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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 12 Apr 2024 15:58:35 +0000Router Congestion Control
https://resolver.caltech.edu/CaltechETD:etd-06142004-161237
Authors: {'items': [{'id': 'Gao-Xiaojie', 'name': {'family': 'Gao', 'given': 'Xiaojie'}, 'show_email': 'NO'}]}
Year: 2004
DOI: 10.7907/VZ4D-4047
<p>Congestion is a natural phenomenon in any network queuing system, and is unavoidable if the queuing system is operated at capacity. In this thesis, we study how to set the rules of a queuing system so that all the users have a self interest in controlling congestion when it happens.</p>
<p>Queueing system is a crucial component in effective router congestion control since it determines the way packets from different sources interact with each other. If packets are dropped by the queueing system indiscriminately, in some cases, the effect can be to encourage senders to actually increase their transmission rates, worsening the congestion, and destabilizing the system.</p>
<p>We approach this problem from game theory. We look on each flow as a competing player in the game; each player is trying to get as much bandwidth as possible. Our task is to design a game at the router that will protect low-volume flows and punish high-volume ones. Because of the punishment, being high-volume will be counter productive, so flows will tend to use a responsive protocol as their transport-layer protocol. The key aspect of our solution is that by sending no packets from high-volume flows in case of congestion, it gives these flows an incentive to use a more responsive protocol.</p>
<p>In the thesis, we will describe several implementations of our solution, and show that we achieve the desired game-theoretic equilibrium while also maintaining bounded queue lengths and responding to changes in network flow conditions. Finally, we accompany the theoretical analysis with network simulations under a variety of conditions.</p>
https://thesis.library.caltech.edu/id/eprint/2582On A Capacitated Multivehicle Routing Problem
https://resolver.caltech.edu/CaltechETD:etd-10292007-212511
Authors: {'items': [{'id': 'Gao-Xiaojie', 'name': {'family': 'Gao', 'given': 'Xiaojie'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/3Y9X-EW47
<p>The Vehicle Routing Problem (VRP) is a discrete optimization problem with high industrial relevance and high computational complexity. The problem has been extensively studied since it was introduced by Dantzig and Ramser. In a VRP, we are given a number of customers with known delivery requirements and locations (assumed to be vertices in a network). A fleet of vehicles with limited capacity is available. The objective is to design routes and customer assignments to minimize the total time or distance traveled to serve the demands. Because of its practical significance, this problem has been widely studied.</p>
<p>In this thesis, we present a version of the VRP motivated by mobile sensor networks which we call the Capacitated Multivehicle Routing Problem (CMVRP). In our framework, there are multiple geographically disperse vehicles each equipped with a limited energy supply. The vehicle consumes energy as it moves around and it also consumes energy while serving jobs. This situation models a network of mobile sensors where locomotion and computation all drain the limited capacity battery onboard. Our objective is to determine the minimum amount of energy required to serve all jobs, which takes into account both the service requirement and the travel overhead. We present a constant factor approximation algorithm. Furthermore, we study the on-line problem where job demands arrive sequentially and present a distributed algorithm that serves all jobs using only a constant factor more energy than the off-line solution.</p>https://thesis.library.caltech.edu/id/eprint/4307On Matrix Factorization and Scheduling for Finite-Time Average-Consensus
https://resolver.caltech.edu/CaltechTHESIS:05022010-193157687
Authors: {'items': [{'email': 'kokevin@gmail.com', 'id': 'Ko-Chih-Kai', 'name': {'family': 'Ko', 'given': 'Chih-Kai'}, 'show_email': 'NO'}]}
Year: 2010
DOI: 10.7907/GCT7-5Y66
We study the problem of communication scheduling for finite-time average-consensus in arbitrary connected networks. Viewing this consensus problem as a factorization of 1/n 11<sup>T</sup> by network-admissible families of matrices, we prove the existence of finite factorizations, provide scheduling algorithms for finite-time average consensus, and derive almost tight lower bounds on the size of the minimal factorization.https://thesis.library.caltech.edu/id/eprint/5763Clustering Affine Subspaces: Algorithms and Hardness
https://resolver.caltech.edu/CaltechTHESIS:07052012-191337554
Authors: {'items': [{'email': 'luw0315@gmail.com', 'id': 'Lee-Euiwoong', 'name': {'family': 'Lee', 'given': 'Euiwoong'}, 'show_email': 'NO'}]}
Year: 2012
DOI: 10.7907/VF38-NT60
<p>We study a generalization of the famous k-center problem where each object is an affine subspace of dimension Δ, and give either the first or significantly improved algorithms and hardness results for many combinations of parameters. This generalization from points (Δ=0) is motivated by the analysis of incomplete data, a pervasive challenge in statistics: incomplete data objects in R<sup>d</sup> can be modeled as affine subspaces. We give three algorithmic results for different values of k, under the assumption that all subspaces are axis-parallel, the main case of interest because of the correspondence to missing entries in data tables.<br />
1) k=1: Two polynomial time approximation schemes which runs in poly(Δ, 1/ε)nd.<br />
2) k=2: O(Δ<sup>1/4</sup>)-approximation algorithm which runs in poly(n,d,Δ)<br />
3) General k: Polynomial time approximation scheme which runs in 2<sup>O(Δk log k(1+1/ε<sup>2</sup>))</sup>nd</p>
<p>
We also prove nearly matching hardness results; in both the general (not necessarily axis-parallel) case (for k ≥ 2) and in the axis-parallel case (for k ≥ 3), the running time of an approximation algorithm with any approximation ratio cannot be polynomial in even one of k and Δ, unless P = NP. Furthermore, assuming that the 3-SAT problem cannot be solved subexponentially, the dependence on both k and Δ must be exponential in the general case (in the axis-parallel case, only the dependence on k drops to 2<sup>Ω√k)</sup>). The simplicity of the first and the third algorithm suggests that they might be actually used in statistical applications. The second algorithm, which demonstrates a theoretical gap between the axis-parallel and general case for k=2, displays a strong connection between geometric clustering and classical coloring problems on graphs and hypergraphs, via a new Helly-type theorem.</p>https://thesis.library.caltech.edu/id/eprint/7171A Nearly-Quadratic Gap Between Adaptive and Non-Adaptive Property Testers
https://resolver.caltech.edu/CaltechTHESIS:11302011-091414252
Authors: {'items': [{'email': 'jhurwitz@cs.caltech.edu', 'id': 'Hurwitz-Jeremy-Scott', 'name': {'family': 'Hurwitz', 'given': 'Jeremy Scott'}, 'show_email': 'NO'}]}
Year: 2012
DOI: 10.7907/W178-HP57
<p>We show that for all integers t ≥ 8 and arbitrarily small ε > 0, there exists a graph property Π (which depends on ε) such that ε-testing Π has non-adaptive query complexity Q = Θ(q<sup>2-2/t</sup>), where q = Õ(ε<sup>-1</sup>) is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties ([GR07]). This also gives evidence that the canonical transformation of Goldreich and Trevisan ([GT03]) is essentially optimal when converting an adaptive property tester into a non-adaptive property tester.</p>
<p>To do so, we consider the property of being decomposable into a disjoint union of subgraphs, each of which is a (possibly unbalanced) blow-up of a given base-graph H. In [GR09], Goldreich and Ron proved that when H is a simple t-cycle, the non-adaptive query complexity is Ω(ε<sup>-2+2/t</sup>, even under the promise that G has maximum degree O(εN). In this thesis, we prove a matching upper bound for the non-adaptive complexity and a tight (up to a polylogarithmic factor) upper bound on the adaptive complexity.</p>
<p>Specifically, we show that for all H, testing whether G is a collection of blow-ups of H and has maximum degree O(εN) requires only O(ε<sup>-1</sup>lg<sup>3</sup>ε<sup>-1</sup>) adaptive queries or O(ε<sup>-2+1/(δ+2)</sup>+ε<sup>-2+2/W</sup>) non-adaptive queries, where δ = Δ(H) is the maximum degree of H and W< |H|<sup>2</sup> is a bound on the size of witnesses against H.</p>https://thesis.library.caltech.edu/id/eprint/6741Behavior of O(log n) Local Commuting Hamiltonians
https://resolver.caltech.edu/CaltechTHESIS:05272016-141059835
Authors: {'items': [{'email': 'jenishc@gmail.com', 'id': 'Mehta-Jenish-C', 'name': {'family': 'Mehta', 'given': 'Jenish C.'}, 'show_email': 'YES'}]}
Year: 2016
DOI: 10.7907/Z9V98611
<p>We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H over n qubits, where H is a sum of k-local terms acting non-trivially on O(log n) qubits, and all the k-local terms commute, and show the following - </p>
<p>1. We show that a specific case of O(log n) local commuting hamiltonians over the hypercube is in NP using the Bravyi-Vyalyi Structure theorem.</p>
<p>2. We give a simple proof of a generalized area law for commuting hamiltonians (which seems to be a folklore result) in all dimensions, and deduce the case for O(log n) local commuting hamiltonians.</p>
<p>3. We show that traversing the ground space of O(log n) local commuting hamiltonians is QCMA complete.</p>
<p>The first two behaviours seem to indicate that deciding whether the ground space energy of O(log n)-local commuting hamiltonians is low or high might be in NP or possibly QCMA, though the last behaviour seems to indicate that it may indeed be the case that O(log n)-local commuting hamiltonians are QMA complete. </p>https://thesis.library.caltech.edu/id/eprint/9792Combinatorial and Algebraic Propeties of Nonnegative Matrices
https://resolver.caltech.edu/CaltechTHESIS:06062022-043503154
Authors: {'items': [{'email': 'jenishc@gmail.com', 'id': 'Mehta-Jenish-Chetan', 'name': {'family': 'Mehta', 'given': 'Jenish Chetan'}, 'show_email': 'YES'}]}
Year: 2022
DOI: 10.7907/3vxb-6778
<p>We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories.</p>
<p>1. We show the first quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental theorem which has been used within diverse areas of mathematics. The Perron-Frobenius theorem shows that any irreducible nonnegative matrix <i>R</i> will have a largest positive eigenvalue <i>r</i>, and every other eigenvalue <i>λ</i> is such that Re<i>λ</i> < <i>R</i> and |λ| ≤ <i>r</i>. We capture the notion of irreducibility through the widely studied notion of edge expansion <i>φ</i> of <i>R</i> which intuitively measures how well-connected the underlying digraph of <i>R</i> is, and show a quantitative relation between the spectral gap Δ = 1-Re<i>λ</i>/<i>r</i> (where <i>λ</i> ≠ <i>r</i> has the largest real part) of <i>R</i> to the edge expansion <i>φ</i> as follows.</p>
<p>(1/15) • [(Δ(<i>R</i>))/n] ≤ <i>φ</i>(<i>R</i>) ≤ √[2 • Δ(<i>R</i>)].</p>
<p>This also provides a more general result than the Cheeger-Buser inequalities since it applies to any nonnegative matrix.</p>
<p>2. We study constructions of specific nonsymmetric matrices (or nonreversible Markov Chains) that have small edge expansion but large spectral gap, taking us in a direction more novel and unexplored than studying symmetric matrices with constant edge expansion that have been extensively studied. We first analyze some known but less studied Markov Chains, and then provide a novel construction of a nonreversible chain for which</p>
<p><i>φ</i>(<i>R</i>) ≤ [(Δ(<i>R</i>))/√<i>n</i>],</p>
<p>obtaining a bound exponentially better than known bounds. We also present a candidate construction of matrices for which</p>
<p><i>φ</i>(<i>R</i>) ≤ 2[(Δ(<i>R</i>))/<i>n</i>]</p>
<p>which is the most beautiful contribution of this thesis. We believe these matrices have properties remarkable enough to deserve study in their own right.</p>
<p>3. We connect the edge expansion and spectral gap to other combinatorial properties of nonsymmetric matrices. The most well-studied property is mixing time, and we provide elementary proofs of the relation between mixing time and the edge expansion, and also other bounds relating the mixing time of a nonreversible chain to the spectral gap and to its additive symmetrization. Further, we provide a unified view of the notion of capacity and normalized capacity, and show the monotonicity of capacity of nonreversible chains amongst other results for nonsymmetric matrices. We finally discuss and prove interesting lemmas about different notions of expansion and show the first results for tensor walks or nonnegative tensors.</p>https://thesis.library.caltech.edu/id/eprint/14949The Identification of Discrete Mixture Models
https://resolver.caltech.edu/CaltechTHESIS:02072023-112938936
Authors: {'items': [{'email': 'sgord512@gmail.com', 'id': 'Gordon-Spencer-Lane', 'name': {'family': 'Gordon', 'given': 'Spencer Lane'}, 'orcid': '0000-0002-7101-2370', 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/ebf5-0b48
In this thesis we discuss a variety of results on learning and identifying discrete mixture models, i.e., distributions that are a convex combination of k from a known class C of distributions. We first consider the case where C is the class of binomial distributions, before generalizing to the case of product distributions. We provide a necessary condition for identifiability of mixture of products distributions as well as a generalization to structured mixtures over multiple latent variables.https://thesis.library.caltech.edu/id/eprint/15101