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Time-Space Tradeoffs and Query Complexity in Statistics, Coding Theory, and Quantum Computing

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Time-Space Tradeoffs and Query Complexity in Statistics, Coding Theory, and Quantum Computing Time-Space Tradeoffs and Query Complexity in Statistics, Coding Theory, and Quantum Computing Widad Machmouchi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2013 Reading Committee: Paul Beame, Chair Anna Karlin Anup Rao Program Authorized to Offer Degree: Computer Science and Engineering University of Washington Abstract Time-Space Tradeoffs and Query Complexity in Statistics, Coding Theory, and Quantum Computing Widad Machmouchi Chair of the Supervisory Committee: Professor Paul Beame Computer Science and Engineering Computational complexity is the field that studies the computational resources needed by algorithms to correctly solve computational problems. In this thesis, we derive lower bounds on the compu- tational complexity of different problems for different classes of algorithms. The computational resources we consider are the running time, the memory usage and the input query complexity; i.e., the number of positions in the input that the algorithm reads. While deriving upper bounds involves constructing algorithms that run efficiently in their computational resources, lower bounds for a computational problem establish limits on how fast or space-efficient any algorithm solving that problem can be. We consider two main themes of lower bounds: time-space tradeoffs and query complexity lower bounds. Time-space tradeoffs define a relation between the running time and space of any algorithm solving a certain problems. They give a complete picture of the hardness of a problem by deriving a lower bound on one resource when the other is limited. Query complex- ity measures the depth of a decision tree computing the problem. We derive several lower bounds and upper bounds for different problems on different types of classical computational models and on quantum computers, where the queries to the input are represented by quantum operations operating on the qubits of the algorithm: • We derive the largest time-space tradeoff known for a randomized algorithm solving an ex- plicit problem. We consider the pointer jumping problem, also known as the outdegree-1 graph accessibility problem and derive a T 2 Ω(n log(n=S) log log (n=S)) time-space trade- off for any randomized oblivious algorithm computing it. Oblivious algorithms have the prop- erty that the sequence of operations performed is independent of the value of the input. We derive a similar lower bound for randomized oblivious algorithms with boolean inputs. • Frequency moments and order statistics are attributes computed on a sequence of numbers and are extremely relevant in various data analysis settings. We consider computing several copies of these functions on overlapping sequences of numbers by considering a window of a fixed length and sliding it over the input; these versions are called sliding-window versions of these functions. We give hardness separations between the sliding-window versions of these statistics. On one hand, we give a TS 2 Ω(n2) lower bounds for any algorithm computing sliding-window frequency moments. To complement our lower bound, we give a TS 2 O~(n2) comparison-based algorithm for sliding-window frequency moments. On the other hand we describe a T 2S 2 O~(n3) algorithm for sliding-window element distinctness. For order statistics, we derive a similar lower bound of TS 2 Ω(n2) for any algorithm computing a sliding-widow version of the median, while for the sliding window version of maximum (and minimum), we construct an O(n log n)-time, logarithmic space algorithm. • We study the increase in complexity on quantum computers when considering sliding-window frequency moments. We show that quantum computers do not require the same increase in complexity as their classical counterparts by constructing a quantum algorithm for sliding- window version of the zeroth frequency moments running in time O(n3=2) and logarithmic space, thus beating the classical lower bound for this problem. • In the area of error-correcting codes, we study the query complexity of error detection and correction algorithms. We prove that linear codes over large alphabets that are small in size and concentrated in terms of their codeword weights have testers and correctors that require only a constant number of queries to a word received across a noisy channel. Such codes are called locally-testable and self-correctable. As random codes of small size have the property that their codeword weights are concentrated in a small range of values, they are also locally- testable and self-correctable with high probability. • For linear codes with information-efficient encoding, we study the tradeoff between the time and space of their encoders, and their error-correction property. We derive properties on generator matrices of asymptotically good linear codes that guarantee an optimal time-space tradeoff of TS 2 Ω(n2) for any algorithm encoding them. Moreover, we prove a first step in an approach to establish these properties for all generator matrices of such codes. • To compare classical computers to their quantum counterparts, we consider the query com- plexity of quantum algorithms computing frequency moments and the ONTO function that takes as input a function and determines whether it is surjective. We prove that any quantum computer computing the ONTO function or the parity of the zeroth frequency moment requires a linear number of queries to the input. Since the ONTO function with boolean inputs has a simple polynomial-size, constant-depth circuit, we obtain an almost linear lower bound on the query complexity of AC0, a special class of circuits that have polynomial size, constant depth and unbounded fan-in for their gates. TABLE OF CONTENTS Page List of Figures . iv Chapter 1: Introduction . 1 1.1 Time-Space Tradeoffs . 2 1.2 Query Complexity Lower Bounds . 6 1.3 Overview of the Results . 8 1.4 Organization of the Thesis . 11 Part I: Time-Space Tradeoffs . 13 Chapter 2: Computational Models . 14 2.1 Branching Programs . 16 2.2 Yao’s minimax Principle . 23 2.3 Multiparty Communication Complexity . 24 2.4 Doob Martingales and the Azuma-Hoeffding Inequality . 26 2.5 Lower Bounds for Branching Programs . 27 Chapter 3: Randomized Oblivious Branching Programs . 35 3.1 Pointer Jumping and Generalized Inner Product . 36 3.2 Contributions . 36 3.3 Techniques . 37 3.4 Related Work . 38 3.5 Reducing Permuted Problems to 1GAP ....................... 38 3.6 Time-Space Tradeoff Lower Bound for P ermuted-GIP and 1GAP . 41 3.7 Time-Space Tradeoff Lower Bound for Randomized Oblivious Boolean Branching Programs . 48 i 3.8 Discussion and Open Questions . 54 Chapter 4: Sliding Window Functions . 55 4.1 Frequency Moments, Element Distinctness, and Order Statistics . 55 4.2 Sliding Windows . 56 4.3 Contributions and Techniques . 57 4.4 Related Work . 59 4.5 Element Distinctness and Small-Space Collision-Finding . 60 4.6 Element Distinctness over Sliding Windows . 68 4.7 Frequency Moments over Sliding Windows . 74 4.8 Order Statistics over Sliding Windows . 85 4.9 Discussion and Open Questions . 86 Part II: Error Correcting codes . 89 Chapter 5: Coding Theory Preliminaries . 90 5.1 Coding Theory Basic Notions . 92 5.2 Query Complexity of Error Detection and Correction . 95 5.3 Time and Space Complexity of Encoding . 99 Chapter 6: Local-Testability and Self-Correctability of q-ary Sparse Linear Codes . 102 6.1 Contributions and Techniques . 102 6.2 q-ary Krawtchouk Polynomials and the MacWilliams Identity . 103 6.3 Weight distribution of duals of sparse codes . 106 6.4 Local Testing . 108 6.5 Self-correctability . 113 6.6 Applications to random linear codes . 116 6.7 Discussion and Open Questions . 117 Chapter 7: Sufficient Conditions For Optimal Time-Space Tradeoffs For Encoders . 118 7.1 From Branching Programs to Rigid Matrices . 118 7.2 Removing Low-Dimension Projections from Asymptotically Good Codes . 120 7.3 Comparison to Subsequent Work . 123 ii Part III: Quantum Computing . 125 Chapter 8: Quantum Lower Bounds . 126 8.1 The Quantum Query Model . 128 8.2 AC0 .......................................... 129 8.3 Previous Work on Quantum Query Complexity . 130 8.4 Contributions . 132 8.5 The Quantum Adversary Method . 133 8.6 Quantum Query Lower Bounds for the 2-TO-1-VS-ALMOST-2-TO-1 Function . 135 8.7 A Near Optimal Query Lower Bound for AC0 . 138 8.8 Formula Size Lower Bound for the ONTO Function . 139 8.9 Parity of Sliding-Window F0 ............................. 140 8.10 Discussion and Open Questions . 141 Chapter 9: Conclusion . 143 9.1 Direction for Further Study . 144 Bibliography . 146 iii LIST OF FIGURES Figure Number Page 3.1 A width 4 branching program computing GIP3;4. .................... 40 4.1 Collision-finding with multiple sources ......................... 62 4.2 Transitions from the Markov chain in the proof of Lemma 4.3 . 66 8.1 AC0 circuit computing the ONTO function . 139 iv ACKNOWLEDGMENTS First and foremost, this work would not have been possible without my advisor Paul Beame. His mentoring, good advice, and encouragement throughout my years at UW were invaluable for my academic development. His extensive knowledge and creativity provided me with the right guidance through research paths that were too windy to lead anywhere or had the right idea to pursue. I am grateful for all the effort and care he showed me during the Princeton visit and advising me during the years where Venkat was away. I am also thankful for Venkat Guruswami for advising me through my first few years at UW. Although we did not work together for long, I was lucky to experience his enthusiasm and brilliance and thankful for all the guidance he offered me. I would like to thank Anup Rao and Anna Karlin for their comments and for Rekha Thomas and Dan Suciu for being on my committee.
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