By Kurt Mehlhorn (auth.), Tiziana Calamoneri, Irene Finocchi, Giuseppe F. Italiano (eds.)
This ebook constitutes the refereed complaints of the sixth Italian convention on Algorithms and Computation, CIAC 2006, held in Rome, Italy, in may possibly 2006.
The 33 revised complete papers awarded including three invited papers have been rigorously reviewed and chosen from eighty submissions. one of the issues addressed are sequential, parallel and dispensed algorithms, facts buildings, approximation algorithms, randomized algorithms, online algorithms, graph algorithms, research of algorithms, set of rules engineering, algorithmic video game idea, computational biology, computational complexity, conversation networks, computational geometry, cryptography, discrete optimization, graph drawing, mathematical programming, and quantum algorithms.
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Additional resources for Algorithms and Complexity: 6th Italian Conference, CIAC 2006, Rome, Italy, May 29-31, 2006. Proceedings
The algorithm then checks whether b , e indeed In-Place Randomized Slope Selection 33 contains the k-th smallest intersection point. If this is not the case, the process is repeated for b, e but using a new sample R, else the algorithm iterates with the reﬁned strip b , e . Since the above technique reﬁnes the strip b, e based upon a non-constant number of randomly selected samples, it is referred to as randomized interpolation search, and Matouˇsek observes that it can be used as a randomized substitute for Megiddo’s parametric search technique .
Merging A1 and A2 seems to corrupt the information recorded in DL , but fortunately the information of what goes where can be computed on the ﬂy while running CountAndRecord. To see this, observe that the 2 As noted earlier , the in-place divide-and-conquer scheme can be modiﬁed easily to correctly handle instances where the problem size is not a power of two. In-Place Randomized Slope Selection 37 Algorithm 2. Algorithm CountAndRecord(A1 , A2 , b, e , c, DI ) for incrementing the count c of intersections seen so far by the number of intersections induced by lines in A1 and A2 while recording all relevant intersections in DI .
Since the duality transform is done read-only by simply reinterpreting the coordinate-based representation of a point as the slope-intercept representation of a line, we assume that our input is given as a set P of lines in the plane. Since the arrangement induced by a set of n lines contains Θ(n2 ) intersections, it is infeasible (both with respect to the desired O(n log n) running time and with respect to the in-place setting) to compute all intersections. Instead, the algorithm of Matouˇsek  maintains a vertical strip b, e := [b, e] × IR ⊂ IR2 that is guaranteed to contain the k-th smallest intersection point.