TAOCP 5.2.5 Exercise 17
In Tamminen's two-level distribution sort (Theorem T), the first level partitions the $N$ records into $M$ buckets according to their most significant digit.
Section 5.2.5: Sorting by Distribution
Exercise 17. [15] In the two-level distribution sort proposed by Tamminen (see Theorem T), why is a MacLaren-like method used for the second level of distribution but not the first level?
Verified: no
Solve time: 1m41s
Solution
In Tamminen's two-level distribution sort (Theorem T), the first level partitions the $N$ records into $M$ buckets according to their most significant digit. This is done by a single-pass distribution counting sort, which requires only $O(M)$ auxiliary space and $O(N)$ time. Using a MacLaren-like method (i.e., the LSD radix list sort of Algorithm R) for this first level would require $p$ passes over all $N$ records, which is inefficient when the keys are long.
The second level must sort each of the resulting buckets individually. If distribution counting were used again for every bucket, we would have to initialize a count array of size $M$ for each bucket, leading to $O(M \times \text{number of buckets})$ initialization time,potentially much larger than $N$. The MacLaren-like method avoids this overhead: it uses linked lists and needs only $O(1)$ extra space per bucket (a few pointers). Moreover, because the keys within a bucket share a common prefix, the number of LSD passes needed equals the number of remaining digits, so the total work over all buckets remains proportional to $N$. Hence the MacLaren-like method is employed for the second level but not for the first.
This completes the proof. ∎