Performance engineering case study: heap construction

The behaviour of three methods for constructing a binary heap on a computer with a hierarchical memory is studied. The methods considered are the original one proposed by Williams [1964], in which elements are repeatedly inserted into a single heap; the improvement by Floyd [1964], in which small heaps are repeatedly merged to bigger heaps; and a recent method proposed, e.g., by Fadel et al. [1999] in which a heap is built layerwise. Both the worst-case number of instructions and that of cache misses are analysed. It is well-known that Floyd's method has the best instruction count. Let <i>N</i> denote the size of the heap to be constructed, <i>B</i> the number of elements that fit into a cache line, and let <i>c</i> and <i>d</i> be some positive constants. Our analysis shows that, under reasonable assumptions, repeated insertion and layerwise construction both incur at most <i>cN/B</i> cache misses, whereas repeated merging, as programmed by Floyd, can incur more than (<i>dN</i> log<inf>2</inf> <i>B</i>)/<i>B</i> cache misses. However, for our memory-tuned versions of repeated insertion and repeated merging the number of cache misses incurred is close to the optimal bound <i>N</i>/<i>B</i>. In addition to these theoretical findings, we communicate many practical experiences which we hope to be valuable for others doing experimental algorithmic work.