DOCSLIB.ORG

Submodular and Sparse Optimization Methods for Machine Learning and Communication

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Submodular and Sparse Optimization Methods for Machine Learning and Communication 博士論文 Submodular and Sparse Optimization Methods for Machine Learning and Communication （機械学習と通信のための劣モジュラ・スパース最適化手法） 相馬 輔 Submodular and Sparse Optimization Methods for Machine Learning and Communication Tasuku Soma February 11, 2016 Abstract This dissertation considers optimization problems arising from machine learning and commu- nication, and devises efficient algorithms by exploiting submodular and sparse optimization techniques. The first subject of this dissertation is submodular function maximization over the inte- ger lattice. Maximizing a monotone submodular set function has been extensively studied in machine learning, social science, and related areas, since it encompasses a variety of problems considered in these areas and efficient greedy algorithms are available. However, we often face real scenarios that are beyond these existing models based on submodular set functions. For example, the existing models cannot capture the budget allocation problem in which we have to decide how much budget should be set aside. In this dissertation, we overcome this funda- mental limitation of the existing models by exploiting submodularity over the integer lattice. We demonstrate that our new framework can naturally and concisely capture these difficult real scenarios, and provide efficient approximation algorithms for various constraints. In addition to theoretical guarantees, we conduct numerical experiments to establish practical efficiency of our model and algorithms. Then we move on matrix completion problems. Roughly speaking, matrix completion prob- lems are to determine missing entries in a given partial matrix under some criteria. Our first result is a new algorithm for constructing a multicast code for a wireless network using max- rank matrix completion. In the main ingredient of our algorithm, we employ the theory of mixed matrices, matroids, and submodular optimization. Second, we propose a new problem, the low-rank basis problem for a matrix subspace. Originally this problem comes from com- binatorial optimization, but it turns out that this problem has various applications, including image separation and data compression. We present an efficient heuristic algorithm for this problem, using matroid theory and sparse optimization. The last subject of this dissertation is compressed sensing. The main goal of compressed sensing is to recover a high-dimensional sparse signal from a few linear measurements. While the problem setup is quite simple, compressed sensing has been widely applied to machine learning, statistics, and signal processing. In this dissertation, we consider stable signal recovery in compressed sensing and show that the sum-of-squares method yields a new polynomial time recovery method for Rademacher sensing matrices. Our result sheds light on the power of sum-of-squares method on nonconvex problem arising from compressed sensing. i Acknowledgments I have no doubt on that this dissertation would have been impossible without various support of many people. First of all, I am grateful for my supervisor, Satoru Iwata, who invited me to the world of academic research. He has been continuously giving many research ideas, suggestions, and advice during my Ph.D. program with great patience. My days in the Ph.D. course would have been completely different without his infinite support. I also would like to acknowledge that many parts of this dissertation are based on joint works with my collaborators. Especially, I would like to thank Yuichi Yoshida. His endless mathe- matical ideas and great depth of knowledge on computer science have been truly remarkable and inspired me a lot. The idea of the first part of this dissertation originates from my first joint project with Naonori Kakimura, Kazuhiro Inaba, Ken-ichi Kawarabayashi. Their research ideas and valuable discussions with them were really helpful and they took me to the research field of machine learning. Chapter8 benefited from many discussions and suggestions of Yuji Nakatsukasa and Andr´e Uschmajew during our joint work. They raised my small mathematical problem into really fascinating one, invited me to an exciting research area of sparse optimization, and greatly expanded the scope of my research. I also would like to thank members and former members of our laboratory, especially Ku- nihiro Sadakane, Akiko Takeda, Yutaro Yamaguchi, Yu Yokoi, Naoki Ito, Shuichi Katsumata, and Kaito Fujii. Their valuable comments in our seminar and supports in my daily life in the laboratory have been really appreciated. Furthermore, I would like to thank my parents for their support and love. Spending ten years as a student was not short, but they always encouraged me to continue my research. Finally, I thank Japan Society for Promotion of Science, JST ERATO Kawarabayashi Large Graph Project, JST CREST Iwata team, and the University of Tokyo for their financial sup- port. ii Contents 1 Introduction 1 1.1 Monotone Submodular Function Maximization...................1 1.2 Matrix Completion...................................3 1.3 Compressed Sensing..................................3 1.4 Organization of This Thesis..............................4 1.4.1 Publications Contained in This Dissertation.................5 1.5 Notation.........................................5 2 Submodular Functions, Matroids, and Polymatroids7 2.1 Submodular Functions.................................7 2.1.1 Extensions of Submodular Function.....................8 2.1.2 Submodular Function Minimization.....................9 2.2 Matroids and Polymatroids..............................9 2.2.1 Matroids.................................... 10 2.2.2 Polymatroids.................................. 11 3 Submodularity over the Integer Lattice and Budget Allocation 13 3.1 Monotone Submodular Optimization: Literature Overview............. 13 3.1.1 Monotone Submodular Function Maximization............... 15 3.1.2 Submodular Cover............................... 18 3.2 Optimal Budget Allocation: Motivating Problem.................. 18 3.3 Beyond Set Functions: Submodularity over the Integer Lattice.......... 19 3.3.1 Submodularity in Optimal Budget Allocation................ 21 3.4 Our Framework..................................... 22 3.4.1 Budget Allocation Problem with a Competitor............... 23 3.4.2 Generalized Sensor Placement Model..................... 24 3.4.3 Other Applications............................... 24 3.5 Facts on Submodular Function over the Integer Lattice............... 25 3.5.1 Useful Lemmas................................. 25 3.5.2 Reducibility of DR-submodular Functions.................. 26 4 Monotone Submodular Function Maximization over the Integer Lattice 27 4.1 Overview of this Chapter............................... 27 4.1.1 Technical Contribution............................. 28 4.1.2 Related Work.................................. 28 4.1.3 Preliminaries.................................. 29 4.2 Cardinality Constraint for DR-Submodular Function................ 29 4.3 Cardinality Constraint for Lattice Submodular Function.............. 30 4.4 Polymatroid Constraint for DR-Submodular Function............... 33 iii 4.4.1 Continuous Extension for Polymatroid Constraints............. 33 4.4.2 Continuous Greedy Algorithm for Polymatroid Constraint......... 34 4.4.3 Rounding.................................... 38 4.5 Knapsack Constrant for DR-Submodular Function................. 38 4.5.1 Multilinear extension for knapsack constraints............... 39 4.5.2 Algorithm.................................... 40 4.5.3 Subroutine for handling small items..................... 40 4.5.4 Complete algorithm.............................. 43 4.6 Knapsack Constrant for Lattice Submodular Function............... 48 5 The Diminishing Return Submodular Cover Problem 51 5.1 Algorithm for the DR-submodular Cover....................... 52 5.2 Discussion........................................ 55 5.3 Experiments....................................... 56 5.3.1 Experimental Setting.............................. 56 5.3.2 Experimental Results............................. 58 6 Introduction to Matrix Completion 60 6.1 Matrices with Indeterminates and Matrix Subspace................. 60 6.2 Max-Rank Matrix Completion............................ 61 6.2.1 Lov´asz’sRandomized Algorithm....................... 61 6.2.2 Deterministic Algorithms for Max-Rank Matrix Completion........ 61 6.2.3 Simultaneous Max-Rank Matrix Completion................ 62 6.2.4 Hardness of Max-Rank Matrix Completion................. 62 6.3 Low-Rank Matrix Completion............................. 62 6.3.1 Nuclear Norm Relaxation........................... 63 6.3.2 Random Matrix and Observation Model................... 64 7 Faster Algorithm for Multicasting in Linear Deterministic Relay Network 65 7.1 Introduction....................................... 65 7.1.1 Our Contribution................................ 66 7.1.2 Related Work.................................. 67 7.2 Preliminaries...................................... 67 7.2.1 Mixed Matrix.................................. 67 7.2.2 Mixed Matrix Completion........................... 68 7.2.3 Cauchy-Binet Formula............................. 69 7.3 Flow Model....................................... 69 7.4 Algorithm........................................ 70 7.5 Complexity Excluding Unicast Computation..................... 72 7.6 Concluding Remarks.................................. 73 8 The Low-Rank Basis Problem for a Matrix Subspace 74 8.1 Introduction....................................... 74 8.1.1 Our Contribution...............................
Load more

Recommended publications
  • Formalizing Randomized Matching Algorithms
    Formalizing Randomized Matching Algorithms Dai Tri Man Leˆ and Stephen A. Cook Department of Computer Science, University of Toronto fledt, [email protected] Abstract—Using Jerˇabek’s´ framework for probabilistic rea- set to another. Using this method, Jerˇabek´ developed tools for soning, we formalize the correctness of two fundamental RNC2 describing algorithms in ZPP and RP. He also showed in [13], algorithms for bipartite perfect matching within the theory VPV [14] that the theory VPV+sWPHP(L ) is powerful enough to for polytime reasoning. The first algorithm is for testing if a FP bipartite graph has a perfect matching, and is based on the formalize proofs of very sophisticated derandomization results, Schwartz-Zippel Lemma for polynomial identity testing applied e.g. the Nisan-Wigderson theorem [23] and the Impagliazzo- to the Edmonds polynomial of the graph. The second algorithm, Wigderson theorem [12]. (Note that Jerˇabek´ actually used the due to Mulmuley, Vazirani and Vazirani, is for finding a perfect single-sorted theory PV1+sWPHP(PV), but these two theories matching, where the key ingredient of this algorithm is the are isomorphic.) Isolating Lemma. In [15], Jerˇabek´ developed an even more systematic ap- proach by showing that for any bounded P=poly definable set, I. INTRODUCTION there exists a suitable pair of surjective “counting functions” 1 There is a substantial literature on theories such as PV, S2 , which can approximate the cardinality of the set up to a poly- VPV, V 1 which capture polynomial time reasoning [8], [4], nomially small error. From this and other results he argued [17], [7].
    [Show full text]
  • Coded Sparse Matrix Multiplication
    Coded Sparse Matrix Multiplication Sinong Wang∗, Jiashang Liu∗ and Ness Shroff∗† ∗Department of Electrical and Computer Engineering yDepartment of Computer Science and Engineering The Ohio State University fwang.7691, liu.3992 [email protected] Abstract r×t In a large-scale and distributed matrix multiplication problem C = A|B, where C 2 R , the coded computation plays an important role to effectively deal with “stragglers” (distributed computations that may get delayed due to few slow or faulty processors). However, existing coded schemes could destroy the significant sparsity that exists in large-scale machine learning problems, and could result in much higher computation overhead, i.e., O(rt) decoding time. In this paper, we develop a new coded computation strategy, we call sparse code, which achieves near optimal recovery threshold, low computation overhead, and linear decoding time O(nnz(C)). We implement our scheme and demonstrate the advantage of the approach over both uncoded and current fastest coded strategies. I. INTRODUCTION In this paper, we consider a distributed matrix multiplication problem, where we aim to compute C = A|B from input matrices A 2 Rs×r and B 2 Rs×t for some integers r; s; t. This problem is the key building block in machine learning and signal processing problems, and has been used in a large variety of application areas including classification, regression, clustering and feature selection problems. Many such applications have large- scale datasets and massive computation tasks, which forces practitioners to adopt distributed computing frameworks such as Hadoop [1] and Spark [2] to increase the learning speed.
    [Show full text]
  • Algebraic Algorithms in Combinatorial Optimization
    Algebraic Algorithms in Combinatorial Optimization CHEUNG, Ho Yee A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Computer Science and Engineering The Chinese University of Hong Kong September 2011 Thesis/Assessment Committee Professor Shengyu Zhang (Chair) Professor Lap Chi Lau (Thesis Supervisor) Professor Andrej Bogdanov (Committee Member) Professor Satoru Iwata (External Examiner) Abstract In this thesis we extend the recent algebraic approach to design fast algorithms for two problems in combinatorial optimization. First we study the linear matroid parity problem, a common generalization of graph matching and linear matroid intersection, that has applications in various areas. We show that Harvey's algo- rithm for linear matroid intersection can be easily generalized to linear matroid parity. This gives an algorithm that is faster and simpler than previous known al- gorithms. For some graph problems that can be reduced to linear matroid parity, we again show that Harvey's algorithm for graph matching can be generalized to these problems to give faster algorithms. While linear matroid parity and some of its applications are challenging generalizations, our results show that the al- gebraic algorithmic framework can be adapted nicely to give faster and simpler algorithms in more general settings. Then we study the all pairs edge connectivity problem for directed graphs, where we would like to compute minimum s-t cut value between all pairs of vertices. Using a combinatorial approach it is not known how to solve this problem faster than computing the minimum s-t cut value for each pair of vertices separately.
    [Show full text]
  • Coded Computation for Speeding up Distributed Machine Learning
    CODED COMPUTATION FOR SPEEDING UP DISTRIBUTED MACHINE LEARNING DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Sinong Wang Graduate Program in Electrical and Computer Engineering The Ohio State University 2019 Dissertation Committee: Ness B. Shroff, Advisor Atilla Eryilmaz Abhishek Gupta Andrea Serrani c Copyrighted by Sinong Wang 2019 ABSTRACT Large-scale machine learning has shown great promise for solving many practical ap- plications. Such applications require massive training datasets and model parameters, and force practitioners to adopt distributed computing frameworks such as Hadoop and Spark to increase the learning speed. However, the speedup gain is far from ideal due to the latency incurred in waiting for a few slow or faulty processors, called \straggler" to complete their tasks. To alleviate this problem, current frameworks such as Hadoop deploy various straggler detection techniques and usually replicate the straggling task on another available node, which creates a large computation overhead. In this dissertation, we focus on a new and more effective technique, called coded computation to deal with stragglers in the distributed computation problems. It creates and exploits coding redundancy in local computation to enable the final output to be recoverable from the results of partially finished workers, and can therefore alleviate the impact of straggling workers. However, we observe that current coded computation techniques are not suitable for large-scale machine learning application. The reason is that the input training data exhibit both extremely large-scale targeting data and a sparse structure. However, the existing coded computation schemes destroy the sparsity and creates large computation redundancy.
    [Show full text]
  • Matchings in Graphs 1 Definitions 2 Existence of a Perfect Matching
    Matchings in Graphs Lecturer: Prajakta Nimbhorkar Meeting: 7 Scribe: Karteek 18/02/2010 In this lecture we look at an \efficient" randomized parallel algorithm for each of the follow- ing: • Checking the existence of perfect matching in a bipartite graph. • Checking the existence of perfect matching in a general graph. • Constructing a perfect matching for a given graph. • Finding a min-weight perfect matching in the given weighted graph. • Finding a Maximum matching in the given graph. where “efficient" means an NC-algorithm or an RNC-algorithm. 1 Definitions Definition 1 NC: The set of problems that can be solved within poly-log time by a polynomial number of processors working in parallel. Definition 2 RNC: The set of problems for which there is a poly-log time algorithm which uses polynomially many processors in parallel and in addition is allowed polynomially many random bits and the following properties hold: 1 • If the correct answer is YES, the algorithm returns YES with probabilty ≥ 2 . • If the correct answer is NO, the algorithm always returns NO. 2 Existence of a perfect matching 2.1 Bipartite Graphs Here we consider the decision problem of checking if a bipartite graph has a perfect matching. Let the given graph be G = (V; E). Let V =A[B. Since we are looking for perfect matchings, we only look at graphs where jAj = jBj = n, because otherwise, there cannot be a perfect matching. Fact 3 In a bipartite graph as above, every perfect matching can be seen as a permutation from Sn and every permutation from Sn can be seen as perfect matching.
    [Show full text]
  • A Random Linear Network Coding Approach to Multicast Tracey Ho, Member, IEEE, Muriel Médard, Senior Member, IEEE, Ralf Koetter, Senior Member, IEEE, David R
    IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 10, OCTOBER 2006 4413 A Random Linear Network Coding Approach to Multicast Tracey Ho, Member, IEEE, Muriel Médard, Senior Member, IEEE, Ralf Koetter, Senior Member, IEEE, David R. Karger, Associate Member, IEEE, Michelle Effros, Senior Member, IEEE, Jun Shi, and Ben Leong Abstract—We present a distributed random linear network coding approach for transmission and compression of informa- tion in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves ca- pacity with probability exponentially approaching I with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error ex- ponents for linear Slepian–Wolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two Fig. 1. An example of distributed random linear network coding. and $ examples of practical scenarios: distributed network operation are the source processes being multicast to the receivers, and the coefficients and networks with dynamically varying connections. Our deriva- are randomly chosen elements of a finite field. The label on each link represents the process being transmitted on the link. tion of these results also yields a new bound on required field size for centralized network coding on general multicast networks. Index Terms—Distributed compression, distributed networking, This family of problems includes traditional single-source mul- multicast, network coding, random linear coding.
    [Show full text]
  • Advanced Algorithms, a Graduate-Level Course Taught by Anupam Gupta at Carnegie Mellon University in Fall 2020
    ADVANCEDALGORITHMS notes for cmu 15-850 (fall 2020) lecturer: anupam gupta About this document This document contains the course notes for 15-850:Advanced Algorithms, a graduate-level course taught by Anupam Gupta at Carnegie Mellon University in Fall 2020. Parts of these notes were written by the students of previous versions of the class (based on the lectures) and then edited by the professor. The names of the student scribes will appear soon, as will missing details and bug fixes, more chapters, exercises, and (better) figures. The notes have not been thoroughly checked for accuracy, espe- cially attributions of results. They are intended to serve as study resources and not as a substitute for professionally prepared publica- tions. We apologize for any inadvertent inaccuracies or misrepresen- tations. More information about the course, including problem sets and references, can be found on the course website: https://www.cs.cmu.edu/~15850/ The style files (as well as the text on this page!) are mildly adapted from the ones developed by Yufei Zhao (MIT), for his notes on Graph Theory and Additive Combinatorics. As some of you may guess, the LATEX template used for these notes is called tufte-book. Contents I Discrete Algorithms 9 1 Minimum Spanning Trees 11 1.1 Minimum Spanning Trees: History............. 11 1.2 The Classical Algorithms.................. 13 1.3 Fredman and Tarjan’s O(m log∗ n)-time Algorithm... 15 1.4 A Linear-Time Randomized Algorithm.......... 18 1.5 Optional: MST Verification................. 21 1.6 The Ackermann Function.................. 25 1.7 Matroids............................ 25 2 Arborescences: Directed Spanning Trees 27 2.1 Arborescences........................
    [Show full text]
  • Network Flow Algorithms for Wireless Networks and Design and Analysis of Rate Compatible LDPC Codes Cuizhu Shi Iowa State University
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2011 Network flow algorithms for wireless networks and design and analysis of rate compatible LDPC codes Cuizhu Shi Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Electrical and Computer Engineering Commons Recommended Citation Shi, Cuizhu, "Network flow algorithms for wireless networks and design and analysis of rate compatible LDPC codes" (2011). Graduate Theses and Dissertations. 11972. https://lib.dr.iastate.edu/etd/11972 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Network flow algorithms for wireless networks and design and analysis of rate compatible LDPC codes by Cuizhu Shi A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Electrical Engineering Program of Study Committee: Aditya Ramamoorthy, Major Professor Zhengdao Wang Nicola Elia Sang W. Kim Ryan Martin Iowa State University Ames, Iowa 2011 Copyright c Cuizhu Shi, 2011. All rights reserved. ii TABLE OF CONTENTS LIST OF TABLES . v LIST OF FIGURES . vi ACKNOWLEDGEMENTS . viii ABSTRACT . ix CHAPTER 1. OVERVIEW . 1 1.1 Introduction on Wireless Information Flow . .3 1.1.1 The Known Facts . .4 1.1.2 The Unknown . .6 1.2 A Step Further { the Deterministic Channel Model .
    [Show full text]
  • Coded Sparse Matrix Multiplication
    Coded Sparse Matrix Multiplication Sinong Wang 1 Jiashang Liu 1 Ness Shroff 2 Abstract and the master node has to collect the results from all work- ers to output matrix C. As a result, a major performance In a large-scale and distributed matrix multiplica- bottleneck is the latency in waiting for a few slow or faulty tion problem C = A|B, where C 2 r×t, the R processors – called “stragglers” to finish their tasks (Dean coded computation plays an important role to ef- & Barroso, 2013). To alleviate this problem, current frame- fectively deal with “stragglers” (distributed com- works such as Hadoop deploy various straggler detection putations that may get delayed due to few slow techniques and usually replicate the straggling task on an- or faulty processors). However, existing coded other available node. schemes could destroy the significant sparsity that exists in large-scale machine learning problems, Recently, forward error correction or coding techniques pro- and could result in much higher computation over- vide a more effective way to deal with the “straggler” in head, i.e., O(rt) decoding time. In this paper, we the distributed tasks (Dutta et al., 2016; Lee et al., 2017a; develop a new coded computation strategy, we Tandon et al., 2017; Yu et al., 2017; Li et al., 2018; Wang call sparse code, which achieves near optimal re- et al., 2018). It creates and exploits coding redundancy in covery threshold, low computation overhead, and local computation to enable the matrix C recoverable from linear decoding time O(nnz(C)). We implement the results of partial finished workers, and can therefore alle- our scheme and demonstrate the advantage of the viate some straggling workers.
    [Show full text]
  • Exact Covers Via Determinants Andreas Björklund
    Exact Covers via Determinants Andreas Björklund To cite this version: Andreas Björklund. Exact Covers via Determinants. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Inria Nancy Grand Est & Loria, Mar 2010, Nancy, France. pp.95-106. inria-00455349 HAL Id: inria-00455349 https://hal.inria.fr/inria-00455349 Submitted on 10 Feb 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symposium on Theoretical Aspects of Computer Science 2010 (Nancy, France), pp. 95-106 www.stacs-conf.org EXACT COVERS VIA DETERMINANTS ANDREAS BJORKLUND¨ E-mail address: [email protected] Abstract. Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-Dimensional Match- ing problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in O∗(2n(k−2)/k) time. The O∗() notation hides factors polynomial in n and k. The general Exact Cover by k-Sets problem asks the same when the partition constraint is dropped and arbitrary hyperedges of cardinality k are permitted.
    [Show full text]
  • Admin Review String Matching
    Admin Arora talk. No class Monday. Review Fingerprinting: Universe of size u • Map to random fingerprint in universe of size v u • ≤ probability of collision 1=v • Freivald's technique verify matrix multiplication AB = C • check ABr = Cr for random r in 0; 1 n • f g probability of success 1=2 • works to check any matrix identity, not just product • useful if matrices \implicit" like AB • mapping size-n2 matrices to size-n vectors • In general, many ways to fingerprint explicitly represented objects. But for implicit objects, different methods have different strengths and weaknesses. We'll fingerprint 3 ways: vector multiply • number mod a random prime • polynomial evaluation at a random point • String matching Checksums: Alice and Bob have bit strings of length n • Think of n bit integers a, b • take a prime number p, compare a mod p and b mod p with log p bits. • trouble if a = b (mod p). How avoid? How likely? • { c = a b is n-bit integer. − 1 { so at most n prime factors. { How many prime factors less than k? Θ(k= ln k) { so take 2n2 log n limit { number of primes about n2 { So on random one, 1=n error prob. { O(log n) bits to send. { implement by add/sub, no mul or div! How find prime? { Well, a randomly chosen number is prime with prob. 1= ln n, { so just try a few. { How know its prime? Simple randomized test (later) Pattern matching in strings m-bit pattern • n-bit string • work mod prime p of size at most t • prob.
    [Show full text]
  • Lecture 11 — January 28, 2014 1 Overview 2 Ideas from Linear Algebra
    Advanced Graph Algorithms Jan-Apr 2014 Lecture 11 | January 28, 2014 Lecturer: Saket Sourabh Scribe: Prafullkumar P Tale 1 Overview In this lecture we will revisit the problem of finding perfect matching in non-bipartite graphs. We will study the algebraic approach to solve this problem. In the earlier lectures, we have seen the algorithm to decide whether a graph has perfect matching or not. In this, we will present O(nw+2) and O(nw+1) order algorithm to find the perfect matching. Before attacking perfect matching problem, we will state few well known results from Linear Algebra. 2 Ideas from Linear Algebra Theorem 1. If M is real skew symmetric matrix, then det(M) = [P f(M)]2 (1) where determinant and pfaffian of matrix A is given as follows n X Y det(M) = sgn(σ) Mi,σ(i) σ2Sn i=1 X pf(M) = sgn(µ)mµ µ #crossing where Sn is permutation group, µ is matching on f1; 2; : : : ; ng and sgn(µ) = (−1) . Definition 2 (Rank). For any given matrix M, rank of M is defined as dimension of largest square sub-matrix Msub, such that det(Msub) 6= 0. Notation Let M be an n × n matrix and X; Y be subsets of f1; 2; : : : ; ng. M[X; Y ] or MXY will denote the submatrix of M obtained by choosing the rows of M corresponding to indices in X and columns corresponding to indices in Y . Similarly M XY will donate the submatrix of M obtained by deleting rows of M corresponding to indices in X and columns corresponding to indices in Y .
    [Show full text]