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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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Cited by 6 (3 self)
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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.
Nearoptimal hot potato routing on trees
 in Proceedings of EuroPar 2004, LNCS 3149
, 2004
"... In hotpotato (deflection) routing, nodes in the network have no buffers for packets in transit, so that some conflicting packets must be deflected away from their destinations. In this work, we study onetomany batch routing problems on arbitrary tree topologies with n nodes. The routing time of a ..."
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Cited by 5 (3 self)
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In hotpotato (deflection) routing, nodes in the network have no buffers for packets in transit, so that some conflicting packets must be deflected away from their destinations. In this work, we study onetomany batch routing problems on arbitrary tree topologies with n nodes. The routing time of a routing algorithm is the time for the last packet to reach its destination. Denote by rt ∗ the optimal routing time for a given routing problem. We construct the first hotpotato routing algorithms whose routing times are asymptotically nearoptimal; that is, the incurred routing times are within polylogarithmic factors from optimal. More specifically, we present: 1. A deterministic algorithm whose routing time is O(δ · rt ∗ · lg n), where δ is the maximum node degree; thus, for boundeddegree trees, the routing time becomes O(rt ∗ · lg n). 2. A randomized algorithm whose routing time is O(rt ∗ · lg2 n) with high probability; randomization is used for adjusting packet priorities. Both algorithms are local, hence distributed, and greedy; so, routing decisions are made locally, and packets are advanced towards their destinations whenever possible, respectively.
Universal Bufferless Routing
, 2004
"... In a routing problem, a set of packets must be routed from their sources to their destinations along specified paths in a connected network. The celebrated result of Leighton, Maggs and Rao (1988) established, nonconstructively, the existence of a routing schedule which uses constant size bffers an ..."
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In a routing problem, a set of packets must be routed from their sources to their destinations along specified paths in a connected network. The celebrated result of Leighton, Maggs and Rao (1988) established, nonconstructively, the existence of a routing schedule which uses constant size bffers and routes the packets in optimal time. Since then, constructive algorithms, as well as generalizations to distributed, buffered routing schedules have been developed. A long standing open problem...
Direct Routing
 In Proceedings of the 12th Annual European Symposium on Algorithms ESA 2004
, 2003
"... Direct routing is a special case of bu#erless routing in which packets are not allowed to conflict with each other. The task is to compute the injection times of the packets so that they don't conflict. A well known lower bound on the routing time of any algorithm is # C + D), where the conge ..."
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Cited by 4 (4 self)
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Direct routing is a special case of bu#erless routing in which packets are not allowed to conflict with each other. The task is to compute the injection times of the packets so that they don't conflict. A well known lower bound on the routing time of any algorithm is # C + D), where the congestion C is the maximum number of paths that use any edge, and the dilation D is the maximum length of any path.
Optimal algorithms for packet routing on trees
 Basser Dept of Computer Science, University of Sydney
, 1994
"... In this paper, we study the permutation packet routing problem on trees. We show that every permutation can be routed on a tree of n vertices in n, 1 routing steps. We provide an algorithm which produces in O(n 2) time a schedule that needs O(n 2) bits for its description. Moreover, we describe an o ..."
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Cited by 3 (2 self)
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In this paper, we study the permutation packet routing problem on trees. We show that every permutation can be routed on a tree of n vertices in n, 1 routing steps. We provide an algorithm which produces in O(n 2) time a schedule that needs O(n 2) bits for its description. Moreover, we describe an online algorithm that completes the routing of any permutation in n,1 routing steps by using at each vertex v bu ering area of size at most 2d(v), where d(v) is the degree of vertex v. Our results provide upper bounds on the number of routing steps required to route a permutation on an arbitrary connected graph G since the routing can be done by using only the edges of a spanning tree of G.
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
Greedy Õ(C+D) HotPotato Routing on Trees
, 2003
"... In hotpotato (deflection) routing, nodes in the network have no bu#ers for packets in transit. A hotpotato routing algorithm is greedy if packets are advanced from their sources toward their destinations whenever possible. The dilation D is the longest distance a packet has to travel; the conges ..."
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Cited by 1 (1 self)
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In hotpotato (deflection) routing, nodes in the network have no bu#ers for packets in transit. A hotpotato routing algorithm is greedy if packets are advanced from their sources toward their destinations whenever possible. The dilation D is the longest distance a packet has to travel; the congestion C is the maximum number of packets that traverse any edge. The routing time of a routingalgorithm is the time for the last packet to reach its destination. A well known lower bound on the routing time is # C +D).
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.
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.
Lower Bounds for Onetoone Packet Routing on Trees using HotPotato Algorithms
, 2000
"... In this paper we consider hotpotato packet routing of onetoone routing patterns on trees. For all sufficiently large n we construct a onetoone packet routing problem on an nnode tree for which an oblivious greedy hotpotato algorithm requires at least 2n \Gamma o(n) time steps. This lower boun ..."
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In this paper we consider hotpotato packet routing of onetoone routing patterns on trees. For all sufficiently large n we construct a onetoone packet routing problem on an nnode tree for which an oblivious greedy hotpotato algorithm requires at least 2n \Gamma o(n) time steps. This lower bound is also shown to be valid for the minimumdistance heuristic. We also establish a lower bound of 2( d\Gamma3 d\Gamma2 )n \Gamma o(n) for trees of maximum d. As an upper bound, we apply the charging argument of Borodin et al. [12] to show that any greedy hotpotato algorithm routes a onetoone routing pattern on an nnode tree within 2(n \Gamma 1) steps. 1 Introduction In a packet routing problem we are given a synchronous network represented by a connected undirected graph and a set of packets distributed over the nodes of the graph. Each packet has an origin and a destination node, and the aim is to route each packet to its destination in as few steps as possible, subject to each edge...