Locality-preserving allocations Problems and coloured Bin Packing

We study the following problem, introduced by Chung et al. in 2006. We are given, online or offline, a set of coloured items of different sizes, and wish to pack them into bins of equal size so that we use few bins in total (at most $\alpha$ times optimal), and that the items of each colour span few bins (at most $\beta$ times optimal). We call such allocations $(\alpha, \beta)$-approximate. As usual in bin packing problems, we allow additive constants and consider $(\alpha,\beta)$ as the asymptotic performance ratios. We prove that for \eps>0, if we desire small $\alpha$, no scheme can beat (1+\eps, \Omega(1/\eps))-approximate allocations and similarly as we desire small $\beta$, no scheme can beat (1.69103, 1+\eps)-approximate allocations. We give offline schemes that come very close to achieving these lower bounds. For the online case, we prove that no scheme can even achieve $(O(1),O(1))$-approximate allocations. However, a small restriction on item sizes permits a simple online scheme that computes (2+\eps, 1.7)-approximate allocations

Approximation Algorithms for Geometric Covering Problems for Disks and Squares

Geometric covering is a well-studied topic in computational geometry. We study three covering problems: Disjoint Unit-Disk Cover, Depth-(≤ K) Packing and Red-Blue Unit-Square Cover. In the Disjoint Unit-Disk Cover problem, we are given a point set and want to cover the maximum number of points using disjoint unit disks. We prove that the problem is NP-complete and give a polynomial-time approximation scheme (PTAS) for it. In Depth-(≤ K) Packing for Arbitrary-Size Disks/Squares, we are given a set of arbitrary-size disks/squares, and want to find a subset with depth at most K and maximizing the total area. We prove a depth reduction theorem and present a PTAS. In Red-Blue Unit-Square Cover, we are given a red point set, a blue point set and a set of unit squares, and want to find a subset of unit squares to cover all the blue points and the minimum number of red points. We prove that the problem is NP-hard, and give a PTAS for it. A "mod-one" trick we introduce can be applied to several other covering problems on unit squares

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.Peer reviewe

The effects of packing structure on the effective thermal conductivity of granular media: A grain scale investigation

Structural characteristics are considered to be the dominant factors in determining the effective properties of granular media, particularly in the scope of transport phenomena. Towards improved heat management, thermal transport in granular media requires an improved fundamental understanding. In this study, the effects of packing structure on heat transfer in granular media are evaluated at macro- and grain-scales. At the grain-scale, a gas-solid coupling heat transfer model is adapted into a discrete-element-method to simulate this transport phenomenon. The numerical framework is validated by experimental data obtained using a plane source technique, and the Smoluschowski effect of the gas phase is found to be captured by this extension. By considering packings of spherical SiO2 grains with an interstitial helium phase, vibration induced ordering in granular media is studied, using the simulation methods developed here, to investigate how disorder-to-order transitions of packing structure enhance effective thermal conductivity. Grain-scale thermal transport is shown to be influenced by the local neighbourhood configuration of individual grains. The formation of an ordered packing structure enhances both global and local thermal transport. This study provides a structure approach to explain transport phenomena, which can be applied in properties modification for granular media.Comment: 11 figures, 29 page

Engineering Pathways of Polymersome Production and Characterisation

Polymeric vesicles, also known as polymersomes, have been widely studied for their use in biological applications and have shown great potential for their application as delivery systems. Current polymersome production methods usually require a long period of time to produce highly monodisperse samples. Therefore, there is an unmet need for a process able to speed up the production of polymersomes. In this work, an engineered top-down process based on the hydration of polymer coated glass beads is presented, along with the computation of particle molecular parameters and their TEM characterisation, based on the staining agent brightness intensity. Polymersomes prepared by the beads hydration method were characterised by TEM and DLS techniques, showing a high vesicle throughput with a narrow average diameter in just a few hours, compared to the 4-6 weeks of hydration needed for alternative hydration techniques. Particle characterisation was carried out with an asymmetric flow fi eld-flow fractionation (AF4) system on PMPC25-b-PDPAn (n=43,68,100) self-assemblies. AF4 allowed the simultaneous particle analysis by DLS and SLS, making it feasible to infer particle morphology and the computation of the block copolymer packing factor, by applying the corresponding geometrical model. For the shortest PDPA chain (n=43), AF4 suggested a disk-like micelle morphology and a 0.4 packing factor whilst for the remaining PDPA chain lengths (n=68,100) AF4 indicated the presence of both disk-like micelles and vesicle-like morphologies with a packing factor value characteristic of bilayer assemblies (0.5). Finally, a TEM brightness intensity is described with the brightness intensity of the staining agent used during the TEM grids preparation. Due to the bilayer comprised within a vesicle, the number of MPC monomers is greater, compared to a disk micelle, therefore the contrast produced by the staining agent would be more signi cant, making it possible to differentiate disks from vesicles. TEM intensity analysis con firmed the presence of only one particle morphology for PMPC25-b-PDPA43, as suggested by AF4, and showed a correlation between size and morphology for PMPC25-b-PDPA68 and PMPC25-b-PDPA100. In conclusion, a novel and more effective method for the production of polymersomes is described, which decreases the waiting time for vesicle formation. AF4 system validated the growth of disk-like micelles into vesicles for block copolymers with a packing factor value of 0.5. Moreover, a TEM analysis based on particle brightness intensity aided in the differentiation of particle morphology, considering that disk micelles can show a wide range of sizes which makes it challenging to categorise them in a common TEM micrograph

Packing Unit Disks

Given 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