NDLI: Star reduction among minimal length addition chains

Content Provider	Springer Nature Link
Author	Bahig, Hatem M.
Copyright Year	2011
Abstract	An addition chain for a natural number n is a sequence $${1=a_0 < a_1 < \cdots < a_r=n}$$ of numbers such that for each 0 < i ≤ r, a i = a j + a k for some 0 ≤ k ≤ j < i. The minimal length of an addition chain for n is denoted by ℓ(n). If j = i − 1, then step i is called a star step. We show that there is a minimal length addition chain for n such that the last four steps are stars. Then we conjecture that there is a minimal length addition chain for n such that the last $${\lfloor\frac{\ell(n)}{2}\rfloor}$$ -steps are stars. We verify that the conjecture is true for all numbers up to 218. An application of the result and the conjecture to generate a minimal length addition chain reduce the average CPU time by 23–29% and 38–58% respectively, and memory storage by 16–18% and 26–45% respectively for m-bit numbers with 14 ≤ m ≤ 22.
Ending Page	352
Page Count	18
Starting Page	335
File Format	PDF
ISSN	0010485X
e-ISSN	14365057
Journal	Computing
Issue Number	4
Volume Number	91
Language	English
Publisher	Springer Vienna
Publisher Date	2011-01-05
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Minimal length addition chain Addition chain Computer Science Calculation of integer sequences Branch and bound algorithm General Star chain
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics Numerical Analysis Computer Science Applications Software

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