k-means is a simple and flexible clustering algorithm that has remained in common use for 50+ years. In this thesis, we discuss the algorithm in general, its advantages, weaknesses and how its ability to locate clusters can be enhanced with a suitable initialization method. We formulate appropriate requirements for the (batched) UnifRandom, k-means++ and Kaufman initialization methods and compare their performance on real and generated data through simulations. We find that all three methods (followed by the k-means procedure) are able to accurately locate at least up to nine well-separated clusters, but the appropriately batched UnifRandom and the Kaufman methods are both significantly more computationally expensive than the k-means++ method already for K = 5 clusters in a dataset of N = 1000 points.

A packet buffer for the protocol processor is a large memory space that holds incoming data packets for an application. Data packets for each application are stored in the form of FIFO queues in the packet buffer. Packets are dropped when the buffer is full. An efficient buffer management algorithm is required to manage the buffer space among the different FIFO queues and to avoid heavy packet loss. This thesis develops a simulation model for the packet buffer and studies the performance of conventional buffer management algorithms when applied to packet buffer. This thesis proposes a new buffer management algorithm, Dynamic Algorithm with Different Thresholds (DADT) to improve the packet loss ratio. This algorithm takes advantage of the different packet sizes for each application and proportionally allocates buffer space for each queue. The performance of the DADT algorithm is dependent upon the packet size distribution in a network traffic load. Three different network traffic loads are considered for our simulations. For the average network traffic load, the DADT algorithm shows an improvement of 6.7 % in packet loss ratio over the conventional dynamic buffer management algorithm. For the high and actual network traffic loads, the DADT algorithm shows an improvement of 5.45 % and 3.6 % in packet loss ratio respectively. Based on the simulation results, the DADT algorithm outperforms the conventional buffer management algorithms for various network traffic loads.

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<p>We investigate the function ρ_<em>d</em>(<em>n</em>) = max { <em>r</em>(<em><strong>x</strong></em>) | <em><strong>x</strong></em> is a (<em>d</em>, <em>n</em>)-string } where <em>r</em>(<em><strong>x</strong></em>) is the number of runs in the string <em><strong>x</strong></em>, and a (<em>d</em>, <em>n</em>)-string is a string with length <em>n</em> and exactly <em>d</em> distinct symbols. Our investigation is motivated by the conjecture that ρ_<em>d</em>(<em>n</em>) ≤ <em>n</em>-<em>d</em>. We present and discuss fundamental properties of the ρ_<em>d</em>(<em>n</em>) function. The values of ρ_<em>d</em>(<em>n</em>) are presented in the (<em>d</em>, <em>n</em>-<em>d</em>)-table with rows indexed by <em>d</em> and columns indexed by <em>n</em>-<em>d</em> which reveals the regularities of the function. We introduce the concepts of the r-cover and core vector of a string, yielding a novel computational framework for determining ρ_<em>d</em>(<em>n</em>) values. The computation of the previously intractable instances is achieved via first computing a lower bound, and then using the structural properties to limit our exhaustive search only to strings that can possibly exceed this number of runs. Using this approach, we extended the known maximum number of runs in binary string from 60 to 74. In doing so, we find the first examples of run-maximal strings containing four consecutive identical symbols. Our framework is also applied for an arbitrary number of distinct symbols, <em>d</em>. For example, we are able to determine that the maximum number of runs in a string with 23 distinct symbols and length 46 is 23. Further, we discuss the structural properties of a shortest (<em>d</em>, <em>n</em>)-string <em><strong>x</strong></em> such that <em>r</em>(<em><strong>x</strong></em>) > <em>n</em>-<em>d</em>, should such a string exist.</p> / Doctor of Philosophy (PhD)

Let F be a free group of finite rank. Given words u, v ∈ F, J.H.C. Whitehead solved the decision problem of finding an automorphism φ ∈ Aut(F), carrying u to v. He used topological methods to produce an algorithm. Higgins and Lyndon gave a very concise proof of the problem based on the works of Rapaport. We provide a detailed account of Higgins and Lyndon’s proof of the peak reduction lemma and the restricted version of Whitehead’s theorem, for cyclic words as well as for sets of cyclic words, with a full explanation of each step. Then, we give an inductive proof of Whitehead’s minimization theorem and describe Whitehead’s decision algorithm. Noticing that Higgins and Lyndon’s work is limited to the cyclic words, we extend their proofs to ordinary words and sets of ordinary words. In the last chapter, we mention an example given by Whitehead to show that the decision problem for finitely generated subgroups is more difficult and outline an approach due to Gersten to overcome this difficulty. We also give an extensive literature survey of Whitehead’s algorithm / Thesis / Master of Science (MSc)