Supriya Krishnamurthy

A statistical theory of chord under churn

Most earlier studies of DHTs under churn have either depended on simulations as the primary investigation tool, or on establishing bounds for DHTs to function. In this paper, we present a complete analytical study of churn using a master-equation-based approach, used traditionally in non-equilibrium statistical mechanics to describe steady-state or transient phenomena. Simulations are used to verify all theoretical predictions. We demonstrate the application of our methodology to the Chord system. For any rate of churn and stabilization rates, and any system size, we accurately predict the fraction of failed or incorrect successor and finger pointers and show how we can use these quantities to predict the performance and consistency of lookups under churn. We also discuss briefly how churn may actually be of different types and the implications this will have for the functioning of DHTs in general.

Circumspect descent prevails in solving random constraint satisfaction problems

We study the performance of stochastic local search algorithms for random instances of the K-satisfiability (K-SAT) problem. We present a stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT is a focused algorithm in the sense that it focuses on variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large K-SAT instances almost surely in linear time, up to high clause-to-variable ratios α; for example, for K = 4 we observe linear-time performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms.

An analytical study of a structured overlay in the presence of dynamic membership

In this paper, we present an analytical study of dynamic membership (aka churn) in structured peer-to-peer networks. We use a fluid model approach to describe steady-state or transient phenomena and apply it to the Chord system. For any rate of churn and stabilization rates and any system size, we accurately account for the functional form of the probability of network disconnection as well as the fraction of failed or incorrect successor and finger pointers. We show how we can use these quantities to predict both the performance and consistency of lookups under churn. All theoretical predictions match simulation results. The analysis includes both features that are generic to structured overlays deploying a ring as well as Chord-specific details and opens the door to a systematic comparative analysis of, at least, ring-based structured overlay systems under churn.

Comparing Maintenance Strategies for Overlays

In this paper, we present an analytical tool for understanding the performance of structured overlay networks under churn based on the master-equation approach of physics. We motivate and derive an equation for the average number of hops taken by lookups during churn, for the chord network. We analyse this equation in detail to understand the behaviour with and without churn. We then use this understanding to predict how lookups will scale for varying peer population as well as varying the sizes of the routing tables. We also consider a change in the maintenance algorithm of the overlay, from periodic stabilisation to a reactive one which corrects fingers only when a change is detected. We generalise our earlier analysis to understand how the reactive strategy compares with the periodic one.

Clustering of solutions in hard satisfiability problems

We study numerically the solution space structure of random 3-SAT problems close to the SAT/UNSAT transition. This is done by considering chains of satisfiability problems, where clauses are added ...

Brief announcement: the accuracy of tree-based counting in dynamic networks

We study a simple Bellman-Ford-like protocol which performs network size estimation over a tree-shaped overlay. A continuous time Markov model is constructed which allows key protocol characteristics to be estimated under churn, including the expected number of nodes at a given (perceived) distance to the root and, for each such node, the expected (perceived) size of the subnetwork rooted at that node. We validate the model by simulations, using a range of network sizes, node degrees, and churn-to-protocol rates, with convincing results.