A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 8.5 External Sorting 309replaces several (potentially) sequential passes with a single random access pass. Ifthe processing would not be sequential anyway (such as when all processing is ona single disk drive), no time is lost by doing so.Multiway merging can greatly reduce the number of passes required. If thereis room in memory to store one block for each run, then all runs can be mergedin a single pass. Thus, replacement selection can build initial runs in one pass,and multiway merging can merge all runs in one pass, yielding a total cost of twopasses. However, for truly large files, there might be too many runs for each to geta block in memory. If there is room to allocate B blocks for a B-way merge, andthe number of runs R is greater than B, then it will be necessary to do multiplemerge passes. In other words, the first B runs are merged, then the next B, andso on. These super-runs are then merged by subsequent passes, B super-runs at atime.How big a file can be merged in one pass? Assuming B blocks were allocated tothe heap for replacement selection (resulting in runs of average length 2B blocks),followed by a B-way merge, we can process on average a file of size 2B 2 blocksin a single multiway merge. 2B k+1 blocks on average can be processed in k B-way merges. To gain some appreciation for how quickly this grows, assume thatwe have available 0.5MB of working memory, and that a block is 4KB, yielding128 blocks in working memory. The average run size is 1MB (twice the workingmemory size). In one pass, 128 runs can be merged. Thus, a file of size 128MBcan, on average, be processed in two passes (one to build the runs, one to do themerge) with only 0.5MB of working memory. A larger block size would reducethe size of the file that can be processed in one merge pass for a fixed-size workingmemory; a smaller block size or larger working memory would increase the filesize that can be processed in one merge pass. With 0.5MB of working memory and4KB blocks, a file of size 16 gigabytes could be processed in two merge passes,which is big enough for most applications. Thus, this is a very effective algorithmfor single disk drive external sorting.Figure 8.11 shows a comparison of the running time to sort various-sized filesfor the following implementations: (1) standard Mergesort with two input runs andtwo output runs, (2) two-way Mergesort with large initial runs (limited by the sizeof available memory), and (3) R-way Mergesort performed after generating largeinitial runs. In each case, the file was composed of a series of four-byte records(a two-byte key and a two-byte data value), or 256K records per megabyte of filesize. We can see from this table that using even a modest memory size (two blocks)to create initial runs results in a tremendous savings in time. Doing 4-way mergesof the runs provides another considerable speedup, however large-scale multi-way