Many algorithms and heuristics work well on real data, despite having poor complexity under the standard worst-case measure. Smoothed analysis [36] is a step towards a theory that explains the behavior of algorithms in practice. It is based on the assumption that inputs to algorithms are subject to random perturbation and modification in their formation. A concrete example of such a smoothed analysis is a proof that the simplex algorithm for linear programming usually runs in polynomial time, when its input is subject to modeling or measurement noise. 1. MODELING REAL DATA “My experiences also strongly confirmed my previous opinion that the best theory is inspired by practice and the best practice is inspired by theory. ” [Donald E. Knuth: “Theory and Practice”, Theoretical Computer Science, 90 (1), 1–15, 1991.] Algorithms are high-level descriptions of how computational tasks are performed. Engineers and experimentalists design and implement algorithms, and generally consider them a success if they work in practice. However, an algorithm that works well in one practical domain might perform poorly in another. Theorists also design and analyze algorithms, with the goal of providing provable guarantees about their performance. The traditional goal of theoretical computer science is to prove that an algorithm performs well This material is based upon work supported by the National

Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity. We investigate what happens in between by analysing the smoothed height of binary search trees: Randomly perturb a given (adversarial) sequence and then take the expected height of the binary search tree generated by the resulting sequence. As perturbation models, we consider partial permutations, partial alterations, and partial deletions. On the one hand, we prove tight lower and upper bounds of roughly Θ((1 − p) · n/p) for the expected height of binary search trees under partial permutations and partial alterations, where n is the number of elements and p is the smoothing parameter. This means that worst-case instances are rare and disappear under slight perturbations. On the other hand, we examine how much a perturbation can increase the height of a binary search tree, i.e. how much worse well balanced instances can become.

complex networks. This has been the case since time immemorial. But only recently has technology allowed us to record the structure of these huge networks and use this information in decision making. Hyperlinking web pages together and ranking them based on network structure has already revolutionized the way we manage information (“Just google it”). This is only the beginning. To benefit from our new wealth in networked data, we need to really understand networks. We need accurate measurements, relevant models, and efficient algorithms. This is the focus of my research. It sits at the intersection of mathematics and computer science, and incorporates elements of economics as well. Current work: Network sampling, modeling, algorithms, and economics I use the term network sampling to refer to the process of gathering network data from the natural or artificial environment. The mathematical field of random graph theory was greatly invigorated by the observation of real-world graphs in the late 1990s [25, 6, 3]. In some of these works, such as the analysis of the power grid of the western United States, the network structure is known with little or no error. But in many other cases, we only

Abstract In a load balancing network each processor has an initial collection of unit-size jobs, tokens, and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially effects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis) by starting from an arbitrary assignment of balancer directions, then flipping each assignment with probability α independently. For a large class of balancing networks our result implies that after O(log n) rounds the discrepancy is whp O((1/2−α) log n+log log n). This matches and generalizes the known bounds for α = 0 and α = 1/2. 1