William Gasarch

William Gasarch
University of Maryland, College Park | UMD, UMCP, University of Maryland College Park · Department of Computer Science

PhD, Computer Science, Harvard, 1980

About

333
Publications
18,478
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1,605
Citations
Introduction
Additional affiliations
August 1985 - present
University of Maryland, College Park
Position
  • Professor (Full)

Publications

Publications (333)
Article
Full-text available
A c-coloring of the grid GN,M = [N] × [M] is a mapping of GN,M into [c] such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed to find a 4-coloring of G17,17. Though a coloring was produced, finding it proved to be difficult. This raises the question of whether there is some complexity lower bound. Consi...
Article
This book really is written for people who know NOTHING about quantum mechanics and don't know much math, hence I was able to give it a fair review. Since I have sometimes heard people talk about quantum mechanics I could also recognize some of the discussions in the book as well known controversies in quantum mechanics.
Article
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Erdős proved that for every infinite \(X \subseteq \mathbb {R}^d\) there is \(Y \subseteq X\) with \(|Y|=|X|\), such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-the...
Article
In 2001 I (innocently!) asked Lane if I could write an article for his SIGACT News Complexity Theory Column that would be a poll of what computer scientists (and others) thought about P=?NP and related issues. It was to be an objective record of subjective opinions. I asked (by telegraph in those days) over 100 theorists. Exactly 100 responded, whi...
Article
This issue's Open Problem Column is by Stephen Fenner, Fred Green, and Steven Homer. It is titled: Fixed-Parameter Extrapolation and Aperiodic Order: Open Problems. It is not on xedparameter tractability. It is on a generalization of the geometric notion of convexity.
Preprint
Erd\"{o}s proved that for every infinite $X \subseteq \mathbb{R}^d$ there is $Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from $Y$ have distinct distances, and he gave partial results for general $a$-ary volume. In this paper, we search for the strongest possible canonization results for $a$-ary volume, making use of general model-...
Article
There have been two polls asking theorists (and others) what they thought of P vs NP (and other questions) [4, 5]. Both were written by William Gasarch and appeared in the Complexity Column of SIGACT News, edited by Lane A. Hemaspaandra. They were in 2002 and 2012. Since William Gasarch is fond of Van der Waerden's theorem, you would think the next...
Article
You have $m$ muffins and $s$ students. You want to divide the muffins into pieces and give the shares to students such that every student has $\frac{m}{s}$ muffins. Find a divide-and-distribute protocol that maximizes the minimum piece. Let $f(m,s)$ be the minimum piece in the optimal protocol. We prove that $f(m,s)$ exists, is rational, and findin...
Article
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Let D be an integral domain, d in N at least 1, a0 a nonzero element of D. There are two players, Wanda and Nora. (Wanda stands for wants root, Nora stands for no root). One of the players is Player I and the other is Player II. Consider the following game: (1) Players I and II (with Player I going first) alternately choose coefficients from D for...
Chapter
Many programs allow the user to input data several times during its execution. If the program runs forever the user may input data infinitely often. A program terminates if it terminates no matter what the user does. We discuss various ways to prove that program terminates. The proofs use well orderings, Ramsey Theory, and Matrices. These technique...
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In this column, we survey the factorization problem, discussing the algorithmic ideas as well as the applications to other problems. We then discuss the challenges ahead, in particular focusing on the goal of obtaining deterministic factoring algorithms. While deterministic PIT algorithms have been developed for various restricted circuit classes,...
Article
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Let A be a finite subset of $\nat$. Then NIM(A;n) is the following 2-player game: initially there are $n$ stones on the board and the players alternate removing $a\in A$ stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts...
Article
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It is well known that $\sum_{p\le n} 1/p =ln(ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge ln(ln(n)) + O(1)$. This is based on Euler's proof that $\sum_p 1/p$ diverges. We then present three proofs that $\sum_{p\le n} 1/p \le ln(ln(n)) + O(1)$ The first one, due to Mertens,...
Article
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A division of a cake by n people is envy free if everyone thinks they got the biggest pieces. Note that peoples tastes can differ. There is a discrete protocol for envy free division for n=3 which takes at most 5 cuts. For n=4 and beyond there is a protocol but the number of cuts it takes is unbounded. In particular the number of cuts depends on pe...
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Let a1, a2, ..., aL be coin denominations. Assume you have an unlimited number of each coin. How many ways can you make n cents with these coins? Schur's theorem gives the answer asymptotically and also yields the coefficient of the dominant term. We give a complete proof that a high school student who has not had calculus can understand.
Article
Alice and Bob want to know if two strings of length n are almost equal. That is, do the strings differ on at most a bits? Let 0 ≤ a ≤ n-1. We show (1) any deterministic protocol-as well as any error-free quantum protocol (C∗ version)-for this problem requires at least n-2 bits of communication, and (2) a lower bound of n/2-1 for error-free Q∗ quant...
Article
There are languages A such that there is a Pushdown Automata (PDA) that recognizes A which is much smaller than any Deterministic Pushdown Automata (DPDA) that recognizes A. There are languages A such that there is a Linear Bounded Automata (Linear Space Turing Machine, henceforth LBA) that recognizes A which is much smaller than ny PDA that recogn...
Article
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A set is low if A' \le_T HALT. A set is superlow if A' \le_tt HALT. A set is superduperlow if A' \le_btt HALT. While it was known that any superduperlow is decidable it does not seem to be well known. We include two unpublished proofs of this result: One from Carl Jockush and one from Frank Stephan.
Article
A fundamental problem in computer science is stated informally as: Given a problem, how hard is it? We measure hardness by looking at the following question: Given a set A what is the fastest algorithm to determine if "x ∈ A?" We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, so...
Article
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This is a book review column that has reviews of many books, including the one requested: The Satisfiability Problem: Algorithms and Analyses by Schoning and Toran
Article
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Let COLk be the set of all k-colorable graphs. It is easy to show that if a<b then COLa \le COLb (poly time reduction). Using the Cook-Levin theorem it is easy to show that if 3 \le a< b then COLb \le COLa. However this proof is insane in that it translates a graph to a formula and then the formula to a graph. We give a simple proof that COLk \le C...
Article
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Given a dollar, how many ways are there to make change using pennies, nickels, dimes, and quarters? What if you are given a different amount of money? What if you use different coin denominations? The answers to problems of this type are usually given in terms of generating functions. For several special cases of coin denominations we derive closed...
Article
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GurariEitan. An introduction to the theory of computation. Principles of computer science series. Computer Science Press, Rockville, Md., 1989, xii + 314 pp. - Volume 56 Issue 1 - William I. Gasarch
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Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the non-zero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erd\H{o}s in 1957, the function $h_{2,d}(n)$ has bee...
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Mathematics has been around for thousands of years, and this has given it plenty of opportunities to become very complicated.
Article
An affine-invariant property over a finite field is a property of functions over Fn/p that is closed under all affine transformations of the domain. This class of properties includes such well-known beasts as low-degree polynomials, polynomials that ...
Article
Let {p1,...,pn}⊆Rd. We think of d≪n. How big is the largest subset X of points such that all of the distances determined by elements of (X2) are different? We show that X is at least Ω((n1/(6d)(logn)1/3)/d1/3). This is not the best known; however the technique is new. Assume that no 3 of the original points are in the line. How big is the largest s...
Article
The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that ...
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Let P be a set of n points in R^d. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n^{1/6d}) This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of P such th...
Article
This text is meant to be an introduction to a recent strategy introduced by Bourgain and Gamburd (following a work of Helfgott) for proving graph-expansion. The strategy is designed for graphs H that are defined using some underlying group G. The strategy ...
Article
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If E is a linear homogenous equation and c a natural then the Rado number $R_c(E)$ is the least N so that any c-coloring of the positive integers from 1 to N contains a monochromatic solution. Rado characterized for which E R_c(E) always exists. The original proof of Rado's theorem gave enormous bounds on R_c(E) (when it existed). In this paper we...
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Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erd...
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This opinionated essay discusses the role of intellectual and instrumental values in scientific fields, and the way these values evolve due to internal and external forces. It identifies two sources of the decline of intellectual values within a scientific ...
Article
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A c-coloring of G(n,m)=n x m is a mapping of G(n,m) into {1,...,c} such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed via the internet to find a 4-coloring of G(17,17). This attracted considerable attention from the popular mathematics community. A coloring was produced; however, finding it proved to...
Article
Full-text available
0.1 What is Ramsey Theory and why did we write this book?.. 7 0.2 What is a purely combinatorial proof?............. 8 0.3 Who could read this book?.................... 9 0.4 Abbreviations used in this book................. 9
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What is the difference between struggling for achievements and competing for success? What is the effect of competitions on a scientic field? What are the specific implications on TOC? In this opinionated essay, I address these questions and related ...
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It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every $\aleph_0$-coloring of the reals there exists distinct reals $e_1,e_2,e_3,e_4$ such that $e_1+e_2=e_3+e_4$?} Is it true? Erdos...

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