Approximation of Staircases by Staircases

| The simplest nontrivial monotone functions are \staircases." The problem arises: what is the best approximation of some monotone function f(x) by a staircase with M jumps? In particular: what if f(x) is itself a staircase with N, N > M, steps? This paper considers algorithms for solving, and theorems relating to, this problem. All of the algorithms we propose are space-optimal up to a constant factor and and also runtime-optimal except for at most a logarithmic factor. One application of our results is to \data compression" of probability distributions. We nd yet another remarkable property of Monge's inequality, called the \concave cost as a function of zigzag number theorem." This property leads to new ways to get speedups in certain 1-dimensional dynamic programming problems satisfying this inequality. Deenition 1 of a monotone function] A real-valued function f(x) of x, deened for x real, is monotone if a b implies f(a) f(b). Deenition 2 of a staircase function] A \staircase with M jumps," M 2 f0; 1; 2; : ::g, is a monotone function of x, deened for 0 x 1, such that f(x) assumes exactly M + 1 distinct values. We have restricted ourselves to the interval 0; 1] for convenience, but naturally everything we say is easily translated to any other interval. One of the most important ways in which monotone functions arise is as cumulative distribution functions (\CDFs") of probability distributions. One of the most important ways in which N-jump staircase functions arise is as the approximations to unknown CDFs gotten by sampling N times. It is then often desirable to have the probability distribution around so that one can use it, but impractical to keep all the (huge amount of) sampling data. This leads to the question of how to \compress" an N-jump staircase CDF into an M-jump staircase CDF with M < N. This is the fundamental problem (cf. deenition 10) considered in this paper. Another way in which the N-staircase by M-staircase approximation problem arises is when one wants to nd the best staircase approximation of any monotone function f(x). If nothing in particular is known about f(x), except that a black box is available which will input x and output f(x), then we might as well pretend that f(x) is a staircase which ts the values f(x i) at whatever x i we have asked the black box. Indeed, there are two \extreme" …