A Linear - Time Algorithm That Locates

The problem of locating local maxima and minima of a function from approximate measurement results is vital for many physical applications: In spectral analysis, chemical species are identiied by locating local maxima of the spectra. In radioastronomy, sources of celestial radio emission, and their subcomponents, are identiied by locating local maxima of the measured brightness of the radio sky. Elementary particles are identiied by locating local maxima of the experimental curves. i = y i + ". The set F of all the functions f(x) that satisfy this property can be considered as a function interval (this deenition was, in essence, rst proposed by R. Moore). We say that an interval I locates a local maximum if all functions f 2 F attain a local maximum at some point from I. So, the problem is to generate intervals I 1 ; :::; I k that locate local maxima. Evidently, if I locates a local maximum, then any bigger interval J I also locates this maximum. We want to nd the smallest possible location I. We (Charlottesville, VA) for discussing possible astronomical applications, and to anonymous referees for important improvements. 1 propose an algorithm that nds the smallest possible locations in linear time (i.e., in time that is Cn for some C). 1 Why is This Problem Important for Applications The problem is really important. In many applications, the following problem arises: we know that a physical quantity y is a function of some other physical quantity x, y = f(x), and we want to know the local maxima of this function f. In spectral analysis, chemical species are identiied by locating local maxima of the spectra. In radioastronomy, sources of celestial radio emission and their subcompo-nents are identiied by locating local maxima of the measured brightness of the radio sky. In other words, we are interested in the local maxima of the brightness distribution, i.e., of the function y(x) that describes how the intensity y of the signal depends on the position x of the point from which we receive this signal. Elementary particles are identiied by locating local maxima of the experimental curves that describe (crudely speaking) the scattering intensity y as a function of energy x. Local maxima and minima are also used in the methods that accelerate the convergence of the measurement result to the real value of a physical variable, and thus allow the user to …