Comparing Elastic-Degenerate Strings: Algorithms, Lower Bounds, and Applications

An elastic-degenerate (ED) string T is a sequence of n sets T[1], . . ., T[n] containing m strings in total whose cumulative length is N. We call n, m, and N the length, the cardinality and the size of T, respectively. The language of T is defined as L(T) = {S1 · · · Sn : Si ∈ T[i] for all i ∈ [1, n]}. ED strings have been introduced to represent a set of closely-related DNA sequences, also known as a pangenome. The basic question we investigate here is: Given two ED strings, how fast can we check whether the two languages they represent have a nonempty intersection? We call the underlying problem the ED String Intersection (EDSI) problem. For two ED strings T1 and T2 of lengths n1 and n2, cardinalities m1 and m2, and sizes N1 and N2, respectively, we show the following: There is no O((N1N2)1−ϵ)-time algorithm, thus no O ((N1m2 + N2m1)1−ϵ)-time algorithm and no O ((N1n2 + N2n1)1−ϵ)-time algorithm, for any constant ϵ > 0, for EDSI even when T1 and T2 are over a binary alphabet, unless the Strong Exponential-Time Hypothesis is false. There is no combinatorial O((N1 + N2)1.2−ϵf(n1, n2))-time algorithm, for any constant ϵ > 0 and any function f, for EDSI even when T1 and T2 are over a binary alphabet, unless the Boolean Matrix Multiplication conjecture is false. An O(N1 log N1 log n1 + N2 log N2 log n2)-time algorithm for outputting a compact (RLE) representation of the intersection language of two unary ED strings. In the case when T1 and T2 are given in a compact representation, we show that the problem is NP-complete. An O(N1m2 + N2m1)-time algorithm for EDSI. An Õ(N1ω−1n2 + N2ω−1n1)-time algorithm for EDSI, where ω is the exponent of matrix multiplication; the Õ notation suppresses factors that are polylogarithmic in the input size. We also show that the techniques we develop have applications outside of ED string comparison

Path computation in multi-layer networks: Complexity and algorithms

Carrier-grade networks comprise several layers where different protocols coexist. Nowadays, most of these networks have different control planes to manage routing on different layers, leading to a suboptimal use of the network resources and additional operational costs. However, some routers are able to encapsulate, decapsulate and convert protocols and act as a liaison between these layers. A unified control plane would be useful to optimize the use of the network resources and automate the routing configurations. Software-Defined Networking (SDN) based architectures, such as OpenFlow, offer a chance to design such a control plane. One of the most important problems to deal with in this design is the path computation process. Classical path computation algorithms cannot resolve the problem as they do not take into account encapsulations and conversions of protocols. In this paper, we propose algorithms to solve this problem and study several cases: Path computation without bandwidth constraint, under bandwidth constraint and under other Quality of Service constraints. We study the complexity and the scalability of our algorithms and evaluate their performances on real topologies. The results show that they outperform the previous ones proposed in the literature.Comment: IEEE INFOCOM 2016, Apr 2016, San Francisco, United States. To be published in IEEE INFOCOM 2016, \<http://infocom2016.ieee-infocom.org/\&g

Learning Character Strings via Mastermind Queries, with a Case Study Involving mtDNA

We study the degree to which a character string, $Q$, leaks details about itself any time it engages in comparison protocols with a strings provided by a querier, Bob, even if those protocols are cryptographically guaranteed to produce no additional information other than the scores that assess the degree to which $Q$ matches strings offered by Bob. We show that such scenarios allow Bob to play variants of the game of Mastermind with $Q$ so as to learn the complete identity of $Q$. We show that there are a number of efficient implementations for Bob to employ in these Mastermind attacks, depending on knowledge he has about the structure of $Q$, which show how quickly he can determine $Q$. Indeed, we show that Bob can discover $Q$ using a number of rounds of test comparisons that is much smaller than the length of $Q$, under reasonable assumptions regarding the types of scores that are returned by the cryptographic protocols and whether he can use knowledge about the distribution that $Q$ comes from. We also provide the results of a case study we performed on a database of mitochondrial DNA, showing the vulnerability of existing real-world DNA data to the Mastermind attack.Comment: Full version of related paper appearing in IEEE Symposium on Security and Privacy 2009, "The Mastermind Attack on Genomic Data." This version corrects the proofs of what are now Theorems 2 and 4