Multiplicative Drift Analysis

We introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. Our multiplicative version of the classical drift theorem allows easier analyses in the often encountered situation that the optimization progress is r...

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Veröffentlicht in:Algorithmica Jg. 64; H. 4; S. 673 - 697
Hauptverfasser: Doerr, Benjamin, Johannsen, Daniel, Winzen, Carola
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer-Verlag 01.12.2012
Springer Verlag
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Algorithms
Data Structures and Information Theory
Mathematics of Computing
Neural and Evolutionary Computing
Theory of Computation
Drift analysis
Randomized search heuristics
Evolutionary algorithms
Runtime analysis
ISSN:0178-4617, 1432-0541
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Zusammenfassung:We introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. Our multiplicative version of the classical drift theorem allows easier analyses in the often encountered situation that the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem of how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O ( n log n ), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1+ o (1))1.39e n ln n , again using multiplicative drift analysis. We also prove a corresponding lower bound of (1− o (1))e n ln n which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours in graphs.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-012-9622-x