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On the Power of Discontinuous Approximate Computations 1 Karl Aberer Bruno Codenotti

Summary: On the Power of Discontinuous Approximate Computations 1
Karl Aberer Bruno Codenotti
International Computer Science Institute Istituto di Elaborazione dell'Informazione
1947 Center Street, Suite 600, Berkeley, CA 94704 Consiglio Nazionale delle Ricerche, Pisa, Italy
email: aberer@icsi.berkeley.edu email: codenotti@iei.pi.cnr.it
Abstract. Comparision operations are used in algebraic computations to avoid degeneracies, but are
also used in numerical computations to avoid huge roundo errors. On the other hand, the classes of
algorithmsusing only arithmetic operations are the most studied in complexitytheory, and are used, e.g.,
to obtain fast parallel algorithms for numerical problems. In this paper, we study, by using a simulation
argument, the relative power of di erent sets of operations for computing with approximations. We
prove that comparisions can be simulated e ciently and with the same error bounds for most inputs
by arithmetic operations when divisions are present. To develop our simulation strategy we combine
notions imported from approximation theory and topology with complexity and error bounds.
1 Introduction
In this paper we show that approximate computations over the reals do not su er very much
from the lack of conditional statements, provided that division is allowed. On the other hand
we show that the set of operations without division is very poor, if one wants to achieve the
goal of keeping the roundo error small.
The problem studied in this paper has relations with several di erent research elds, and we
attack it by using techniques and ideas related , e.g., to algebraic complexity tools used to re-


Source: Aberer, Karl - Faculté Informatique et Communications, Ecole Polytechnique Fédérale de Lausanne


Collections: Computer Technologies and Information Sciences