Randomized search heuristics like simulated annealing and evolutionary algorithms are applied successfully in many different situations. However, the theory on these algorithms is still in its infancy. Here it is discussed how and why such a theory should be developed. Afterwards, some fundamental results on evolutionary algorithms are presented in order to show how theoretical results on randomized search heuristics can be proved and how they contribute to the understanding of evolutionary algorithms.

Many experiments have shown that evolutionary algorithms are useful randomized search heuristics for optimization problems. In order to learn more about the reasons for their e#ciency and in order to obtain proven results on evolutionary algorithms it is necessary to develop a theory of evolutionary algorithms. Such a theory is still in its infancy. A major part of a theory is the analysis of di#erent variants of evolutionary algorithms on selected functions. Several results of this kind have been obtained during the last years. Here important analytical tools are presented, discussed, and applied to well-chosen example functions.

As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. This paper presents a probabilistic language, called λ○, whose expressive power is beyond discrete distributions. Rich expressiveness of λ ○ is due to its use of sampling functions, i.e., mappings from the unit interval (0.0, 1.0] to probability domains, in specifying probability distributions. As such, λ ○ enables programmers to formally express and reason about sampling methods developed in simulation theory. The use of λ ○ is demonstrated with three applications in robotics: robot localization, people tracking, and robotic mapping. All experiments have been carried out with real robots.

Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is Θ ( 1 / √ n log n). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair. In most applications delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this trade-off for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay trade-off is given by D(n) = Θ(nT (n)), where T (n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of Θ(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be Θ(n log n). The optimal throughput-delay trade-off for mobile networks is also established. To capture physical movement in the real world, a random walk model for node mobility is assumed. It is shown that for throughput of O ( 1 / √ n log n) , which can also be achieved in static networks, the throughput-delay trade-off is the same as in static networks, i.e., D(n) = Θ(nT (n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be Θ(n log n), which is the delay for throughput of Θ(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.

Evolutionary and genetic algorithms (EAs and GAs) are quite successful randomized function optimizers. This success is mainly based on the interaction of di#erent operators like selection, mutation, and crossover. Since this interaction is still not well understood, one is interested in the analysis of the single operators. Jansen and Wegener (2001a) have described so-called real royal road functions where simple steady-state GAs have a polynomial expected optimization time while the success probability of mutation-based EAs is exponentially small even after an exponential number of steps. This success of the GA is based on the crossover operator and a population whose size is moderately increasing with the dimension of the search space. Here new real royal road functions are presented where crossover leads to a small optimization time, although the GA works with the smallest possible population size --- namely 2.

this paper appeared in "Proceedings of the 32nd Annual ACM Symposium on Theory of Computing," Portland, OR, May 2000. 2 Current affiliation: Department of Computer Science, Princeton University, Princeton, NJ 08544. Most of this work was done while the author was at Stanford University and was visiting IBM Almaden Research Center. Research at Stanford was supported by the Pierre and Christine Lamond Fellowship, NSF Grant IIS-9811904.. and NSF Award CCR-9357849, with matching funds from. IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. 3 Most of this work was done while the Ruthor was visiting the IBM Almaden Research Center. 4 Supported in part by a David and Lucre Packard Foundation Fellowship, an A/fred P. Sloan Research Fellowship, an ONR Young Investigator Award, and NSF Faculty Early Career Development Award CCR-9701399

this paper is organized as follows. In Section 2 we consider the hypercube segmentation problem, describe our randomized approximation algorithm, and present is derandomization. In Section 3 we present

In evaluating an algorithm, worst-case analysis can be overly pessimistic.

. Evolutionary algorithms are randomized search heuristics whose general variants have been successfully applied in black box optimization. In this scenario the function f to be optimized is not known in advance and knowledge on f can be obtained only by sampling search points a revealing the value of f(a). In order to analyze the behavior of di#erent variants of evolutionary algorithms on certain functions f , the expected runtime until some optimal search point is sampled and the success probability, i.e., the probability that an optimal search point is among the first sampled points, are of particular interest. Here a simple method for the analysis is discussed and applied to several functions. For specific situations more involved techniques are necessary. Two such results are presented. First, it is shown that the most simple evolutionary algorithm optimizes each pseudo-boolean linear function in an expected time of O(n log n). Second, an example is shown where crosso...

Abstract. We analyze “coin-wallet ” and “balance-wallet ” under partial real-time audit, and compute upper bounds on theft due to the fact that not all the transactions are audited in real time, assuming that everything else is perfect. In particular, we assume that the audit regime holds for innocent payees. Let v be the maximum allowed balance in a wallet, and 0 � µ � 1 be the fraction of transactions that are audited in real time in an audit round. Assume one unit transactions. We show that the upper bound on expected theft for coin-wallet is limµ→0 µ −2, while for plausible (similar) parameter choice the bound for a balance-wallet is O(exp(mvµ)), where1<m. The former is nicely bounded for small transactions, however, the bound for balance-wallet can become huge in those cases where we require very small false alarm probability. We conclude that partial audit, may be suitable for coin-wallets with low denomination coins, and possibly for balance-wallet, when we may tolerate a relatively high false alarm rate, but it may be too risky for balance-wallet, where very low false alarm rate is required.