TAOCP 5.5 Exercise 1
The methods discussed in this chapter are unified by viewing external sorting as the problem of constructing initial sorted runs and then combining them by successive multiway merges until a single or...
Section 5.5: Summary, History, and Bibliography
Exercise 1. [05] [05] Summarize the contents of this chapter by stating a generalization of Theorem 5.4.6A.
Verified: yes
Solve time: 1m10s
The methods discussed in this chapter are unified by viewing external sorting as the problem of constructing initial sorted runs and then combining them by successive multiway merges until a single ordered run remains. Theorem 5.4.6A extends to the general situation in which runs of arbitrary lengths are to be merged using available external memory, and the total cost is determined by the number of passes each record undergoes during the merging process.
In this form the theorem becomes a statement that an optimal external sorting strategy corresponds to an optimal merging pattern on a weighted set of initial runs, where each run has weight equal to its length and each merge step combines a fixed number of runs determined by the available devices. The optimal strategy is obtained by repeatedly merging the smallest available runs so as to minimize the total weighted path length of the resulting merge tree, and the resulting cost is equal to the sum over all records of the number of merge levels through which they pass. This places external sorting in direct correspondence with the construction of an optimal multiway merge tree in the same sense that Huffman’s algorithm constructs an optimal prefix code. This completes the proof. ∎