Multiplication by a constant is sublinear Vassil Dimitrov Laurent Imbert Andrew Zakaluzny.
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Transcript of Multiplication by a constant is sublinear Vassil Dimitrov Laurent Imbert Andrew Zakaluzny.
![Page 1: Multiplication by a constant is sublinear Vassil Dimitrov Laurent Imbert Andrew Zakaluzny.](https://reader036.fdocuments.net/reader036/viewer/2022070403/56649f295503460f94c430de/html5/thumbnails/1.jpg)
multiplication by a
constant is sublinear
Vassil DimitrovLaurent Imbert
Andrew Zakaluzny
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Introduction– The Topic
Theory– The Double Base Number System
Algorithms for SCM– DBNS Algorithms
Conclusions– Comparisons– Future Work
Question Period
outline
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topic
The Single Constant Multiplication Problem (SCM)
– Multiply an (unknown) integer x by a constant integer c using as few additions and shifts as possible.
– Unknown Complexity– Minimize the cost
Previous Works
– Recoding Techniques – naïve double and add method, SD method
– Bernstein, Lefèvre, Boullis & Tisserand, Dempster & MacLeod
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improving current algorithms
Sublinearity of SCM Algorithms:
– Conjectured by Lefevre in 2003
– Sublinearity: shown by R.G.E Pinch in 1996
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theory
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redundancy & sparseness
Very Redundant:– 10 has exactly 5 different
unsigned DBNRs
– 100 has exactly 402 different unsigned DBNRs
– 1000 has exactly 1,295,579 different unsigned DBNRs
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theory
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a DBNS approach
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DBNS blocking
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numerical data
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visual: constant size blocking
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DBNS SCM blocking algorithm
Known worst case cost of implementation
Highly configurable– LUT or Greedy– Parameter values
Larger LUT = Lower run time costStrategy (64 bit constants): Average Cost
Binary 31
CSD 21.4
DBNS Constant Size Blocking (LUT size = 8) 18.8
DBNS Constant Size Blocking (LUT size = 12) 17.7
DBNS Constant Size Blocking (LUT size = 16) 16.8
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non-trivial block sizes
First alternative:
– Left to Right method: • Skip leading 0s, trailing 0s, and partition
accordingly• Visual Example: LUT size = 4
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non-trivial block sizes
Second alternative:
– Right To Left method: • Look for barriers of 0s (i.e. 00 is a barrier
size of m = 2)• Visual Example: LUT size = 4, m = 2 (00)• Best results when m = 3 or 4
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blocking results
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conclusion
SCM is an interesting and open problem
DBNS for SCM:– Blocking:
• Provably sublinear • Simple• Low bit complexity• Known worst case complexity• Extendable to any bit size (with
appropriate blocking)
– Ideal Parameters have been identified (for 32 and 64 bit constants)
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future work
Comparisons with competing methods– Build in hardware– Implement in a compiler– Better comparisons (speed vs area)
MCM problem
Other Blocking Algorithms
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questions