By Mahmuda Ahmed, Iffat Chowdhury, Matt Gibson (auth.), Frank Dehne, Roberto Solis-Oba, Jörg-Rüdiger Sack (eds.)
This e-book constitutes the refereed lawsuits of the thirteenth Algorithms and knowledge constructions Symposium, WADS 2013, held in London, ON, Canada, August 2013. The Algorithms and knowledge buildings Symposium - WADS (formerly "Workshop on Algorithms and information Structures") is meant as a discussion board for researchers within the sector of layout and research of algorithms and knowledge constructions. The forty four revised complete papers offered during this quantity have been conscientiously reviewed and chosen from 139 submissions. The papers current unique study on algorithms and knowledge buildings in all components, together with bioinformatics, combinatorics, computational geometry, databases, pictures, and parallel and disbursed computing.
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Additional resources for Algorithms and Data Structures: 13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings
Repeat with the remainder of ri , and continue until all rectangles have been processed. In the case of 2SP-SSC, the stacking constraint must be respected when placing slices. See Figure 1. H 6 r3 5 4 3 r2 r3 2 F r3 1 r1 0 0 1 2 3 4 Fig. 1. An execution of the First Fit algorithm on a 2SP-SSC instance. Note that r3 is sliced twice, and a smaller height would be achieved without the stacking constraint. It is not hard to show that after placing each rectangle, the diﬀerence between the maximum height H and the ﬂoor F (the maximum height to which the entire strip is ﬁlled) is at most hmax .
Putting all of this together and using hmax ≤ HOPT , the left strip has height at most AL AL AL hmax + − t + ≤ HOPT + − t + HOPT − t = 2HOPT − 2t + . αW αW αW In the right strip, all rectangles have height at most t. Hence the right strip has at most t(1 − α)W empty area, and its height is at most 5 AR (1 − α)HOPT W +t≤ + t = HOPT + t ≤ HOPT (1 − α)W (1 − α)W 3 by choice of t. We have chosen the partition into left and right pieces carefully to ensure that AL is “just right”. In the ﬁrst case, AR = (1 − α)HOPT W , which implies that AL ≤ αHOPT W .
Then the left side of the strip in 0 and thus must contain a total area of S ∗ must contain at least the pieces in RL 0 0 at least AL . It follows that AL ≤ HOPT αW , and hence the claim holds. A0 L With this claim, the left strip has height 2HOPT − 2t + αW ≤ 3HOPT − 2t ≤ 5 H by choice of t, and we have now proved the approximation bound. OPT 3 Smart-Grid Electricity Allocation via Strip Packing with Slicing 35 Lastly, we remove the assumption that the value of HOPT is given. By replacing HOPT with a user-supplied value HGUESS in the algorithm, it is easy to check that the algorithm has the following behavior: if HGUESS ≥ HOPT , the solution returned has height at most (5/3)HGUESS .
Algorithms and Data Structures: 13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings by Mahmuda Ahmed, Iffat Chowdhury, Matt Gibson (auth.), Frank Dehne, Roberto Solis-Oba, Jörg-Rüdiger Sack (eds.)