We study comparison sorting in the evolving data model, introduced by Anagnostopoulos, Kumar, Mah-dian and Upfal (2011), where the true total order changes while the sorting algorithm is processing the input. More precisely, each comparison operation of the algorithm is followed by a sequence of evolution steps, where an evolution step perturbs the rank of a random item by a “small” random value. The goal is to maintain an ordering that remains close to the true order over time. Previous works have analyzed adaptations of classic sorting algorithms, assuming that an evolution step changes the rank of an item by just one, and that a fixed constant number$b$of evolution steps take place between two comparisons. In fact, the only previous result achieving optimal linear total deviation, by Besa Vial, Devanny, Eppstein, Goodrich and Johnson (2018a), applies just for$b=1$. We analyze a very simple sorting algorithm suggested by Mahdian (2014), which samples a random pair of adjacent items in each step and swaps them if they are out of order. We show that the algorithm achieves and maintains, with high probability, optimal total deviation,$O(n)$, and optimal maximum deviation,$O(\log n)$, under very general model settings. Namely, the perturbation introduced by each evolution step is sampled from a general distribution of bounded moment generating function, and we just require that the average number of evolution steps between two sorting steps be bounded by an (arbitrary) constant, where the average is over a linear number of steps. The key ingredients of our proof are a novel potential function argument that inserts “gaps” in the list of items, and a general analysis framework which separates the analysis of sorting from that of the evolution steps, and is applicable to a variety of settings for which previous approaches do not apply. Our results settle conjectures and open problems in the three aforementioned works, and provide theoretical support that simple quadratic algorithms are optimal and robust for sorting evolving data, as empirically observed by Besa Vial, Devanny, Eppstein, Goodrich and Johnson (2018b).

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range (2,3). In particular, we focus on the expected times for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that up to multiplicative constants: the cover time is n(log n)², the maximum hitting time is nlog n, and the average hitting time is n. The first two results hold in expectation and a.a.s. and the last in expectation (with respect to the HRG). We prove these results by determining the effective resistance either between an average vertex and the well-connected "center" of HRGs or between an appropriately chosen collection of extremal vertices. We bound the effective resistance by the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane on which we overlay a forest-like structure.

We show that in the random hyperbolic graph model as formalized by Gugelmann, Panagiotou, and Peter (2012) in the most interesting range of $\frac12 < \alpha < 1$ the size of the second largest component is $\Theta((\log n)^{1/(1-\alpha)})$. Our research is motivated by the question raised by Bode, Fountoulakis, and Müller (2013) regarding the uniqueness of linear size components in random hyperbolic graphs, which naturally leads to the question regarding the size of the second largest component. We also show that for $\alpha=\frac12$ with constant probability the corresponding size is $\Theta(\log n)$, whereas for $\alpha=1$ it is $\Omega(n^{\delta})$ for some $\delta > 0$.

The two most intensively studied communication paradigms for spreading rumors are the so-called PUSH and PULL algorithms. The previous analysis of these protocols assumed that every node could process all such push/pull operations within a single step, which could be unrealistic in practical situations. We propose a new framework for the analysis of rumor spreading accommodating buffers, in which a node can process only few push/pull messages at a time. We develop time complexity upper and lower bounds for randomized rumor spreading in the new framework, and compare the results with analogous ones in the classical setting. Our results highlight that there might be a very significant performance loss if messages are processed at each network node in first-in first-out order.

Motivated by the recent emergence of the so-called opportunistic communication networks, we consider the issue of adaptivity in the most basic continuous time (asynchronous) rumor spreading process. In our setting a rumor has to be spread to a population; the service provider can push it at any time to any node in the network and has unit cost for doing this. On the other hand, as usual in rumor spreading, nodes share the rumor upon meeting and this imposes no cost on the service provider. Rather than fixing a budget on the number of pushes, we consider the cost version of the problem with a fixed deadline and ask for a minimum cost strategy that spreads the rumor to every node. A non-adaptive strategy can only intervene at the beginning and at the end, while an adaptive strategy has full knowledge and intervention capabilities. Our main result is that in the homogeneous case (where every pair of nodes randomly meet at the same rate) the benefit of adaptivity is bounded by a constant. This requires a subtle analysis of the underlying random process that is of interest in its own right. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We formalize the notion and initiate the investigation of approximate testing for arbitrary forms of the error term. Until now only the case of absolute error had been addressed ignoring the fact that often only the most significant figures of a numerical calculation are valid. This work considers approximation errors whose magnitude grows with the size of the input to the program. We demonstrate the viability of this new concept by addressing the basic and benchmark problem of self-- testing for the class of linear and polynomial functions. We obtain stronger versions of results of Ergun, Ravi Kumar, and Rubinfeld [EKR96] by exploiting elegant techniques from Hyers--Ulam stability theory. 1 Introduction The following is a quote from Knuth [Knu98, Ch. 4, x 2.2]: Floating point computation is by nature inexact, and programmers can easily misuse it so that the computed answers consist almost entirely of "noise." One of the principal problems of numerical analysis is to determine how ac...

We study the vulnerability of two implementations of the Data Encryption Standard (DES) cryptosystem under a timing attack. A timing attack is a method, recently proposed by Paul Kocher, that is designed to break cryptographic systems. It exploits the engineering aspects involved in the implementation of cryptosystems and might succeed even against cryptosys-tems that remain impervious to sophisticated cryptanalytic techniques. A timing attack is, essentially, a way of obtaining some users private information by carefully measuring the time it takes the user to carry out cryptographic operations. In this work, we analyze two implementations of DES. We show that a timing attack yields the Hamming weight of the key used by both DES implementations. Moreover, the attack is computationally inexpensive. We also show that all the design characteristics of the target system, necessary to carry out the timing attack, can be inferred from timing measurements.

Let Dist(f,g)=Pr/sub u/[f(u)/spl ne/g(u)] denote the relative distance between functions f,g mapping from a group G to a group H, and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f,g). Given a function f:G/spl rarr/H we let Err(f)=Pr/sub u,/spl upsi//[f(u)+f(/spl upsi/)/spl ne/f(u+/spl upsi/)] denote the rejection probability of the Blum-Luby-Rubinfeld (1993) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular lower bounds on Err(f) in terms of Dist(f). We discuss when the underlying groups are G=GF(2)/sup n/ and H=GF(2). In this case, the collection of linear functions describe a Hadamard code of block length 2/sup n/ and for an arbitrary function f mapping GF(2)/sup n/ to GF(2) the distance Dist(l) measures its distance to a Hadamard code. Err(f) is a parameter that is "easy to measure" and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation. Improved analyses translate into better nonapproximability results. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete in some. We present functions L,U:[0,1]/spl rarr/[0,1] such that for all x /spl isin/ [0,1] we have L(x)/spl les/Err(f)/spl les/U(x) whenever Dist(f)=x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range and close on the rest. Part of our strengthening is obtained by showing a new connection between linearity testing and Fourier analysis.

Let Dist(f,g)=Pr/sub u/ [f(u)/spl ne/g(u)] denote the relative distance between functions f,g mapping from a group G to a group H, and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f,g). Given a function f:G/spl rarr/H we let Err(f)=Pr/sub u/,v[f(u)+f(v)/spl ne/f(u+v)] denote the rejection probability of the BLR (Blum-Luby-Rubinfeld) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular the study of lower bounds on Err(f) in terms of Dist(f). The case we are interested in is when the underlying groups are G=GF(2)/sup n/ and H=GF(2). The corresponding test is used in the construction of efficient PCPs and thence in the derivation of hardness of approximation results, and, in this context, improved analyses translate into better non-approximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete (i.e. tight) in some. In particular we present functions L,U:[0,1]/spl rarr/[0,1] such that for all x/spl isin/[0,1] we have L(x)