Dorit Ron

Quantitative Analysis of the Full Bitcoin Transaction Graph

The Bitcoin scheme is a rare example of a large scale global payment system in which all the transactions are publicly accessible (but in an anonymous way). We downloaded the full history of this scheme, and analyzed many statistical properties of its associated transaction graph. In this paper we answer for the first time a variety of interesting questions about the typical behavior of users, how they acquire and how they spend their bitcoins, the balance of bitcoins they keep in their accounts, and how they move bitcoins between their various accounts in order to better protect their privacy. In addition, we isolated all the large transactions in the system, and discovered that almost all of them are closely related to a single large transaction that took place in November 2010, even though the associated users apparently tried to hide this fact with many strange looking long chains and fork-merge structures in the transaction graph.

Graph minimum linear arrangement by multilevel weighted edge contractions

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a linear-time algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short running time of the algorithm enabled experiments with very large graphs.

Multigrid Solvers and Multilevel Optimization Strategies

An optimization problem is the task of minimizing (or maximizing — for definiteness we discuss minimization) a certain real-valued “objective functional” (or “cost” , or “energy” , or “performance index”, etc.) E(x), possibly under a set of equality and/or inequality constraints, where x = (x 1, …, x n ) is a vector (often the discretization of one or several functions) of unknown variables (real or complex numbers, and/or integers, and/or Ising spins, etc.). A general process for solving such problems is the point-by-point minimization, in which one changes only one variable x j (or few of them) at a time, lowering E as much as possible in each such step. More generally, the process accepts any candidate change of one or few variables if it causes a drop in energy (бE < 0).

Multilevel algorithms for linear ordering problems

Linear ordering problems are combinatorial optimization problems that deal with the minimization of different functionals by finding a suitable permutation of the graph vertices. These problems are widely used and studied in many practical and theoretical applications. In this paper, we present a variety of linear--time algorithms for these problems inspired by the Algebraic Multigrid approach, which is based on weighted-edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short (linear) running time of the algorithms enables testing very large graphs.

Optimal multigrid algorithms for calculating thermodynamic limits

Beyond eliminating the critical slowing down, multigrid algorithms can also eliminate the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced in coarse grids during the multigrid cycle. Thermodynamic limits can be calculated to accuracy ɛ in justO(ε-2) computer operations. Examples described in detail and with results of numerical tests are the calculation of the susceptibility, the σ-susceptibility, and the average energy in Gaussian models, and also the determination of the susceptibility and the critical temperature in a two-dimensional Ising spin model. Extension to more advanced models is outlined.

Renormalization Multigrid (RMG): Statistically Optimal Renormalization Group Flow and Coarse-to-Fine Monte Carlo Acceleration

New renormalization-group algorithms are developed with adaptive representations of the renormalized system which automatically express only significant interactions. As the amount of statistics grows, more interactions enter, thereby systematically reducing the truncation error. This allows statistically optimal calculation of thermodynamic limits, in the sense that it achieves accuracy ε in just O(ε−2) random number generations. There are practically no finite-size effects and the renormalization transformation can be repeated arbitrarily many times. Consequently, the desired fixed point is obtained and the correlation-length critical exponent ν is extracted. In addition, we introduce a new multiscale coarse-to-fine acceleration method, based on a multigrid-like approach. This general (non-cluster) algorithm generates independent equilibrium configurations without slow down. A particularly simple version of it can be used at criticality. The methods are of great generality; here they are demonstrated on the 2D Ising model.

A Multilevel Algorithm for the Minimum 2-sum Problem

In this paper we introduce a direct motivation for solving the minimum 2-sum problem, for which we present a linear-time algorithm inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. Our results turned out to be better than previous results, while the short running time of the algorithm enabled experiments with very large graphs. We thus introduce a new benchmark for the minimum 2-sum problem which contains 66 graphs of various characteristics. In addition, we propose the straightforward use of a part of our algorithm as a powerful local reordering method for any other (than multilevel) framework.

A Fast Multigrid Algorithm for Energy Minimization under Planar Density Constraints

The two-dimensional layout optimization problem reinforced by the efficient space utilization demand has a wide spectrum of practical applications. Formulating the problem as a nonlinear minimization problem under planar equality and/or inequality density constraints, we present a linear time multigrid algorithm for solving correction to this problem. The method is demonstrated on various graph drawing (visualization) instances.

Surprising convergence of the Monte Carlo renormalization group for the three-dimensional Ising model

We present a surprisingly simple approach to high-accuracy calculations of the critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.

A Hardware Implementation of the CSP Primitives and its Verification

A design for a hardware interface that implements CSP-like communication primitives is presented. The design is based on a bus scheme that allows processes to “eavesdrop” on messages not directly addressed to them. A temporal logic specification is given for the network and an outline of a verification proof is sketched.