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We define the decision problem data arrangement, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that data arrangement is strongly NP-complete for any fixed d ≥ 2 and explain the… (More)

We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter… (More)

Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ''cross edges'' between the parts. We are interested in sparse random graphs Ž. G with edge probability crn. We show that, if c) ln 4, then the bisection width is ⍀ n n, c r n… (More)

We study the k-core of a random (multi)graph on n ver-tices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the… (More)

- Malwina J Luczak, Colin Mcdiarmid
- 2004

There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where 0 < λ < 1. Upon arrival each customer selects d ≥ 2 servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system… (More)

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high… (More)

- Malwina J Luczak, Colin Mcdiarmid
- 2005

Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, where λ > 0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when d… (More)

We compare the performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks to a path selection algorithm that is based on the balanced allocations principle [4, 18]-the Balanced Dynamic Alternative Routing (BDAR) algorithm. While the standard technique checks alternative routes… (More)

- Malwina J Luczak, Colin Mcdiarmid, Eli Upfal
- 2003

We consider a random sequence of calls between nodes in a complete network with link capacities. Each call first tries the direct link. If that link is saturated, then the 'first-fit dynamic alternative routing algorithm' sequentially selects up to d random two-link alternative routes, and assigns the call to the first route with spare capacity on each… (More)