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dc.contributor.authorLafreniere, Benjamin J. 19:19:50 (GMT) 19:19:50 (GMT)
dc.description.abstractGiven a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation.en
dc.publisherUniversity of Waterlooen
dc.subjectcomputational geometryen
dc.subjectdisk packingen
dc.subjectmaximum independent seten
dc.subjectlower boundsen
dc.subjectdiscrete geometryen
dc.subjectdisk intersection graphsen
dc.titlePacking Unit Disksen
dc.typeMaster Thesisen
dc.subject.programComputer Scienceen of Computer Scienceen
uws-etd.degreeMaster of Mathematicsen

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