Comparison sort

Abstract: We initiate the systematic study of the energy complexity of algorithms (in addition to time and space complexity) based on Landauer's Principle in physics, which gives a lower bound on the amount of energy a system must dissipate if it destroys information. We propose energy-aware variations of three standard models of computation: circuit RAM, word RAM, and transdichotomous RAM. On top of these models, we build familiar high-level primitives such as control logic, memory allocation, and garbage collection with zero energy complexity and only constant-factor overheads in space and time complexity, enabling simple expression of energy-efficient algorithms. We analyze several classic algorithms in our models and develop low-energy variations: comparison sort, insertion sort, counting sort, breadth-first search, Bellman-Ford, Floyd-Warshall, matrix all-pairs shortest paths, AVL trees, binary heaps, and dynamic arrays. We explore the time/space/energy trade-off and develop several general techniques for analyzing algorithms and reducing their energy complexity. These results lay a theoretical foundation for a new field of semi-reversible computing and provide a new framework for the investigation of algorithms.

Abstract: Sample sort, a generalization of quicksort that partitions the input into many pieces, is known as the best practical comparison based sorting algorithm for distributed memory parallel computers. We show that sample sort is also useful on a single processor. The main algorithmic insight is that element comparisons can be decoupled from expensive conditional branching using predicated instructions. This transformation facilitates optimizations like loop unrolling and software pipelining. The final implementation, albeit cache efficient, is limited by a linear number of memory accesses rather than the \(\mathcal{O}\!\left(n\log n\right)\) comparisons. On an Itanium 2 machine, we obtain a speedup of up to 2 over std::sort from the GCC STL library, which is known as one of the fastest available quicksort implementations.

Abstract: We initiate the systematic study of the energy complexity of algorithms (in addition to time and space complexity) based on Landauer's Principle in physics, which gives a lower bound on the amount of energy a system must dissipate if it destroys information. We propose energy-aware variations of three standard models of computation: circuit RAM, word RAM, and transdichotomous RAM. On top of these models, we build familiar high-level primitives such as control logic, memory allocation, and garbage collection with zero energy complexity and only constant-factor overheads in space and time complexity, enabling simple expression of energy-efficient algorithms. We analyze several classic algorithms in our models and develop low-energy variations: comparison sort, insertion sort, counting sort, breadth-first search, Bellman-Ford, Floyd-Warshall, matrix all-pairs shortest paths, AVL trees, binary heaps, and dynamic arrays. We explore the time/space/energy trade-off and develop several general techniques for analyzing algorithms and reducing their energy complexity. These results lay a theoretical foundation for a new field of semi-reversible computing and provide a new framework for the investigation of algorithms.

Abstract: In a neural network of N neuron circuits, having an engaged neuron's calculated p bit wide distance between an input vector and a prototype vector and stored in the weight memory thereof, an aggregate search/sort circuit (517) of N engaged neurons' search/sort circuits. The aggregate search/sort circuit determines the minimum distance among the calculated distances. Each search/sort circuit (502-1) has p elementary search/sort units connected in series to form a column, such that the aggregate circuit is a matrix of elementary search/sort units. The distance bit signals of the same bit rank are applied to search/sort units in each row. A feedback signal is generated by ORing in an OR gate (12.1) all local search/sort output signals from the elementary search/sort units of the same row. The search process is based on identifying zeroes in the distance bit signals, from the MSB's to the LSB's. As a zero is found in a row, all the columns with a one in that row are excluded from the subsequent row search. The search process continues until only one distance, the minimum distance, remains and is available at the output of the OR circuit. The above described search/sort circuit may further include a latch allowing the aggregate circuit to sort remaining distances in increasing order.