Algorithm Theory – SWAT 2014: 14th Scandinavian Symposium by Inge Li Gørtz, R. Ravi

By Inge Li Gørtz, R. Ravi

This ebook constitutes the refereed lawsuits of the 14th overseas Scandinavian Symposium and Workshops on set of rules idea, SWAT 2014, held in Copenhagen, Denmark, in July 2014. The 33 papers have been conscientiously reviewed and chosen from a complete of 134 submissions. The papers current unique learn and canopy quite a lot of themes within the box of layout and research of algorithms and knowledge buildings together with yet now not restricted to approximation algorithms, parameterized algorithms, computational biology, computational geometry and topology, allotted algorithms, external-memory algorithms, exponential algorithms, graph algorithms, on-line algorithms, optimization algorithms, randomized algorithms, streaming algorithms, string algorithms, sublinear algorithms and algorithmic online game theory.

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Extra info for Algorithm Theory – SWAT 2014: 14th Scandinavian Symposium and Workshops, Copenhagen, Denmark, July 2-4, 2014. Proceedings

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For any vector u = (u1 , . . , u2l−1 ) ∈ U , we define a target configuration c(u) that contains ui (m − μ)/(2l − 1) class-i machines, for i = 1, . . , 2l − 1, provided that 2l−1 i=1 ui (m − μ)/(2l − 1) does not exceed μ. More specifically, for any i u = (u1 , . . , u2l−1 ) ∈ U , let π0 = 0 and πi = j=1 uj (m − μ)/(2l − 1) , be the partial sums of the first i entries of u, multiplied by (m − μ)/(2l − 1) , for i = 1, . . , 2l − 1. Let μ = π2l−1 . First construct a vector c (u) = (c1 , . . , cμ ) of length μ that contains exactly ui (m− μ)/(2l − 1) class-i machines.

In Section 4 we give the details of implementing the algorithm on a RAM and in Section 5 we present the packed sorting algorithm. Finally, in Section 6 we discuss how to adapt our algorithm to work with an arbitrary word size. 2 Algorithm In this section we give a high level description of the algorithm. The input is n words x1 , x2 , . . , xn , each containing a w-bit integer from U = {0, 1, . . , 2w − 1}. We assume the elements are distinct. Otherwise we can ensure this by hashing the elements into buckets in expected O(n) time and only sorting a reduced input with one element from each bucket.

1. Example of how nodes are introduced and how they disappear from detail i to i + 1. The bits that are marked by a dotted circle are omitted in the recursion. node when characters are 2 bits because “00” are the first bits on both edges below it. Thus the “00” bits below a should not appear in the next recursion – this is captured by Invariant iii. A similar situation happens at the node b, however since there are two different 2-bit strings below it, we get the inherited node binh . At the node c we see that the order among its edges is determined by the first two bits, thus the last two bits can be discarded.

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