Abstract
Multivariate polynomial interpolation is a key computation in many areas of science and engineering and, in our case, is crucial for the solution of the reverse engineering of genetic networks modeled by finite fields. Faster implementations of such algorithms are needed to cope with the increasing quantity and complexity of genetic data. We present a new algorithm based on Lagrange interpolation for multivariate polynomials that not only identifies redundant variables in the data and generates polynomials containing only nonredundant variables, but also computes exclusively on a reduced data set. Implementation of this algorithm to FPGA led us to identify a systolic array-based architecture useful for performing three interpolation subtasks: Boolean cover, distinctness, and polynomial addition. We present a generalization of these tasks that simplifies their mapping to the systolic array, and control and storage considerations to guarantee correct results for input sequences longer than the array. The subtasks were modeled and implemented to FPGA using the proposed architecture, then used as building blocks to implement the rest of the algorithm. Speedups up to172×and67×were obtained for the subtasks and complete application, respectively, when compared to a software implementation, while achieving moderate resource utilization.
Highlights
Recent years have seen a significant increase in methods and tools to collect genetic data from which important information can be extracted using a number of techniques [1]
Our research group focuses on multivariate finite field gene network (MFFGN) models, in which multiple genes are monitored at each time step and their expression levels are discretized to a predefined set of values {0, 1, 2, . . . , p − 1}, where p is a prime number, that is, the expression level of each gene is an element of the prime field Zp [7, 8]
This paper presents a new methodology based on Lagrange interpolation with two important properties: (1) it identifies redundant variables and generates polynomials containing only nonredundant variables, and (2) it computes exclusively on a reduced data set
Summary
Recent years have seen a significant increase in methods and tools to collect genetic data from which important information can be extracted using a number of techniques [1]. Microarray data collected at various steps in organism development can help geneticists understand its developmental process and response to environmental stimuli [2]. Several models such as ordinary differential equation models [3], continuous models [4], stochastic models [5], and discrete models, of which Boolean models [6] are special cases, have been proposed. Our research group focuses on multivariate finite field gene network (MFFGN) models, in which multiple genes are monitored at each time step and their expression levels are discretized to a predefined set of values {0, 1, 2, . Polynomial interpolation over finite fields has applications in error correcting codes and is a major building block of many numerical methods [9, 10]
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