Yonatan Aumann

An O (log k ) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm

It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves upon the previously best-known bound of O(log2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min-cut ratio, is presented.

Security Against Covert Adversaries: Efficient Protocols for Realistic Adversaries

In the setting of secure multiparty computation, a set of mutually distrustful parties wish to securely compute some joint function of their private inputs. The computation should be carried out in a secure way, meaning that no coalition of corrupted parties should be able to learn more than specified or somehow cause the result to be “incorrect.” Typically, corrupted parties are either assumed to be semi-honest (meaning that they follow the protocol specification) or malicious (meaning that they may deviate arbitrarily from the protocol). However, in many settings, the assumption regarding semi-honest behavior does not suffice and security in the presence of malicious adversaries is excessive and expensive to achieve.In this paper, we introduce the notion of covert adversaries, which we believe faithfully models the adversarial behavior in many commercial, political, and social settings. Covert adversaries have the property that they may deviate arbitrarily from the protocol specification in an attempt to cheat, but do not wish to be “caught” doing so. We provide a definition of security for covert adversaries and show that it is possible to obtain highly efficient protocols that are secure against such adversaries. We stress that in our definition, we quantify over all (possibly malicious) adversaries and do not assume that the adversary behaves in any particular way. Rather, we guarantee that if an adversary deviates from the protocol in a way that would enable it to “cheat” (meaning that it can achieve something that is impossible in an ideal model where a trusted party is used to compute the function), then the honest parties are guaranteed to detect this cheating with good probability. We argue that this level of security is sufficient in many settings.

A statistical theory for quantitative association rules

Association rules are a key data-mining tool and as such have been well researched. So far, this research has focused predominantly on databases containing categorical data only. However, many real-world databases contain quantitative attributes and current solutions for this case are so far inadequate. In this paper we introduce a new definition of quantitative association rules based on statistical inference theory. Our definition reflects the intuition that the goal of association rules is to find extraordinary and therefore interesting phenomena in databases. We also introduce the concept of sub-rules which can be applied to any type of association rule. Rigorous experimental evaluation on real-world datasets is presented, demonstrating the usefulness and characteristics of rules mined according to our definition.

Efficient calculation of interval scores for DNA copy number data analysis

DNA amplifications and deletions characterize cancer genome and are often related to disease evolution. Microarray-based techniques for measuring these DNA copy-number changes use fluorescence ratios at arrayed DNA elements (BACs, cDNA, or oligonucleotides) to provide signals at high resolution, in terms of genomic locations. These data are then further analyzed to map aberrations and boundaries and identify biologically significant structures. We develop a statistical framework that enables the casting of several DNA copy number data analysis questions as optimization problems over real-valued vectors of signals. The simplest form of the optimization problem seeks to maximize phi(I) = Sigmanu(i)/radical|I| over all subintervals I in the input vector. We present and prove a linear time approximation scheme for this problem, namely, a process with time complexity O (nepsilon(-2)) that outputs an interval for which phi(I) is at least Opt/alpha(epsilon), where Opt is the actual optimum and alpha(epsilon) --> 1 as epsilon --> 0. We further develop practical implementations that improve the performance of the naive quadratic approach by orders of magnitude. We discuss properties of optimal intervals and how they apply to the algorithm performance. We benchmark our algorithms on synthetic as well as publicly available DNA copy number data. We demonstrate the use of these methods for identifying aberrations in single samples as well as common alterations in fixed sets and subsets of breast cancer samples.

Efficient calculation of interval scores for DNA copy number data analysis.

DNA amplifications and deletions characterize cancer genome and are often related to disease evolution. Microarray-based techniques for measuring these DNA copy-number changes use fluorescence ratios at arrayed DNA elements (BACs, cDNA, or oligonucleotides) to provide signals at high resolution, in terms of genomic locations. These data are then further analyzed to map aberrations and boundaries and identify biologically significant structures. We develop a statistical framework that enables the casting of several DNA copy number data analysis questions as optimization problems over real-valued vectors of signals. The simplest form of the optimization problem seeks to maximize phi(I) = Sigmanu(i)/radical|I| over all subintervals I in the input vector. We present and prove a linear time approximation scheme for this problem, namely, a process with time complexity O (nepsilon(-2)) that outputs an interval for which phi(I) is at least Opt/alpha(epsilon), where Opt is the actual optimum and alpha(epsilon) --> 1 as epsilon --> 0. We further develop practical implementations that improve the performance of the naive quadratic approach by orders of magnitude. We discuss properties of optimal intervals and how they apply to the algorithm performance. We benchmark our algorithms on synthetic as well as publicly available DNA copy number data. We demonstrate the use of these methods for identifying aberrations in single samples as well as common alterations in fixed sets and subsets of breast cancer samples.

A Statistical Theory for Quantitative Association Rules

Association rules are a key data-mining tool and as such have been well researched. So far, this research has focused predominantly on databases containing categorical data only. However, many real-world databases contain quantitative attributes and current solutions for this case are so far inadequate. In this paper we introduce a new definition of quantitative association rules based on statistical inference theory. Our definition reflects the intuition that the goal of association rules is to find extraordinary and therefore interesting phenomena in databases. We also introduce the concept of sub-rules which can be applied to any type of association rule. Rigorous experimental evaluation on real-world datasets is presented, demonstrating the usefulness and characteristics of rules mined according to our definition.

Fault tolerant data structures

The authors consider the tolerance of data structures to memory faults. They observe that many pointer-based data structures (e.g. linked lists, trees, etc.) are highly nonresilient to faults. A single fault in a linked list or tree may result in the loss of the entire set of data. They present a formal framework for studying the fault tolerance properties of pointer-based data structures, and provide fault tolerant versions of the stack, the linked list, and the dictionary tree.

Borders: An Efficient Algorithm for Association Generation in Dynamic Databases

We consider the problem of finding association rules in a database with binary attributes. Most algorithms for finding such rules assume that all the data is available at the start of the data mining session. In practice, the data in the database may change over time, with records being added and deleted. At any given time, the rules for the current set of data are of interest. The naive, and highly inefficient, solution would be to rerun the association generation algorithm from scratch following the arrival of each new batch of data. This paper describes the Borders algorithm, which provides an efficient method for generating associations incrementally, from dynamically changing databases. Experimental results show an improved performance of the new algorithm when compared with previous solutions to the problem.

Pattern Matching with Swaps

Let a text string T of n symbols and a pattern string P of m symbols from alphabet ? be given. A swapped version T? of T is a length n string derived from T by a series of local swaps (i.e., t???t?+1 and t??+1?t?), where each element can participate in no more than one swap. The pattern matching with swaps problem is that of finding all locations i for which there exists a swapped version T? of T with an exact matching of P in location i of T?. It has been an open problem whether swapped matching can be done in less than O(nm) time. In this paper we show the first algorithm that solves the pattern matching with swaps problem in time o(nm). We present an algorithm whose time complexity is O(nm1/3logmlog?) for a general alphabet ?, where ?=min(m,???).

Function matching: algorithms, applications, and a lower bound

We introduce a new matching criterion - function matching - that captures several different applications. The function matching problem has as its input a text T of length n over alphabet ΣT and a pattern P = P[1]P[2] ... P[m] of length m over alphabet ΣP. We seek all text locations i for which, for some function f : ΣP → ΣT (f may also depend on i), the m-length substring that starts at i is equal to f(P[1])f(P[2]) ... f(P[m]). We give a randomized algorithm which, for any given constant k, solves the function matching problem in time O(n log n) with probability 1/nk of declaring a false positive. We give a deterministic algorithm whose time is O(n|ΣP| logm) and show that it is almost optimal in the newly formalized convolutions model. Finally, a variant of the third problem is solved by means of two-dimensional parameterized matching, for which we also give an efficient algorithm.