Towards optimal multiple constant multiplication: A hypergraph approach

Abstract

In this work a novel approach to the multiple constant multiplication problem, i.e., finding a realization of a number of constant multiplications by using shift and addition with a minimum number of additions, is presented. By using a directed hypergraph, the problem comes down to finding a Steiner hypertree in the graph. The proposed formulation can guarantee an optimal solution, given that an optimal Steiner hypertree is found. However, finding a Steiner tree in a hypergraph is an NP-hard problem. Therefore, we present algorithms for different partial problems and discuss how they can be used to solve the whole problem. An integer linear programming model is used to solve the Steiner hypertree problem. The model can also include additional constraints such as adder depth and fan-out.

Comparison of multiplierless implementation of nonlinear-phase versus linear-phase FIR filters

“…This problem is often referred to as the multiple constant multiplication (MCM) problem. Recently, there has been optimal approaches suggested to solve the MCM problem [4], [5]. However, since the MCM problem is NP-hard it is often preferred to use heuristics.…”

Techniques for avoiding sign-extension in multiple constant multiplication

“…When the coefficients are constant it is possible to realize the multiplications as a network of shifts, adders, and subtracters 1 . The problem of realizing this network such that one or several figures of merit, such as area, delay, or power, has attracted numerous researchers for more than a decade [1]- [15]. This problem is often referred to as the multiple constant multiplication (MCM) problem and the resulting network is often called a multiplier block.…”

“…Methods to solve the MCM problem can roughly be divided into three different classes: sub-expression sharing techniques [3]- [5], [8], [10], [12], adder graph techniques [1], [2], [6], [9], [13]- [15], and difference based techniques [7], [11]. So far, most work has focused on reducing the number of adders, the adder cost, possibly reducing/controlling the maximum number of cascaded adders, the adder depth [6], [9], [10], [12].…”

FIR filter optimization for video processing on FPGAs

“…The depth-first search is able to solve MCM instances in a reasonable time but cannot handle different constraints or cost metrics. Another interesting method to optimally solve the MCM problem was given by Gustafsson who transferred the MCM problem to the problem of finding a Steiner hypertree in a directed hypergraph [24]. He used an optimal 0-1 ILP formulation which is very generic and can be flexibly adopted to different cost metrics (at adder or logic level) and different constraints (adder depth and fan-out).…”