Laureates Meet Young Researchers in Heidelberg
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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. -
Great Ideas in Computing
Great Ideas in Computing University of Toronto CSC196 Winter/Spring 2019 Week 6: October 19-23 (2020) 1 / 17 Announcements I added one final question to Assignment 2. It concerns search engines. The question may need a little clarification. I have also clarified question 1 where I have defined (in a non standard way) the meaning of a strict binary search tree which is what I had in mind. Please answer the question for a strict binary search tree. If you answered the quiz question for a non strict binary search tree withn a proper explanation you will get full credit. Quick poll: how many students feel that Q1 was a fair quiz? A1 has now been graded by Marta. I will scan over the assignments and hope to release the grades later today. If you plan to make a regrading request, you have up to one week to submit your request. You must specify clearly why you feel that a question may not have been graded fairly. In general, students did well which is what I expected. 2 / 17 Agenda for the week We will continue to discuss search engines. We ended on what is slide 10 (in Week 5) on Friday and we will continue with where we left off. I was surprised that in our poll, most students felt that the people advocating the \AI view" of search \won the debate" whereas today I will try to argue that the people (e.g., Salton and others) advocating the \combinatorial, algebraic, statistical view" won the debate as to current search engines. -
FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
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. -
Affine Grassmannians in A1-Algebraic Topology
AFFINE GRASSMANNIANS IN A1-ALGEBRAIC TOPOLOGY TOM BACHMANN Abstract. Let k be a field. Denote by Spc(k)∗ the unstable, pointed motivic homotopy category and A1 1 by R ΩGm : Spc(k)∗ →Spc(k)∗ the (A -derived) Gm-loops functor. For a k-group G, denote by GrG the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence A1 A1 R ΩGm G ≃ GrG. We use this to compute the motive M(R ΩGm G) ∈ DM(k, Z[1/e]). 1. Introduction This note deals with the subject of A1-algebraic topology. In other words it deals with with the ∞- category Spc(k) of motivic spaces over a base field k, together with the canonical functor Smk → Spc(k) and, importantly, convenient models for Spc(k). Since our results depend crucially on the seminal papers [1, 2], we shall use their definition of Spc(k) (which is of course equivalent to the other definitions in the literature): start with the category Smk of smooth (separated, finite type) k-schemes, form the universal homotopy theory on Smk (i.e. pass to the ∞-category P(Smk) of space-valued presheaves on k), and A1 then impose the relations of Nisnevich descent and contractibility of the affine line k (i.e. localise P(Smk) at an appropriate family of maps). The ∞-category Spc(k) is presentable, so in particular has finite products, and the functor Smk → Spc(k) preserves finite products. Let ∗ ∈ Spc(k) denote the final object (corresponding to the final k-scheme k); then we can form the pointed unstable motivic homotopy category Spc(k)∗ := Spc(k)/∗. -
P Versus NP Richard M. Karp
P Versus NP Computational complexity theory is the branch of theoretical computer science concerned with the fundamental limits on the efficiency of automatic computation. It focuses on problems that appear to Richard M. Karp require a very large number of computation steps for their solution. The inputs and outputs to a problem are sequences of symbols drawn from a finite alphabet; there is no limit on the length of the input, and the fundamental question about a problem is the rate of growth of the number of required computation steps as a function of the length of the input. Some problems seem to require a very rapidly growing number of steps. One such problem is the inde- pendent set problem: given a graph, consisting of points called vertices and lines called edges connect- ing pairs of vertices, a set of vertices is called independent if no two vertices in the set are connected by a line. Given a graph and a positive integer n, the problem is to decide whether the graph contains an independent set of size n. Every known algorithm to solve the independent set problem encounters a combinatorial explosion, in which the number of required computation steps grows exponentially as a function of the size of the graph. On the other hand, the problem of deciding whether a given set of vertices is an independent set in a given graph is solvable by inspection. There are many such dichotomies, in which it is hard to decide whether a given type of structure exists within an input object (the existence problem), but it is easy to decide whether a given structure is of the required type (the verification problem). -
An Intuitive Introduction to Motivic Homotopy Theory Vladimir Voevodsky
What follows is Vladimir Voevodsky's snapshot of his Fields Medal work on motivic homotopy, plus a little philosophy and \from my point of view the main fun of doing mathematics" Voevodsky (2002). Voevodsky uses extremely simple leading ideas to extend topological methods of homotopy into number theory and abstract algebra. The talk takes the simple ideas right up to the edge of some extremely refined applications. The talk is on-line as streaming video at claymath.org/video/. In a nutshell: classical homotopy studies a topological space M by looking at maps to M from the unit interval on the real line: [0; 1] = fx 2 R j 0 ≤ x ≤ 1 g as well as maps to M from the unit square [0; 1]×[0; 1], the unit cube [0; 1]3, and so on. Algebraic geometry cannot copy these naively, even over the real numbers, since of course the interval [0; 1] on the real line is not defined by a polynomial equation. The only polynomial equal to 0 all over [0; 1] is the constant polynomial 0, thus equals 0 over the whole line. The problem is much worse when you look at algebraic geometry over other fields than the real or complex numbers. Voevodsky shows how category theory allows abstract definitions of the basic apparatus of homotopy, where the category of topological spaces is replaced by any category equipped with objects that share certain purely categorical properties with the unit interval, unit square, and so on. For example, the line in algebraic geometry (over any field) does not have \endpoints" in the usual way but it does come with two naturally distinguished points: 0 and 1. -
Homotopy Theoretic Models of Type Theory
Homotopy Theoretic Models of Type Theory Peter Arndt1 and Krzysztof Kapulkin2 1 University of Oslo, Oslo, Norway [email protected] 2 University of Pittsburgh, Pittsburgh, PA, USA [email protected] Abstract. We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power so that we can soundly inter- pret dependent products and sums in it while also have purely intensional iterpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of models that are examined in the paper. 1 Introduction Starting with the Hofmann-Streicher groupoid model [HS98] it has become clear that there exist deep connections between Intensional Martin-L¨ofType Theory ([ML72], [NPS90]) and homotopy theory. These connections have recently been studied very intensively. We start by summarizing this work | a more complete survey can be found in [Awo10]. It is well-known that Identity Types (or Equality Types) play an important role in type theory since they provide a relaxed and coarser notion of equality between terms of a given type. For example, assuming standard rules for type Nat one cannot prove that n: Nat ` n + 0 = n: Nat but there is still an inhabitant n: Nat ` p: IdNat(n + 0; n): Identity types can be axiomatized in a usual way (as inductive types) by form, intro, elim, and comp rules (see eg. [NPS90]). A type theory where we do not impose any further rules on the identity types is called intensional. -
The P Versus Np Problem
THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a robust class and has equivalent definitions over a large class of computer models. Here we follow standard practice and define the class P in terms of Turing machines. Formally the elements of the class P are languages. Let Σ be a finite alphabet (that is, a finite nonempty set) with at least two elements, and let Σ∗ be the set of finite strings over Σ. -
A Memorable Trip Abhisekh Sankaran Research Scholar, IIT Bombay
A Memorable Trip Abhisekh Sankaran Research Scholar, IIT Bombay It was my first trip to the US. It had not yet sunk in that I had been chosen by ACM India as one of two Ph.D. students from India to attend the big ACM Turing Centenary Celebration in San Francisco until I saw the familiar face of Stephen Cook enter a room in the hotel a short distance from mine; later, Moshe Vardi recognized me from his trip to IITB during FSTTCS, 2011. I recognized Nitin Saurabh from IMSc Chennai, the other student chosen by ACM-India; 11 ACM SIG©s had sponsored students and there were about 75 from all over the world. Registration started at 8am on 15th June, along with breakfast. Collecting my ©Student Scholar© badge and stuffing in some food, I entered a large hall with several hundred seats, a brightly lit podium with a large screen in the middle flanked by two others. The program began with a video giving a brief biography of Alan Turing from his boyhood to the dynamic young man who was to change the world forever. There were inaugural speeches by John White, CEO of ACM, and Vint Cerf, the 2004 Turing Award winner and incoming ACM President. The MC for the event, Paul Saffo, took over and the panel discussions and speeches commenced. A live Twitter feed made it possible for people in the audience and elsewhere to post questions/comments which were actually taken up in the discussions. Of the many sessions that took place in the next two days, I will describe three that I found most interesting. -
References Is Provisional, and Will Grow (Or Shrink) As the Course Progresses
Note: This list of references is provisional, and will grow (or shrink) as the course progresses. It will never be comprehensive. See also the references (and the book) in the online version of Weibel's K- Book [40] at http://www.math.rutgers.edu/ weibel/Kbook.html. References [1] Armand Borel and Jean-Pierre Serre. Le th´eor`emede Riemann-Roch. Bull. Soc. Math. France, 86:97{136, 1958. [2] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133{150, 1975. [3] A. K. Bousfield and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80{130. Springer, Berlin, 1978. [4] Kenneth S. Brown and Stephen M. Gersten. Algebraic K-theory as gen- eralized sheaf cohomology. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 266{ 292. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. [5] William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thoma- son. Algebraic K-theory eventually surjects onto topological K-theory. Invent. Math., 66(3):481{491, 1982. [6] Eric M. Friedlander and Daniel R. Grayson, editors. Handbook of K- theory. Vol. 1, 2. Springer-Verlag, Berlin, 2005. [7] Ofer Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59{70. Amer. Math. Soc., Providence, RI, 1992. [8] Henri Gillet and Daniel R. -
The Unstable Slice Filtration
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 11, November 2014, Pages 5991–6025 S 0002-9947(2014)06116-3 Article electronically published on May 22, 2014 THE UNSTABLE SLICE FILTRATION PABLO PELAEZ Abstract. The main goal of this paper is to construct an analogue of Vo- evodsky’s slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants; this is motivated by the ex- istence of an equivalence of categories between the orthogonal components for Voevodsky’s slice filtration and the birational motivic stable homotopy cate- gories constructed by the author in 2013. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the sta- ble setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting. 1. Introduction Our main goal is to construct a tower in the motivic unstable homotopy category with similar properties to Voevodsky’s slice tower [26] in the motivic stable homo- topy category. In this section we recall Voevodsky’s definition for the slice filtration and explain why our construction in the unstable setting is a natural analogue. Definitions and notation. In this paper X will denote a Noetherian separated base scheme of finite Krull dimension, SchX the category of schemes of finite type over X and SmX the full subcategory of SchX consisting of smooth schemes over X regarded as a site with the Nisnevich topology. All the maps between schemes will be considered over the base X.GivenY ∈ SchX , all the closed subsets Z of Y will be considered as closed subschemes with the reduced structure.