Topological Complexity of the Range Searching

نویسنده

  • Dima Grigoriev
چکیده

We prove an existence of a topological decision tree which solves the range searching problem for a system of real polynomials, in other words, the tree finds all feasible signs vectors of these polynomials, with the (topological) complexity logarithmic in the number of signs vectors. This answers the problem posed in [FK98]. 1 Range Searching Problem Let polynomials f1, . . . , fm ∈ R[X1, . . . , Xn]. Our purpose is to solve the range searching problem [FK98] by means of topological decision trees (TDT) [S87]. Namely, TDT allows tests of the form ”P (x) > 0?” for arbitrary polynomials P ∈ R[X1, . . . , Xn] (thus, we ignore the cost of the computations). We say that a TDT solves the range searching problem for the polynomials f1, . . . , fm if any two input points x, y ∈ R n with different signs vectors (sgn(f1), . . . , sgn(fm))(x) 6= (sgn(f1), . . . , sgn(fm))(y) arrive to different leaves of the TDT. As usual, sgn could attain three values. By the topological complexity of a TDT we mean its depth. Denote by N the number of all feasible signs vectors (sgn(f1), . . . , sgn(fm))(x). It is well known (see e.g. [G88]) that N ≤ (md) where deg(fi) ≤ d, 1 ≤ i ≤ n, or in [BPR96] a better bound N ≤ ((2 + ( m n ) )d). The following result answers the problem posed in section 4.2 [FK98].

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عنوان ژورنال:
  • J. Complexity

دوره 16  شماره 

صفحات  -

تاریخ انتشار 2000