We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiX i with integer coefficients |ai| < 2 L , this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.
Abstract. We show how to route on very large public transportation networks (up to half a billion arcs) with average query times of a few milliseconds. We take into account many realistic features like: traffic days, walking between stations, queries between geographic locations instead of a source and a target station, and multi-criteria cost functions. Our algorithm is based on two key observations: (1) many shortest paths share the same transfer pattern, i.e., the sequence of stations where a change of vehicle occurs; (2) direct connections without change of vehicle can be looked up quickly. We precompute the respective data; in practice, this can be done in time linear in the network size, at the expense of a small fraction of non-optimal results. We have accelerated public transportation routing on Google Maps with a system based on our ideas. We report experimental results for three data sets of various kinds and sizes.
Abstract. The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial q(x) = q n x n + . . . + q 0 with root separation ρ, coefficients |q n | ≥ 1 and |q i | ≤ 2 τ , it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ) 2 ) bit operations.
Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points where a linear or quadratic function is non-negative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).
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