Simulated Annealing Meta-heuristic for Addition Chain Optimization

In this work, a simulated annealing (SA) algorithm is implemented in the Python programming language with the aim of minimizing addition chains of the "star-chain" type. The strategies for generating and mutating individuals are similar to those used by the evolutionary programming (EP) and genetic algorithms (GA) methods found in the literature [1]-[3]. The proposed variant is the acceptance mechanism that is based on the simulated annealing meta-heuristic (SA). The hypothesis is that with the proposed acceptance mechanism, diversity is obtained in the search-space through a simple strategy that allows finding better solutions compared to the deterministic method Optimized Window. The simulations were performed with exponents in the range 218-234 and were compared with the results reported in [3], where a GA is proposed to get optimal addition chains. It is concluded that the proposed algorithm is able to find chains of shorter length than those found with the Optimized Window method and with a performance similar to that of the GA proposed in [3].

An Efficient Multicore Algorithm for Minimal Length Addition Chains

A minimal length addition chain for a positive integer m is a finite sequence of positive integers such that (1) the first and last elements in the sequence are 1 and m, respectively, (2) any element greater than 1 in the sequence is the addition of two earlier elements (not necessarily distinct), and (3) the length of the sequence is minimal. Generating the minimal length addition chain for m is challenging due to the running time, which increases with the size of m and particularly with the number of 1s in the binary representation of m. In this paper, we introduce a new parallel algorithm to find the minimal length addition chain for m. The experimental studies on multicore systems show that the running time of the proposed algorithm is faster than the sequential algorithm. Moreover, the maximum speedup obtained by the proposed algorithm is 2.5 times the best known sequential algorithm.