## Abstract

We present a simple randomized algorithm which solves linear programs with time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying

*n*constraints and*d*variables in expected$$\min \{ O(d^2 2^d n),e^{2\sqrt {dIn({n \mathord{\left/ {\vphantom {n {\sqrt d }}} \right. \kern-\nulldelimiterspace} {\sqrt d }})} + O(\sqrt d + Inn)} \}$$

*n*given linear inequalities in*d*variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of$$O(d^2 n + e^{O(\sqrt {dInd} )} ).$$

The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of*n* points in*d*-space, computing the distance of two*n*-vertex (or*n*-facet) polytopes in*d*-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).

## Key words

Computational geometry Combinatorial optimization Linear programming Smallest enclosing ball Smallest enclosing ellipsoid Randomized incremental algorithms## Preview

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