We present a simple randomized construction of size O(n~3) and depth 5.3 log n + O(1) monotone circuits for the majority function on n variables. This result can be viewed as a reduction in the size and a partial derandomization of Valiant's construction of an O(n~(5.3)) monotone formula. On the other hand, compared with the deterministic monotone circuit obtained from the sorting network of Ajtai, Komlos, and Szemeredi, our circuit is much simpler and has depth O(log n) with a small constant. The techniques used in our construction incorporate fairly recent results showing that expansion yields performance guarantee for the belief propagation message passing algorithms for decoding low-density parity-check (LDPC) codes, [3]. As part of the construction, we obtain optimal-depth linear-size monotone circuits for the promise version of the problem, where the number of 1's in the input is promised to be either less than one third, or greater than two thirds. We also extend these improvements to general threshold functions. At last, we show that the size can be further reduced at the expense of increased depth, and obtain a circuit for the majority of size and depth about n~(1+2~(1/2)) and 9.9 log n.
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