) ´, where T is the packet delivery time of our algorithm, T ∗ is the optimal delivery time, n is the size of the network, and N is the number of packets. At the heart of our result is a new deterministic technique for constructing a universal bufferless algorithm by emulating a storeandforward algorithm on a transformation of the network. The main idea is to replace packet buffering in the transformed network with packet circulation in regions of the original network. The cost of the emulation on the packet delivery time is proportional to the buffer sizes used by the storeandforward algorithm. We obtain the advertised result by using a storeandforward algorithm with logarithmic sized buffers. The resulting bufferless algorithm is constructive and it can be implemented in a distributed way.
Potentialfunctionbased Analysis of an offline Heap Construction Algorithm
, 2000
"... In this paper we examine the problem of heap construction on a rooted tree T from a packet routing perspective. Each node of T initially contains a packet which has a keyvalue associated with it. The aim of the heap construction algorithm is to route the packets along the edges of the tree so that ..."
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In this paper we examine the problem of heap construction on a rooted tree T from a packet routing perspective. Each node of T initially contains a packet which has a keyvalue associated with it. The aim of the heap construction algorithm is to route the packets along the edges of the tree so that, at the end of the routing, the tree is heap ordered with respect to the key values associated with the packets. We consider the case where the routing is performed according to the matching model and we present and analyse an offline algorithm that heap orders the tree within 2h(T) routing steps, where h(T) is the height of tree T. The main contribution of the paper is the novel analysis of the algorithm based on potential functions. It is our belief that potential functions will be the main vehicle in analysing fast nonrecursive routing algorithms.
Parametric Permutation Routing via Matchings
, 1996
"... The problem of routing permutations on graphs via matchings is considered, and we present a general algorithm which can be parameterized by dierent heuristics. This leads to a framework which makes the analysis simple and local. 1 Introduction The routing problem we consider is the following: W ..."
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The problem of routing permutations on graphs via matchings is considered, and we present a general algorithm which can be parameterized by dierent heuristics. This leads to a framework which makes the analysis simple and local. 1 Introduction The routing problem we consider is the following: We are given an undirected connected graph with n nodes and a permutation of the nodes. Each node u contains one packet which must be routed to (u). The routing is carried out in a sequence of steps. In one step, each packet can either remain at its current location, or it can be swapped with a neighbor. Thus, at all times each node has exactly one packet. We are interested in designing an algorithm for this problem with a low complexity measured in the number of steps necessary in the worst case to ensure that all packets are routed to their correct locations independent of the initial conguration. This problem was rst dened and investigated in [ACG93, ACG94], and an upper bound of 3...
Online Matching Routing on Trees
, 1997
"... In this paper we examine online heap construction and online permutation routing on trees under the matching model. Let T be and nnode tree of maximum degree d. By providing online algorithms we prove that: (i) For a rooted tree of height h, online heap construction can be completed within (2d ..."
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In this paper we examine online heap construction and online permutation routing on trees under the matching model. Let T be and nnode tree of maximum degree d. By providing online algorithms we prove that: (i) For a rooted tree of height h, online heap construction can be completed within (2d \Gamma 1)h routing steps. (ii) For an arbitrary tree, online permutation routing can be completed within 4dn routing steps. (iii) For a complete dary tree, online permutation routing can be completed within 2(d \Gamma 1)n+ 2d log 2 n routing steps. Technical Report 514 Basser Department of Computer Science University of Sydney Original: 27 May 1997 1 The work of Dr Symvonis was supported by an ARC Institutional Grant. 27 May 1997 1 Introduction In packet routing problems we are given a network (usually represented by a connected, undirected graph) and a set of packets distributed over the nodes of the network. Each packet has an origin node and a destination node and our aim ...
c ○ 1999 Society for Industrial and Applied Mathematics OPTIMAL BOUNDS FOR MATCHING ROUTING ON TREES ∗
"... Abstract. The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (socalled) caterpillar tree partition, we prove that any permutation on an nnode tree (and thus graph) can be routed in 3 n + O(log n) steps. This answers an open 2 ..."
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Abstract. The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (socalled) caterpillar tree partition, we prove that any permutation on an nnode tree (and thus graph) can be routed in 3 n + O(log n) steps. This answers an open 2
Direct Routing on Trees (Extended Abstract)
 IN PROCEEDINGS OF THE NINTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA 98
, 1998
"... We consider offline permutation routing on trees. We are particularly interested in direct tree routing schedules where packets once started move directly towards their destination. The scheduling of start times ascertains that no two packets will use the same edge in the same direction in the same ..."
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We consider offline permutation routing on trees. We are particularly interested in direct tree routing schedules where packets once started move directly towards their destination. The scheduling of start times ascertains that no two packets will use the same edge in the same direction in the same time step. In O(n log n log log n) time and O(n log n) space, we construct a direct tree routing schedule guaranteed to complete the routing within the general optimum of n  1 steps. In addition, our scheme guarantees that at most two packets arrive at the same node in the same time step. Furthermore, if the length of the route of a given packet is d and the maximum number of other routes intersecting the route in a single node is k then the packet arrives to its destination within d + k steps.