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Information Theory
Information Theory Gurkirat,Harsha,Parth,Puneet,Rahul,Tushant November 8, 2015 1 Introduction[1] Information theory is a branch of applied mathematics, electrical engineering, and computer science involving the quantification of information . Firstly, it provides a general way to deal with any information , be it language, music, binary codes or pictures by introducing a measure of information , en- tropy as the average number of bits(binary for our convenience) needed to store one symbol in a message. This abstraction of information through entropy allows us to deal with all kinds of information mathematically and apply the same principles without worrying about it's type or form! It is remarkable to note that the concept of entropy was introduced by Ludwig Boltzmann in 1896 as a measure of disorder in a thermodynamic system , but serendipitously found it's way to become a basic concept in Information theory introduced by Shannon in 1948 . Information theory was developed as a way to find fundamental limits on the transfer , storage and compression of data and has found it's way into many applications including Data Compression and Channel Coding. 2 Basic definitions 2.1 Intuition behind "surprise" or "information"[2] Suppose someone tells you that the sun will come up tomorrow.This probably isn't surprising. You already knew the sun will come up tomorrow, so they didn't't really give you much information. Now suppose someone tells you that aliens have just landed on earth and have picked you to be the spokesperson for the human race. This is probably pretty surprising, and they've given you a lot of information.The rarer something is, the more you've learned if you discover that it happened. -
FOCS 2005 Program SUNDAY October 23, 2005
FOCS 2005 Program SUNDAY October 23, 2005 Talks in Grand Ballroom, 17th floor Session 1: 8:50am – 10:10am Chair: Eva´ Tardos 8:50 Agnostically Learning Halfspaces Adam Kalai, Adam Klivans, Yishay Mansour and Rocco Servedio 9:10 Noise stability of functions with low influences: invari- ance and optimality The 46th Annual IEEE Symposium on Elchanan Mossel, Ryan O’Donnell and Krzysztof Foundations of Computer Science Oleszkiewicz October 22-25, 2005 Omni William Penn Hotel, 9:30 Every decision tree has an influential variable Pittsburgh, PA Ryan O’Donnell, Michael Saks, Oded Schramm and Rocco Servedio Sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing 9:50 Lower Bounds for the Noisy Broadcast Problem In cooperation with ACM SIGACT Navin Goyal, Guy Kindler and Michael Saks Break 10:10am – 10:30am FOCS ’05 gratefully acknowledges financial support from Microsoft Research, Yahoo! Research, and the CMU Aladdin center Session 2: 10:30am – 12:10pm Chair: Satish Rao SATURDAY October 22, 2005 10:30 The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics Tutorials held at CMU University Center into `1 [Best paper award] Reception at Omni William Penn Hotel, Monongahela Room, Subhash Khot and Nisheeth Vishnoi 17th floor 10:50 The Closest Substring problem with small distances Tutorial 1: 1:30pm – 3:30pm Daniel Marx (McConomy Auditorium) Chair: Irit Dinur 11:10 Fitting tree metrics: Hierarchical clustering and Phy- logeny Subhash Khot Nir Ailon and Moses Charikar On the Unique Games Conjecture 11:30 Metric Embeddings with Relaxed Guarantees Break 3:30pm – 4:00pm Ittai Abraham, Yair Bartal, T-H. -
Property Testing on Boolean Functions Thesis Proposal
Property testing on boolean functions Thesis proposal Eric Blais Computer Science Department Carnegie Mellon University Pittsburgh, PA [email protected] May 20, 2010 Abstract Property testing deals with the following general question: given query access to some combinatorial object, what properties of the ob- ject can one test with only a small number of queries? In this thesis, we will study property testing on boolean functions. Specifically, we will focus on two basic questions in the field: Can we characterize the set of properties on boolean functions that can be tested with a constant number of queries? And can we determine the exact number of queries needed to test some fundamental properties of boolean functions? 1 Contents 1 Introduction 3 2 Definitions and notation 4 2.1 Boolean functions . 4 2.2 Property testing . 5 3 Characterization of testable properties 5 3.1 Related work . 6 3.2 Testing function isomorphism . 7 3.3 Completed work on testing function isomorphism . 7 3.4 Proposed research . 8 4 Exact query complexity bounds 10 4.1 Related work . 10 4.2 Completed work on testing juntas . 11 4.3 Proposed research . 11 5 Other proposed research 12 5.1 Alternative models of property testing . 13 5.2 Application of property testing ideas to other domains . 14 6 Suggested timeline 14 2 1 Introduction This thesis is primarily concerned with the study of boolean functions. Boolean functions play a central role in many areas of computer science: complexity theory [44], machine learning [19], coding theory [43], cryptog- raphy [18], circuit design [34], data structures [30], and combinatorics [16]. -
Probabilistic Methods in Combinatorics
Lecture notes (MIT 18.226, Fall 2020) Probabilistic Methods in Combinatorics Yufei Zhao Massachusetts Institute of Technology [email protected] http://yufeizhao.com/pm/ Contents 1 Introduction7 1.1 Lower bounds to Ramsey numbers......................7 1.1.1 Erdős’ original proof..........................8 1.1.2 Alteration method...........................9 1.1.3 Lovász local lemma........................... 10 1.2 Set systems................................... 11 1.2.1 Sperner’s theorem............................ 11 1.2.2 Bollobás two families theorem..................... 12 1.2.3 Erdős–Ko–Rado theorem on intersecting families........... 13 1.3 2-colorable hypergraphs............................ 13 1.4 List chromatic number of Kn;n ......................... 15 2 Linearity of expectations 17 2.1 Hamiltonian paths in tournaments....................... 17 2.2 Sum-free set................................... 18 2.3 Turán’s theorem and independent sets.................... 18 2.4 Crossing number inequality.......................... 20 2.4.1 Application to incidence geometry................... 22 2.5 Dense packing of spheres in high dimensions................. 23 2.6 Unbalancing lights............................... 26 3 Alterations 28 3.1 Ramsey numbers................................ 28 3.2 Dominating set in graphs............................ 28 3.3 Heilbronn triangle problem........................... 29 3.4 Markov’s inequality............................... 31 3.5 High girth and high chromatic number.................... 31 3.6 Greedy random coloring............................ 32 2 4 Second moment method 34 4.1 Threshold functions for small subgraphs in random graphs......... 36 4.2 Existence of thresholds............................. 42 4.3 Clique number of a random graph....................... 47 4.4 Hardy–Ramanujan theorem on the number of prime divisors........ 49 4.5 Distinct sums.................................. 52 4.6 Weierstrass approximation theorem...................... 54 5 Chernoff bound 56 5.1 Discrepancy.................................. -
2021 Leroy P. Steele Prizes
FROM THE AMS SECRETARY 2021 Leroy P. Steele Prizes The 2021 Leroy P. Steele Prizes were presented at the Annual Meeting of the AMS, held virtually January 6–9, 2021. Noga Alon and Joel Spencer received the Steele Prize for Mathematical Exposition. Murray Gerstenhaber was awarded the Prize for Seminal Contribution to Research. Spencer Bloch was honored with the Prize for Lifetime Achievement. Citation for Mathematical Biographical Sketch: Noga Alon Exposition: Noga Alon Noga Alon is a Professor of Mathematics at Princeton and Joel Spencer University and a Professor Emeritus of Mathematics and The 2021 Steele Prize for Math- Computer Science at Tel Aviv University, Israel. He received ematical Exposition is awarded his PhD in Mathematics at the Hebrew University of Jeru- to Noga Alon and Joel Spencer salem in 1983 and had visiting and part-time positions in for the book The Probabilistic various research institutes, including the Massachusetts Method, published by Wiley Institute of Technology, Harvard University, the Institute and Sons, Inc., in 1992. for Advanced Study in Princeton, IBM Almaden Research Now in its fourth edition, Center, Bell Laboratories, Bellcore, and Microsoft Research The Probabilistic Method is an (Redmond and Israel). He joined Tel Aviv University in invaluable toolbox for both 1985, served as the head of the School of Mathematical Noga Alon the beginner and the experi- Sciences in 1999–2001, and moved to Princeton in 2018. enced researcher in discrete He supervised more than twenty PhD students. He serves probability. It brings together on the editorial boards of more than a dozen international through one unifying perspec- technical journals and has given invited lectures in numer- tive a head-spinning variety of ous conferences, including plenary addresses in the 1996 results and methods, linked to European Congress of Mathematics and in the 2002 Inter- applications in graph theory, national Congress of Mathematicians. -
CURRICULUM VITAE Rafail Ostrovsky
last updated: December 26, 2020 CURRICULUM VITAE Rafail Ostrovsky Distinguished Professor of Computer Science and Mathematics, UCLA http://www.cs.ucla.edu/∼rafail/ mailing address: Contact information: UCLA Computer Science Department Phone: (310) 206-5283 475 ENGINEERING VI, E-mail: [email protected] Los Angeles, CA, 90095-1596 Research • Cryptography and Computer Security; Interests • Streaming Algorithms; Routing and Network Algorithms; • Search and Classification Problems on High-Dimensional Data. Education NSF Mathematical Sciences Postdoctoral Research Fellow Conducted at U.C. Berkeley 1992-95. Host: Prof. Manuel Blum. Ph.D. in Computer Science, Massachusetts Institute of Technology, 1989-92. • Thesis titled: \Software Protection and Simulation on Oblivious RAMs", Ph.D. advisor: Prof. Silvio Micali. Final version appeared in Journal of ACM, 1996. Practical applications of thesis work appeared in U.S. Patent No.5,123,045. • Minor: \Management and Technology", M.I.T. Sloan School of Management. M.S. in Computer Science, Boston University, 1985-87. B.A. Magna Cum Laude in Mathematics, State University of New York at Buffalo, 1980-84. Department of Mathematics Graduation Honors: With highest distinction. Personal • U.S. citizen, naturalized in Boston, MA, 1986. Data Appointments UCLA Computer Science Department (2003 { present): Distinguished Professor of Computer Science. Recruited in 2003 as a Full Professor with Tenure. UCLA School of Engineering (2003 { present): Director, Center for Information and Computation Security. (See http://www.cs.ucla.edu/security/.) UCLA Department of Mathematics (2006 { present): Distinguished Professor of Mathematics (by courtesy). 1 Appointments Bell Communications Research (Bellcore) (cont.) (1999 { 2003): Senior Research Scientist; (1995 { 1999): Research Scientist, Mathematics and Cryptography Research Group, Applied Research. -
Derandomized Graph Products
Derandomized Graph Products Noga Alon ∗ Uriel Feigey Avi Wigderson z David Zuckermanx February 22, 2002 Abstract Berman and Schnitger [10] gave a randomized reduction from approximating MAX- SNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n (for some ). This reduction was further studied by Blum [11], who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1]. The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounding this probability.) This lower bound extends also to the case that different sets of vertices are specified for different time steps of the walk. 1 Introduction We present lower bounds on the probability that all steps of a random walk stay within a specified set of vertices of a graph. We then apply these lower bounds to amplify unapprox- imability results about certain NP-hard optimization problems. Our work was motivated by the problem of approximating the size of the maximum clique in graphs. This motivat- ing problem, which serves also as an example of how our lower bounds can be applied, is described in section 1.1. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv Uni- versity, Tel Aviv, Israel and AT & T Bell Labs, Murray Hill, NJ, 07974, USA. Email: [email protected]. -
14 the Probabilistic Method
14 The Probabilistic Method Probabilistic Graph Theory Theorem 14.1 (Szele) For every positive integer n, there exists a tournament on n vertices with at least n!2¡(n¡1) Hamiltonian paths. Proof: Construct a tournament by randomly orienting each edge of Kn in each direction 1 independently with probability 2 . For any permutation σ of the vertices, let Xσ be the indicator random variable which has value 1 if σ is the vertex sequence of a Hamiltonian path, and 0 otherwise. Let X be the random variable which is the total number of Hamiltonain P paths. Then X = Xσ and we have X ¡(n¡1) E(X) = E(Xσ) = n!2 : σ Thus, there must exist at least one tournament on n vertices which has ¸ n!2¡(n¡1) Hamil- tonain paths as claimed. ¤ 1 Theorem 14.2 Every graph G has a bipartite subgraph H for which jE(H)j ¸ 2 jE(G)j. Proof: Choose a random subset S ⊆ V (G) by independently choosing each vertex to be in S 1 with probability 2 . Let H be the subgraph of G containing all of the vertices, and all edges with exactly one end in S. For every edge e, let Xe be the indicator random variable with value 1 if e 2 E(H) and 0 otherwise. If e = uv, then e will be in H if u 2 S and v 62 S or if 1 u 62 S and v 2 S, so E(Xe) = P(e 2 E(H)) = 2 and we ¯nd X 1 E(jE(H)j) = E(Xe) = 2 jE(G)j: e2E(G) 1 Thus, there must exist at least one bipartite subgraph H with jE(H)j ¸ 2 jE(G)j. -
18.218 Probabilistic Method in Combinatorics, Topic 3: Alterations
Lecturer: Prof. Yufei Zhao Notes by: Andrew Lin 3 Alterations Recall the naive probabilistic method: we found some lower bounds for Ramsey numbers in Section 1.1, primarily for the diagonal numbers. We did this with a basic method: color randomly, so that we color each edge red with probability p and blue with probability 1 − p. Then the probability that we don’t see any red s-cliques or blue t-cliques (with a union bound) is at most n (s ) n (t ) p 2 + (1 − p) 2 ; s t and if this is less than 1 for some p, then there exists some graph on n vertices for which there is no red Ks and blue Kt . So we union bounded the bad events there. Well, the alteration method does a little more than that - here’s a proof that mirrors that of Proposition 1.6. We again color randomly, but the idea now is to delete a vertex in every bad clique (red Ks and blue Kt ). How many edges have we deleted? We can estimate by using linearity of expectation: Theorem 3.1 For all p 2 (0; 1); n 2 N, n (s ) n (t ) R(s; t) > n − p 2 − (1 − p) 2 : s t This right hand side begins by taking the starting number of vertices and then we deleting one vertex for each clique. We’re going to explore this idea of “fixing the blemishes” a little more. 3.1 Dominating sets Definition 3.2 Given a graph G, a dominating set U is a set of vertices such that every vertex not in U has a neighbor in U. -
Discrete Mathematics: Methods and Challenges 121 Ous Interesting Applications
ICM 2002 Vol. I 119–135 · · Discrete Mathematics: Methods and Challenges Noga Alon∗ Abstract Combinatorics is a fundamental mathematical discipline as well as an es- sential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Com- puter Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of mod- ern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas. 2000 Mathematics Subject Classification: 05-02. Keywords and Phrases: Combinatorial nullstellensatz, Shannon capacity, Additive number theory, List coloring, The probabilistic method, Ramsey theory, Extremal graph theory. 1. Introduction The originators of the basic concepts of Discrete Mathematics, the mathemat- ics of finite structures, were the Hindus, who knew the formulas for the number of arXiv:math/0212390v1 [math.CO] 1 Dec 2002 permutations of a set of n elements, and for the number of subsets of cardinality k in a set of n elements, already in the sixth century. The beginning of Combinatorics as we know it today started with the work of Pascal and De Moivre in the 17th century, and continued in the 18th century with the seminal ideas of Euler in Graph Theory, with his work on partitions and their enumeration, and with his interest in latin squares. -
Self-Sorting SSD: Producing Sorted Data Inside Active Ssds
Foreword This volume contains the papers presentedat the Third Israel Symposium on the Theory of Computing and Systems (ISTCS), held in Tel Aviv, Israel, on January 4-6, 1995. Fifty five papers were submitted in response to the Call for Papers, and twenty seven of them were selected for presentation. The selection was based on originality, quality and relevance to the field. The program committee is pleased with the overall quality of the acceptedpapers and, furthermore, feels that many of the papers not used were also of fine quality. The papers included in this proceedings are preliminary reports on recent research and it is expected that most of these papers will appear in a more complete and polished form in scientific journals. The proceedings also contains one invited paper by Pave1Pevzner and Michael Waterman. The program committee thanks our invited speakers,Robert J. Aumann, Wolfgang Paul, Abe Peled, and Avi Wigderson, for contributing their time and knowledge. We also wish to thank all who submitted papers for consideration, as well as the many colleagues who contributed to the evaluation of the submitted papers. The latter include: Noga Alon Alon Itai Eric Schenk Hagit Attiya Roni Kay Baruch Schieber Yossi Azar Evsey Kosman Assaf Schuster Ayal Bar-David Ami Litman Nir Shavit Reuven Bar-Yehuda Johan Makowsky Richard Statman Shai Ben-David Yishay Mansour Ray Strong Allan Borodin Alain Mayer Eli Upfal Dorit Dor Mike Molloy Moshe Vardi Alon Efrat Yoram Moses Orli Waarts Jack Feldman Dalit Naor Ed Wimmers Nissim Francez Seffl Naor Shmuel Zaks Nita Goyal Noam Nisan Uri Zwick Vassos Hadzilacos Yuri Rabinovich Johan Hastad Giinter Rote The work of the program committee has been facilitated thanks to software developed by Rob Schapire and FOCS/STOC program chairs. -
Problems and Results in Extremal Combinatorics, Part I
Problems and results in Extremal Combinatorics, Part I Noga Alon ∗ Abstract Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably somewhat biased, and as the title of the paper suggests I hope to write a related paper in the future. Each section of this paper is essentially self contained, and can be read separately. 1 Introduction Extremal Combinatorics deals with the problem of determining or estimating the maximum or min- imum possible cardinality of a collection of finite objects that satisfies certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades, see, e.g., [10], [29], and their many references. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive survey of the area, but rather a collection of several extremal problems, which are hopefully interesting. The techniques used include combinatorial, probabilistic and algebraic tools.