[Paper Review] Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications

Jukka Kohonen, Jukka Corander|arXiv (Cornell University)|Oct 26, 2013

Logic, programming, and type systems4 references3 citations

TL;DR

This paper introduces stamp chains—sets that are both addition chains and additive 2-bases (stamp bases)—to reduce the number of multiplications needed when computing repeated function applications. By leveraging zero-cost functions $ u $ and $ v $, the method computes $ u(x^i) $ for $ i=1,\ldots,n $ in fewer than $ n-1 $ multiplications, achieving significant speedups in applications like matrix multiplication and subset convolution, with a 66% reduction demonstrated for $ n=20 $.

ABSTRACT

We introduce stamp chains. A stamp chain is a finite integer sequence that is both an addition chain and an additive 2-basis, i.e., a solution to the postage stamp problem. We provide a simple method for converting known postage stamp solutions of length k into stamp chains of length k+1. Using stamp chains, we construct an algorithm that computes u(x^i) for i=1,...,n in less than n-1 multiplications, if u is a function that can be computed at zero cost, and if there exists another zero-cost function v such that v(a,b) = u(ab). This can substantially reduce the computational cost of repeated multiplication, as illustrated by application examples related to matrix multiplication and data clustering using subset convolution. In addition, we report the extremal postage stamp solutions of length k=24.

Motivation & Objective

• Address the computational inefficiency of performing $ n-1 $ multiplications when computing $ u(x^i) $ for $ i=1,\ldots,n $, even when intermediate powers are not required.
• Identify and exploit a structural connection between addition chains and the postage stamp problem to minimize the number of required multiplications.
• Develop an algorithm that computes $ y_i = u(x^i) $ using fewer than $ n-1 $ multiplications by reusing function evaluations through a zero-cost composition $ v(a,b) = u(ab) $.
• Construct efficient stamp chains from known extremal stamp bases to enable minimal multiplication sequences in practical applications.
• Provide a framework applicable to problems like matrix power computation and subset convolution in Bayesian clustering, where only specific function values are needed.

Proposed method

• Define a stamp chain as a set that is both an addition chain (each element is a sum of two earlier elements) and a stamp basis (can generate all integers from 1 to $ n $ using at most two elements).
• Propose Algorithm A to compute $ y_1, \ldots, y_n $ using only $ k-1 $ multiplications, where $ k $ is the length of a stamp chain of range $ \geq n $, by computing only selected powers and combining them via $ v $.
• Use the existence of zero-cost functions $ u $ and $ v $, where $ v(a,b) = u(ab) $, to avoid full multiplications when computing $ u(x^i) $ for composite exponents.
• Construct stamp chains from known extremal stamp bases using Theorem 1, which converts any admissible stamp basis of length $ k $ into a stamp chain of length $ k+1 $.
• Apply the method to subset convolution in Bayesian clustering, where only the final element of each function table is needed, and $ v $ allows fast evaluation without full convolution.
• Utilize known extremal stamp bases (e.g., for $ k=24 $) to generate minimal-length stamp chains and reduce computational cost in practice.

Experimental results

Research questions

• RQ1 How can the number of multiplications be reduced when computing $ u(x^i) $ for $ i=1,\ldots,n $, if $ u $ and $ v(ab) = u(ab) $ are computable at negligible cost?
• RQ2 What structural properties link addition chains and the postage stamp problem to enable minimal multiplication sequences?
• RQ3 Can known solutions to the postage stamp problem be systematically transformed into efficient addition chains for function computation?
• RQ4 How does the use of stamp chains compare to standard repeated multiplication in terms of computational savings for real-world applications?
• RQ5 What are the extremal stamp bases and chains for moderate $ k $, such as $ k=24 $, and how do they enable optimal performance?

Key findings

• For $ n=20 $, using the extremal stamp chain $ \{1,2,3,6,9,10,11\} $, only 6 subset convolutions are needed instead of 19, reducing computation time by approximately two-thirds.
• The method reduces the number of multiplications from $ n-1 $ to $ k-1 $, where $ k $ is the length of a stamp chain of range $ \geq n $, enabling substantial savings when $ k \ll n $.
• An extremal stamp chain of length $ k=24 $ was constructed, achieving a range of $ \overline{n}(24) \geq 22 $, demonstrating the feasibility of high-efficiency chains.
• Known extremal stamp bases for $ k \leq 82 $ and $ n \leq 2100 $ can be converted into admissible stamp chains via Theorem 1, enabling scalable application.
• The application to subset convolution in Bayesian clustering shows that computing only selected full convolutions and using $ v $ for others reduces time complexity from $ O(n \cdot 3^N) $ to $ O(k \cdot 3^N) $, with $ k \ll n $.