TAOCP 5.2.5 Exercise 5
The previous solution’s structural idea is essentially correct, but the running-time analysis must be rebuilt using Knuth’s original definition of the cost components of Program R, in which the parame...
Section 5.2.5: Sorting by Distribution
Exercise 5. [20] [20] What changes are necessary to Program R so that it sorts eight-byte keys instead of three-byte keys? Assume that the most significant bytes of $K_i$ are stored in location KEY+$i$ (1:5), while the three least significant bytes are in location INPUT+$i$ (1:3) as presently. What is the running time of the program, after these changes have been made?
Verified: no
Solve time: 3m46s
The previous solution’s structural idea is essentially correct, but the running-time analysis must be rebuilt using Knuth’s original definition of the cost components of Program R, in which the parameter $F$ is a single global quantity for the entire execution.
1. Required changes to Program R
Program R performs a radix sort using 8-bit digits. The original version assumes 3-byte keys, hence 3 distribution passes. For 8-byte keys, the algorithm must perform 8 passes.
1.1 Number of passes
Replace the outer loop bound
$$ p = 3 \quad \text{by} \quad p = 8. $$
Thus the algorithm performs 8 complete distribution and collection cycles.
1.2 Digit access across INPUT and KEY
The most significant 5 bytes are stored in KEY+(1:5) and the least significant 3 bytes in INPUT+(1:3).
For pass $k$, the digit used is the $(9-k)$-th byte of the key:
$$ a_{9-k}, \quad k = 1,2,\dots,8. $$
This yields:
- $k=1,2,3$: $a_8,a_7,a_6$ from
INPUT+(1:3) - $k=4,5,6,7,8$: $a_5,a_4,a_3,a_2,a_1$ from
KEY+(1:5)
No structural change is required in the pile mechanism. Only the digit-extraction step must select the correct storage array depending on $k$.
1.3 Control modification
The switch table (R3SW in Knuth’s notation) must be extended from 3 cases to 8 cases so that each pass selects the correct byte.
No other changes are required to the queue or pile logic.
2. Correct interpretation of the cost model
In Knuth’s analysis of Program R, the expression
$$ 32N + 48M + 38 - 4F $$
already represents the total cost of a complete execution of one distribution-collection cycle for a fixed digit position, where:
- $N$: number of records processed per pass,
- $M$: number of piles (radix buckets),
- $F$: total number of times an empty pile is encountered during the entire execution of the program.
Crucially, $F$ is not a per-pass quantity. It is accumulated globally over the full run of Program R.
Therefore, cost components linear in the number of passes scale with $p$, while $F$ does not decompose into independent per-pass contributions.
3. Running time for 8-byte keys
Each pass performs the same structural work as in the 3-byte case:
- scanning all $N$ records,
- distributing into $M$ piles,
- collecting from piles,
- fixed control overhead.
Hence the per-pass contribution remains:
$$ 32N + 48M + 38. $$
Since there are now 8 passes, the total contribution of these terms is:
$$ 8(32N + 48M + 38). $$
The term involving empty piles remains unchanged because it counts events over the full execution:
$$ -4F. $$
4. Final result
Running time
$$ T(8) = 8(32N + 48M + 38) - 4F. $$
Expanding constants if desired:
$$ T(8) = 256N + 384M + 304 - 4F. $$
5. Summary of changes
- Increase number of radix passes from 3 to 8.
- Extend digit-selection logic to 8 bytes spanning
INPUTandKEY. - Expand switch table for 8-way digit selection.
- No change to pile structure or distribution mechanism.
- Running time scales linearly with number of passes except for the global $F$ term.
$$ \boxed{T(8) = 8(32N + 48M + 38) - 4F} $$