Volker Stemann

Practical loss-resilient codes

We present randomized constructions of linear-time encodable and decodable codes that can transmit over lossy channels at rates extremely close to capacity. The encod-ing and decoding algorithms for these codes have fast and simple software implementations. Partial implementationsof our algorithms are faster by orders of magnitude than the best software implementations of any previous algorithm forthis problem. We expect these codes will be extremely useful for applications such as real-time audio and video transmission over the Internet, where lossy channels are common and fast decoding is a requirement. Despite the simplicity of the algorithms, their design andanalysis are mathematically intricate. The design requires the careful choice of a random irregular bipartite graph,where the structure of the irregular graph is extremely important. We model the progress of the decoding algorithmby a set of differential equations. The solution to these equations can then be expressed as polynomials in one variable with coefficients determined by the graph structure. Based on these polynomials, we design a graph structure that guarantees successful decoding with high probability

Parallel balanced allocations

Volker !Stemann International Computer Science Institute

A remark on matrix rigidity

Abstract The rigidity of a matrix is defined to be the number of entries in the matrix that have to be changed in order to reduce its rank below a certain value. Using a simple combinatorial lemma, we show that one must alter at least c( n 2 r ) log( n r ) entries of an (n × n)-Cauchy matrix to reduce its rank below r, for some constant c. We apply our combinatorial lemma to matrices obtained from asymptotically good algebraic geometric codes to obtain a similar result for r satisfying 2n (√q − 1) n 4 .

Randomized allocation processes

We investigate various randomized processes allocating balls into bins that arise in applications in dynamic resource allocation and on-line load balancing. We consider the scenario when m balls arriving sequentially are to be allocated into n bins on-line and without using a global controller. Traditionally, the main aim of allocation processes is to place the balls into bins to minimize the maximum load in bins. However in many applications it is equally important to minimize the number of trails performed by the balls (the allocation time). We study adaptive allocation schemes that achieve optimal tradeoffs between the maximum load, the maximum allocation time, and the average allocation time. We investigate allocation processes that may reallocate the balls. We provide a tight analysis of the maximum load of processes that during placing a new ball may reassign the balls in up to d randomly chosen bins. We study infinite processes, in which in each step a random ball is removed and a new ball is placed according to some scheduling rule. We present a novel approach that establishes a tight estimation of the time needed for the infinite process to be in the state near to its equilibrium. Finally, we provide a tight analysis of the maximum load of the off-line process in which each ball may be placed into one of d randomly chosen bins. We apply this result to competitive analysis of on-line load balancing processes.

Exploiting storage redundancy to speed up randomized shared memory simulations

This paper presents and analyses efficient implementations of a so-called direct process on distributed memory machines (DMMs) that yields a simulation of an n-processor PRAM on an n-processor optical crossbar DMM with delay O(log log n), a simulation of an n-processor PRAM on an n-processor arbitrary DMM with delay O(log log n/log log log n), an implementation of a static dictionary on an n-processor arbitrary DMM with parallel access time of O(log*n).

Shared Memory Simulations with Triple-Logarithmic Delay

We consider the problem of simulating a PRAM on a distributed memory machine (DMM). Our main result is a randomized algorithm that simulates each step of an n-processor CRCW PRAM on an n-processor DMM with \(\mathcal{O}\)log log log n log*n) delay, with high probability. This is an exponential improvement on all previously known simulations. It can be extended to a simulation of an (n log log log n log*n)-processor EREW PRAM on an n-processor DMM with optimal delay \(\mathcal{O}\)(log log log n log*n), with high probability. Finally a lower bound of Ω(log log log n/log log log log n) expected time is proved for a large class of randomized simulations that includes all known simulations.

Fast and precise Fourier transforms

Many applications of fast Fourier transforms (FFTs), such as computer tomography, geophysical signal processing, high-resolution imaging radars, and prediction filters, require high-precision output. An error analysis reveals that the usual method of fixed-point computation of FFTs of vectors of length 2/sup l/ leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real-time applications. Several researchers have noted that calculation of FFTs with algebraic integers avoids computational noise entirely. We combine a new algorithm for approximating complex numbers by cyclotomic integers with Chinese remaindering strategies to give an efficient algorithm to compute b-bit precision FFTs of length L. More precisely, we approximate complex numbers by cyclotomic integers in Z[e(2/spl pi/i/2/sup n/)] whose coefficients, when expressed as polynomials in e(2/spl pi/i/2/sup n/), are bounded in absolute value by some integer M. For fixed n our algorithm runs in time O(log(M)), and produces an approximation with worst case error of O(1/M(2/sup n-2/-1)). We prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task. The main tool for designing the algorithms is the use of the cyclotomic units, a subgroup of finite index in the unit group of the cyclotomic field. First implementations of our algorithms indicate that they are fast enough to be used for the design of low-cost high-speed/high-precision FFT chips.

Fault-Tolerant Shared Memory Simulations

We consider the problem of simulating a PRAM on a faulty distributed memory machine (DMM). We focus on dynamic faults, i.e. each processor or memory module independently fails during the simulation of a PRAM step with fixed probability and remains faulty for the rest of the simulation. We build upon randomized hashing-based simulations on non-faulty DMMs from [14], which achieve delay O (log log n), with high probability. We design and analyze routines for handling faults occurring during the simulation. Based on these routines we present simulations on faulty DMMs with the same delay O(log log n) as in the non-faulty case, provided that the failure probability of processors and modules is small enough to guarantee an expected linear number of processors and modules to survive the simulation. Thus the facility of being resilient to memory or processor faults increases the delay of the simulation at most by a constant factor.

Fast and precise computations of discrete Fourier transforms using cyclotomic integers

Many applications of fast fourier transforms (FFTs), such as computer- tomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFTs of vectors of length 2l leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real time applications. Several researchers have noted that calculation of FFTs with algebraic integers avoids computational noise entirely, see, e.g., [3]. We will show that complex numbers can be approximated accurately by cyclotomic integers, and combine this idea with Chinese remaindering strategies in the cyclotomic integers to, roughly, give a O(b1+? L log (L)) algorithm to compute b-bit precision FFTs of length L. The first part of the paper will describe the FFT strategy, assuming good app..