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9085 results in Discrete Mathematics, Information Theory and Coding

Page 81 of 455

  • Drift Analysis and Evolutionary Algorithms Revisited

  • J. LENGLER, A. STEGER
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 4 / July 2018
    Published online by Cambridge University Press:
    20 June 2018, pp. 643-666

  • One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a function f: {0,1}n → ℝ. The algorithm starts with a random search point ξ ∈ {0,1}n, and in each round it flips each bit of ξ with probability c/n independently at random, where c > 0 is a fixed constant. The thus created offspring ξ' replaces ξ if and only if f(ξ') ≥ f(ξ). The analysis of the runtime of this simple algorithm for monotone and for linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.

  • Introduction

  • PAUL BALISTER, BÉLA BOLLOBÁS, IMRE LEADER, ROB MORRIS, OLIVER RIORDAN
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 4 / July 2018
    Published online by Cambridge University Press:
    20 June 2018, p. 441

  • This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

  • 2 - On the Number of ON Cells in Cellular Automata

    • By N. J. A. Sloane, The OEIS Foundation Inc., 11 South Adelaide Ave., Highland Park, NJ 08904, USA
  • Edited by Steve Butler, Iowa State University, Joshua Cooper, University of South Carolina, Glenn Hurlbert, Virginia Commonwealth University
  • Book:
    Connections in Discrete Mathematics
    Published online:
    25 May 2018
    Print publication:
    14 June 2018, pp 13-38

  • Summary

    Abstract

    If a cellular automaton (CA) is started with a single ON cell, how many cells will be ON after n generations? For certain “odd-rule” CAs, including Rule 150, Rule 614, and Fredkin's Replicator, the answer can be found by using the combination of a new transformation of sequences, the run length transform, and some delicate scissor cuts. Several other CAs are also discussed, although the analysis becomes more difficult as the patterns become more intricate.

    Introduction

    When confronted with a number sequence, the first thing is to try to conjecture a rule or formula, and then (the hard part) prove that the formula is correct. This chapter had its origin in the study of one such sequence, 1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, … (A1602391), although several similar sequences will also be discussed.

    These sequences arise from studying how activity spreads in cellular automata (for background see [2, 5, 8, 11, 14, 17, 20, 21, 23, 24, 26]). If we start with a single ON cell, how many cells will be ON after n generations? The preceding sequence arises from the CA known as Fredkin's Replicator [13]. In 2014, Hrothgar sent the author a manuscript [10] studying this CA, and conjectured that the sequence satisfied a certain recurrence. One of the goals of the present chapter is to prove that this conjecture is correct; see (2.31).

    In §2.2 we discuss a general class (the “odd-rule” CAs) to which Fredkin's Replicator belongs, and in §2.3 we introduce an operation on number sequences (the “run length transform”) that helps in understanding the resulting sequences. Fredkin's Replicator, which is based on the Moore neighborhood, is the subject of §2.4, and §2.5 analyzes another odd-rule CA, based on the von Neumann neighborhood with a center cell. Although these two CAs are similar, different techniques are required for establishing the recurrences. Both proofs involve making scissor cuts to dissect the configuration of ON cells into recognizable pieces.

Page 81 of 455