. 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 on-line 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 off-line algorithm that solves the same problem within 2(k \Gamma 1) + dist steps. Versions of both the on-line and the off-line algorithms which avoid live-lock situations are also provided. We establish a close relation between the "matching with consumption" and the hot-potato 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...

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.

We consider off-line 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 \Gamma 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. 1 Introduction In this paper, we consider off-line hot-potato permutation packet routing on trees. We are given a permutation ß of the nodes, and for each node v, we want to send a packet fr...

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 key-value 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 off-line 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 non-recursive routing algorithms.