We propose a non-asymptotic approach to analyze kidney exchange that builds on the random graph model of kidney exchange introduced in Ashlagi, Garmarnik, Rees and Roth's "The need for (long) chains in kidney exchange" (2012). We analyze a two phase procedure where random walks are used to allocate chains, followed by allocation via matching in cycles. Random walks preserve the probabilistic structure of residual graphs, greatly facilitating analysis without sending the number of nodes to infinity. We derive useful analytical bounds that illustrate the performance of our procedure and more general kidney allocation procedures. Our results complement previous asymptotic results for large (limit) graphs on the benefits of using chains in kidney exchange and empirical results based on data from fielded kidney exchanges.

This paper analyzes the worst-case performance of randomized backoff on simple multiple-access channels. Most previous analysis of backoff has assumed a statistical arrival model. For batched arrivals, in which all n packets arrive at time 0, we show the following tight high-probability bounds. Randomized binary exponential backoff has makespan Θ(nlgn), and more generally, for any constant r, r-exponential backoff has makespan Θ(nlog lgr n). Quadratic backoff has makespan Θ((n/lg n) 3/2), and more generally, for r> 1, r-polynomial backoff has makespan Θ((n/lg n) 1+1/r). Thus, for batched inputs, both exponential and polynomial backoff are highly sensitive to backoff constants. We exhibit a monotone superpolynomial subexponential backoff algorithm, called loglog-iterated backoff, that achieves makespan Θ(nlg lgn/lg lglgn). We provide a matching lower bound showing that this strategy is optimal among all monotone backoff algorithms. Of independent interest is that this lower bound was proved with a delay sequence argument. In the adversarial-queuing model, we present the following stability and instability results for exponential backoff and loglogiterated backoff. Given a (λ,T)-stream, in which at most n = λT packets arrive in any interval of size T, exponential backoff is stable for arrival rates of λ = O(1/lgn) and unstable for arrival rates of λ = Ω(lglgn/lg n); loglog-iterated backoff is stable for arrival rates of λ = O(1/(lg lgnlgn)) and unstable for arrival rates of λ = Ω(1/lg n). Our instability results show that bursty input is close to being worst-case for exponential backoff and variants and that even small bursts can create instabilities in the channel.

Tight bounds on the cost of cache-oblivious searching are proved. It is shown that no cache-oblivious search structure can guarantee that a search performs fewer than lg e log/sub B/N block transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the block sizes are limited to be powers of 2. A modified version of the van Emde Boas layout is proposed, whose expected block transfers between any two levels of the memory hierarchy arbitrarily close to [lg e + O(lg lg B/ lgB)] logB N + O(1). This factor approaches lg e /spl ap/ 1.443 as B increases. The expectation is taken over the random placement of the first element of the structure in memory. As searching in the disk access model (DAM) can be performed in log/sub B/N + 1 block transfers, this result shows a separation between the 2-level DAM and cache-oblivious memory-hierarchy models. By extending the DAM model to k levels, multilevel memory hierarchies can be modeled. It is shown that as k grows, the search costs of the optimal k-level DAM search structure and of the optimal cache-oblivious search structure rapidly converge. This demonstrates that for a multilevel memory hierarchy, a simple cache-oblivious structure almost replicates the performance of an optimal parameterized k-level DAM structure.