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Efficient Algorithms with Asymmetric Read and Write Costs
Efficient Algorithms with Asymmetric Read and Write Costs Guy E. Blelloch1, Jeremy T. Fineman2, Phillip B. Gibbons1, Yan Gu1, and Julian Shun3 1 Carnegie Mellon University 2 Georgetown University 3 University of California, Berkeley Abstract In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fun- damentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We con- sider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M, ω)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost ω 1. For FFT and sorting networks we show a lower bound cost of Ω(ωn logωM n), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when ω is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(ω, log n)/ log(ωM)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n × n diamond DAG of Ω(ωn2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. -
Tarjan Transcript Final with Timestamps
A.M. Turing Award Oral History Interview with Robert (Bob) Endre Tarjan by Roy Levin San Mateo, California July 12, 2017 Levin: My name is Roy Levin. Today is July 12th, 2017, and I’m in San Mateo, California at the home of Robert Tarjan, where I’ll be interviewing him for the ACM Turing Award Winners project. Good afternoon, Bob, and thanks for spending the time to talk to me today. Tarjan: You’re welcome. Levin: I’d like to start by talking about your early technical interests and where they came from. When do you first recall being interested in what we might call technical things? Tarjan: Well, the first thing I would say in that direction is my mom took me to the public library in Pomona, where I grew up, which opened up a huge world to me. I started reading science fiction books and stories. Originally, I wanted to be the first person on Mars, that was what I was thinking, and I got interested in astronomy, started reading a lot of science stuff. I got to junior high school and I had an amazing math teacher. His name was Mr. Wall. I had him two years, in the eighth and ninth grade. He was teaching the New Math to us before there was such a thing as “New Math.” He taught us Peano’s axioms and things like that. It was a wonderful thing for a kid like me who was really excited about science and mathematics and so on. The other thing that happened was I discovered Scientific American in the public library and started reading Martin Gardner’s columns on mathematical games and was completely fascinated. -
Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Curriculum Vitae
Curriculum Vitae Prof. Michal Feldman School of Computer Science, Tel-Aviv University Personal Details • Born: Israel, 1976 • Gender: Female • email: [email protected] Education • Ph.D.: Information Management and Systems, University of California at Berkeley, May 2005. Thesis title: Incentives for cooperation in peer-to-peer systems. • B.Sc.: Bar-Ilan University, Computer Science, Summa Cum Laude, June 1999. Academic Positions 2013 - present: Associate Professor, School of Computer Science, Tel-Aviv University, Israel. Associate Professor, School of Business Administration and Center for 2011 - 2013: the Study of Rationality, Hebrew University of Jerusalem, Israel. 2007 - 2011: Senior Lecturer, School of Business Administration and Center for the Study of Rationality, Hebrew University of Jerusalem, Israel. Additional Positions 2011 - 2013: Visiting Researcher (weekly visitor), Microsoft Research New England, Cambridge, MA, USA. 2011 - 2013: Visiting Professor, Harvard School of Engineering and Applied Sciences, Center for Research on Computation and Society, School of Computer Science, Cambridge, MA, USA (Marie Curie IOF program). 2008 - 2011: Senior Researcher (part-time), Microsoft Research in Herzliya, Israel. 2007 - 2013: Member, Center for the Study of Rationality, Hebrew University. 2005 - 2007: Post-Doctoral Fellow (Lady Davis fellowship), Hebrew University, School of Computer Science and Engineering. 2004: Ph.D. Intern, HP Labs, Palo Alto, California, USA. 1 Grants (Funding ID) • European Research Council (ERC) Starting Grant: \Algorithmic Mechanism Design - Beyond Truthfulness": 1.4 million Euro, 2013-2017. • FP7 Marie Curie International Outgoing Fellowship (IOF): \Innovations in Algorithmic Game Theory" (IAGT): 313,473 Euro, 2011-2014. • Israel Science Foundation (ISF) grant. \Equilibria Under Coalitional Covenants in Non-Cooperative Games - Existence, Quality and Computation:" 688,000 NIS (172,000 NIS /year), 2009-2013. -
Limitations and Possibilities of Algorithmic Mechanism Design By
Incentives, Computation, and Networks: Limitations and Possibilities of Algorithmic Mechanism Design by Yaron Singer A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Christos Papadimitriou, Chair Professor Shachar Kariv Professor Scott Shenker Spring 2012 Incentives, Computation, and Networks: Limitations and Possibilities of Algorithmic Mechanism Design Copyright 2012 by Yaron Singer 1 Abstract Incentives, Computation, and Networks: Limitations and Possibilities of Algorithmic Mechanism Design by Yaron Singer Doctor of Philosophy in Computer Science University of California, Berkeley Professor Christos Papadimitriou, Chair In the past decade, a theory of manipulation-robust algorithms has been emerging to address the challenges that frequently occur in strategic environments such as the internet. The theory, known as algorithmic mechanism design, builds on the foundations of classical mechanism design from microeconomics and is based on the idea of incentive compatible pro- tocols. Such protocols achieve system-wide objectives through careful design that ensures it is in every agent's best interest to comply with the protocol. As it turns out, however, implementing incentive compatible protocols as advocated in classical mechanism design the- ory often necessitates solving intractable problems. To address this, algorithmic mechanism design focuses on designing -
A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems*
Electronic Colloquium on Computational Complexity, Report No. 75 (2019) Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems* Lijie Chen Dylan M. McKay Cody D. Murray† R. Ryan Williams MIT MIT MIT Abstract A frontier open problem in circuit complexity is to prove PNP 6⊂ SIZE[nk] for all k; this is a neces- NP sary intermediate step towards NP 6⊂ P=poly. Previously, for several classes containing P , including NP NP NP , ZPP , and S2P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove k C 6⊂ SIZE[n ] for all k, we show that C ⊂ P=poly implies a “collapse” D = C for some larger class D, where we already know D 6⊂ SIZE[nk] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for PNP that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for PNP are equivalent to fixed-polynomial size circuit lower NP NP k NP bounds for P . That is, P 6⊂ SIZE[n ] for all k if and only if (NP ⊂ P=poly implies PH ⊂ i.o.-P=n ). Next, we present new consequences of the assumption NP ⊂ P=poly, towards proving similar re- sults for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. -
Algorithmic System Design Under Consideration of Dynamic Processes
Algorithmic System Design under Consideration of Dynamic Processes Vom Fachbereich Maschinenbau an der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation vorgelegt von Dipl.-Phys. Lena Charlotte Altherr aus Karlsruhe Berichterstatter: Prof. Dr.-Ing. Peter F. Pelz Mitberichterstatter: Prof. Dr. rer. nat. Ulf Lorenz Tag der Einreichung: 19.04.2016 Tag der m¨undlichen Pr¨ufung: 24.05.2016 Darmstadt 2016 D 17 Vorwort des Herausgebers Kontext Die Produkt- und Systementwicklung hat die Aufgabe technische Systeme so zu gestalten, dass eine gewünschte Systemfunktion erfüllt wird. Mögliche System- funktionen sind z.B. Schwingungen zu dämpfen, Wasser in einem Siedlungsgebiet zu verteilen oder die Kühlung eines Rechenzentrums. Wir Ingenieure reduzieren dabei die Komplexität eines Systems, indem wir dieses gedanklich in überschaubare Baugruppen oder Komponenten zerlegen und diese für sich in Robustheit und Effizienz verbessern. In der Kriegsführung wurde dieses Prinzip bereits 500 v. Chr. als „Teile und herrsche Prinzip“ durch Meister Sun in seinem Buch „Die Kunst der Kriegsführung“ beschrieben. Das Denken in Schnitten ist wesentlich für das Verständnis von Systemen: „Das wichtigste Werkzeug des Ingenieurs ist die Schere“. Das Zusammenwirken der Komponenten führt anschließend zu der gewünschten Systemfunktion. Während die Funktion eines technischen Systems i.d.R. nicht verhan- delbar ist, ist jedoch verhandelbar mit welchem Aufwand diese erfüllt wird und mit welcher Verfügbarkeit sie gewährleistet wird. Aufwand und Verfügbarkeit sind dabei gegensätzlich. Der Aufwand bemisst z.B. die Emission von Kohlendioxid, den Energieverbrauch, den Materialverbrauch, … die „total cost of ownership“. Die Verfügbarkeit bemisst die Ausfallzeiten, Lebensdauer oder Laufleistung. Die Gesell- schaft stellt sich zunehmend die Frage, wie eine Funktion bei minimalem Aufwand und maximaler Verfügbarkeit realisiert werden kann. -
The Computational Complexity of Nash Equilibria in Concisely Represented Games∗
Electronic Colloquium on Computational Complexity, Report No. 52 (2005) The Computational Complexity of Nash Equilibria in Concisely Represented Games∗ Grant R. Schoenebeck y Salil P. Vadhanz May 4, 2005 Abstract Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1 Introduction In recent years, there has been a surge of interest at the interface between computer science and game theory. On one hand, game theory and its notions of equilibria provide a rich framework for modelling the behavior of selfish agents in the kinds of distributed or networked environments that often arise in computer science, and offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior. -
54 OP08 Abstracts
54 OP08 Abstracts CP1 Dept. of Mathematics Improving Ultimate Convergence of An Aug- [email protected] mented Lagrangian Method Optimization methods that employ the classical Powell- CP1 Hestenes-Rockafellar Augmented Lagrangian are useful A Second-Derivative SQP Method for Noncon- tools for solving Nonlinear Programming problems. Their vex Optimization Problems with Inequality Con- reputation decreased in the last ten years due to the com- straints parative success of Interior-Point Newtonian algorithms, which are asymptotically faster. In the present research We consider a second-derivative 1 sequential quadratic a combination of both approaches is evaluated. The idea programming trust-region method for large-scale nonlin- is to produce a competitive method, being more robust ear non-convex optimization problems with inequality con- and efficient than its ”pure” counterparts for critical prob- straints. Trial steps are composed of two components; a lems. Moreover, an additional hybrid algorithm is defined, Cauchy globalization step and an SQP correction step. A in which the Interior Point method is replaced by the New- single linear artificial constraint is incorporated that en- tonian resolution of a KKT system identified by the Aug- sures non-accent in the SQP correction step, thus ”guiding” mented Lagrangian algorithm. the algorithm through areas of indefiniteness. A salient feature of our approach is feasibility of all subproblems. Ernesto G. Birgin IME-USP Daniel Robinson Department of Computer Science Oxford University [email protected] -
From the Collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ
From the collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ These documents can only be used for educational and research purposes (“Fair use”) as per U.S. Copyright law (text below). By accessing this file, all users agree that their use falls within fair use as defined by the copyright law. They further agree to request permission of the Princeton University Library (and pay any fees, if applicable) if they plan to publish, broadcast, or otherwise disseminate this material. This includes all forms of electronic distribution. Inquiries about this material can be directed to: Seeley G. Mudd Manuscript Library 65 Olden Street Princeton, NJ 08540 609-258-6345 609-258-3385 (fax) [email protected] U.S. Copyright law test The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or other reproduction is not to be “used for any purpose other than private study, scholarship or research.” If a user makes a request for, or later uses, a photocopy or other reproduction for purposes in excess of “fair use,” that user may be liable for copyright infringement. The Princeton Mathematics Community in the 1930s Transcript Number 11 (PMC11) © The Trustees of Princeton University, 1985 MERRILL FLOOD (with ALBERT TUCKER) This is an interview of Merrill Flood in San Francisco on 14 May 1984. The interviewer is Albert Tucker.