Edward Reingold

ABSTRACT It is shown that when a graph is represented as a binary connection matrix, the problems of finding the shortest path between two nodes of a graph, of determining whether the graph has a cycle, and of determining if a graph is strongly connected each require at leastO(n2) operations. Thus the presently known best algorithms are optimal to within a multiplicative constant.

ABSTRACT In response to a large (currently 2000 students per semester) and increasing enrollment in introductory computer science courses, we have started a project to automate these courses on the PLATO IV Computer-Based Education system at the University of Illinois. The key components of our automated course are: 1. a library of lessons, covering several programming languages, computing techniques, and application areas 2. a completely self-contained interactive programming system for the preparation, execution and debugging of programs written by students in any of the languages covered by the lessons 3. a conversational advice-giving and information retrieval system to guide the student through the library of lessons, based on his goals and past performance. The goals and current status of this project are described.

We provide a set of "natural" requirements for well-orderings of(binary) list structures. We show that the resultant order-type is thesuccessor of the first critical epsilon number.The checker has to verify that the process comes to an end. Hereagain he should be assisted by the programmer giving a furtherdefinite assertion to be verified. This may take the form of a quantitywhich is asserted to decrease continually and vanish when themachine stops. To the pure mathematician it is...

ABSTRACT It is shown that when a graph is represented as a binary connection matrix, the problems of finding the shortest path between two nodes of a graph, of determining whether the graph has a cycle, and of determining if a graph is strongly connected each require at leastO(n2) operations. Thus the presently known best algorithms are optimal to within a multiplicative constant.

ABSTRACT Bounds are obtained for the solution to the divide-and-conquer recurrence \[M(n) = \max_{k_1 + \cdots + k_p = n} (M(k_1) + M(k_2) +\cdots + M(k_p) + \min (f(k_1), \cdots, f(k_p))),\] for nondecreasing functions $f$. Similar bounds are found for the recurrence with "min" replaced by "sum-of-all-but-the-max." Such recurrences appear in the analysis of various algorithms. The bounds follow from analyses of partition trees.

ABSTRACT Given a set of n elements each of which is either red or blue, it is known that in the worst case n-ν(n) pairwise equal/not equal color comparisons are necessary and sufficient to determine the majority color, where ν(n) is the number of 1-bits in the binary representation of n. We prove that 2n 3-8n 9π+O(logn) such comparisons are necessary and sufficient in the average case.

ABSTRACT Given a function F: N + →(X,Y} with the property that if F(n 0 )=Y then F(n)=Y for all n>n 0 , the unbounded search problem is to use tests of the form “is F(i)=X? '' to determine the smallest n such that F(n)=Y; the “cost” of a search algorithm is a function c(n), the number of such tests used when the location of the first Y is n. A solution to this search problem specifies a prefix-free, binary encoding of the positive integers in which the cost c(n) is the number of bits used to encode n. It is shown that the “ultimate algorithm”, of J. L. Bentley and A. Yao [Inform. Processing Letters 5, 82-87 (1976; Zbl 0335.68030)], which is within an additive Θ(lg * n) factor of a lower bound on the cost of this problem, is “far” from optimal in the sense that it is just the second in an infinite sequence of search algorithms, each of which is much closer to optimality than its predecessor. A corresponding sequence of lower bounds is also given, based on Kraft’s inequality, each of which is much stronger than its predecessor. Diagonalizing over this sequence of search algorithms yields an algorithm, which is given explicitly in a Pascal-like notation, that is within an additive factor of α(n)+2 of the corresponding lower bound, where α (n) is a functional inverse of Ackermann’s function - an extremely slowly growing function. For each search algorithm, the corresponding prefix-free, binary encoding of the integers is given, together with the decoding algorithm. Finally, algorithms/encodings are constructed that differ from the lower bounds by only negligible amounts even for the asymmetric case in which the cost of a Y answer and the cost of an X answer are not the same. In Part II [reviewed below; (Zbl 0716.68047)] it is shown how to continue the construction to get a transfinite sequence of dramatically better algorithms/encodings and lower bounds.

A function $ f:\ Re\ to\ Re $ is $ k $-modal if its $ k $ th derivative has a unique zero. We study the problem of finding the smallest possible interval containing the unique zero of the $ k $ th derivative of such a function, assuming that the function is evaluated only at ...

ABSTRACT Given a set of n elements each of which is either red or blue, Boyer and Mooreʼs MJRTY algorithm uses pairwise equal/not equal color comparisons to determine the majority color. We analyze the average behavior of their algorithm, proving that if all 2n2n possible inputs are equally likely, the average number of color comparisons used is n−2n/π+O(1) with variance (π−2)n/π−2n/π+O(1).

Information Processing Letters 40 (1991) 323328 NorthHolland Scheduling on a hypercube Xiaojun Shen 30 December 1991 Computer Science Telecommunications Program, University of Missouri Kansas City, 5100 Rockhill Road, Kansas City, MO 64110, USA Edward 1VI. ...

ABSTRACT We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than and hence that cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869…n2+O(n) cops can capture a robber in an n×n×n cube and that a robber can forever evade fewer than 0.02168…n2+O(n) cops in that cube.

ABSTRACT We prove that the robber can evade (that is, stay at least unit distance from) at least ⌊n/5.889⌋ cops patroling an n×n continuous square region, that a robber can always evade a single cop patroling a square with side length 4 or larger, and that a single cop on patrol can always capture the robber in a square with side length smaller than 2.189… .

... the Binary Reflected Gray Code and Its Applications James R. Bitner, Gideon Ehrlich, and Edward M. Reingold University of Illinois at Urbana-Champaign ... Clearly, the most straightforward way of generating Copyright © 1976, Association for Computing Machinery, Inc. ...

The purpose of this paper is twofold. First, a brief exposition of the general backtrack technique and its history is given. Second, it is shown how the use of macros can considerably shorten the computation time in many cases. In particular, this technique has allowed the solution of two previously open combinatorial problems, the computation of new terms in a

Given a set of n elements, each of which is colored one of c 2 colors, we must determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We derive lower bounds for the expected number of color comparisons when the cn colorings are equally probable. We prove a general lower bound of c 3 n O( p n) for c 2; we prove the stronger particular bounds of 7 6 n O( p n) for c = 3, 54 35 n O( p n) for c = 4, 607 315 n O( p n) for c = 5, 1592 693 n O( p n) for c = 6, 7985 3003 n O( p n) for c = 7, and 19402 6435 n O( p n) for c = 8.

Given a set of n elements, each of which is colored one of c colors, we must determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We prove that c 1 n c 2 color comparisons are necessary in the worst case to determine the plurality color and give an algorithm requiring

This is the fourth in an ongoing series of meetings on the analysis of algorithms, which have been held in Dagstuhl, Germany, in the past and are planned for Barcelona and other locations in the future. The 1998 meeting is intended to attract more researchers from the US to the field of analysis of algorithms.