An Efficient O( $N$ ) Comparison-Free Sorting Algorithm

In this paper, we propose a novel sorting algorithm that sorts input data integer elements on-the-fly without any comparison operations between the data—comparison-free sorting. We present a complete hardware structure, associated timing diagrams, and a formal mathematical proof, which show an overall sorting time, in terms of clock cycles, that is linearly proportional to the number of inputs, giving a speed complexity on the order of O(N). Our hardware-based sorting algorithm precludes the need for SRAM-based memory or complex circuitry, such as pipelining structures, but rather uses simple registers to hold the binary elements and the elements’ associated number of occurrences in the input set, and uses matrix-mapping operations to perform the sorting process. Thus, the total transistor count complexity is on the order of O(N). We evaluate an application-specified integrated circuit design of our sorting algorithm for a sample sorting of N = 1024 elements of size K = 10-bit using 90-nm Taiwan Semiconductor Manufacturing Company (TSMC) technology with a 1 V power supply. Results verify that our sorting requires approximately 4– $6~\mu \text{s}$ to sort the 1024 elements with a clock cycle time of 0.5 GHz, consumes 1.6 mW of power, and has a total transistor count of less than 750 000.