This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point by simply "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure takes expected time close to O(n 1/(d+1)) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.
A method of analyzing time bounds for randomized distributed algorithms is presented, in the context of a new and general framework for describing and reasoning about randomized algorithms. The method consists of proving auxiliary statements of the form U t ! p U 0 , which means that whenever the algorithm begins in a state in set U, with probability p, it will reach a state in set U 0 within time t. The power of the method is illustrated by its use in proving a constant upper bound on the expected time for some process to reach its critical region, in Lehmann and Rabin's Dining Philosophers algorithm.
As increasingly large volumes of sophisticated options are traded in world financial markets, determining a "fair" price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte Carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper we show that pricing an arbitrary path-dependent option is #-P hard. We show that certain types of path-dependent options can be valued exactly in polynomial time. Asian options are pathdependent options that are particularly hard to price. and for these we design deterministic polynomialtime approximate algorithms. We show that the value ofa perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value ofan otherwise identical n-period American put option. In contrast to Monte Carlo methods, our algorithms have guaranteed error bounds that are polynomially small (and in some cases exponentially small) in the maturity n. For the error analysis we derive large-deviation results for random walks that may be of independent interest. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuraq, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, recornmanufacturer, or otherwise does not necessarily constitute or imply its endorsement, mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. ..
This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point by simply "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure takes expected time close to O(n 1/(d+1) ) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.
We consider the following scheduling problem. A system is composed of n processors drawn from a pool of N . The processors can become faulty while in operation and faulty processors never recover. A report is issued whenever a fault occurs. This report states only the existence of a fault, but does not indicate its location. Based on this report, the scheduler can recon gure the system and choose another set of n processors. The system operates satisfactorily as long as at most f of the n selected processors are faulty. We exhibit a scheduling strategy allowing the system to operate satisfactorily until approximately (N=n)f faults are reported in the worst case. Our precise bound is tight.
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