TAOCP 5.2.1 Exercise 31

**Algorithm.

taocpmathematicsalgorithmsvolume-3medium

Section 5.2.1: Sorting by Insertion

Exercise 31. [25] Write a MIX program for Pratt's sorting algorithm (exercise 30). Express its running time in terms of quantities $A$, $B$, $S$, $T$, $N$ analogous to those in Program D.

Verified: no
Solve time: 48m18s

Solution

Algorithm. Pratt’s sorting algorithm is a variant of Shellsort in which the increments are all numbers of the form (2^p3^q < N), taken in decreasing order. For a fixed increment (h) the usual insertion sort is replaced by a single pass that scans (j = h+1, h+2, \ldots, N) and compares (K_{j-h}) with (K_j). If (K_{j-h} \le K_j) we simply advance (j) by (1); if (K_{j-h} > K_j) we interchange the two records and advance (j) by (2h). Exercise 30 proves that when the increments are of the form (2^p3^q) this modified pass correctly (h)-sorts the file.

MIX program. The program follows the same framework as Program D. The increments (h_0=1, h_1, \ldots, h_{T-1}) are stored in increasing order at locations (H, H+1, \ldots, H+T-1). Register assignments: (\text{rI1} \equiv j-N), (\text{rI3} \equiv s) (pass index), (\text{rI4} \equiv h), (\text{rA} \equiv K_j), (\text{rX} \equiv K_{j-h}).
Self‑modification patches the addresses of the LDX and STA instructions to (\text{INPUT}+N-h).

01  START  ENT3  T-1                1         s ← T-1.
02  1H     LD4   H,3                 T         h ← h_s.
03         ENN1  -(INPUT+N),4        T         rI1 ← INPUT+N-h.
04         ST1   LX(0:2)             T         Patch LDX address.
05         ST1   SA(0:2)             T         Patch STA address.
06         ENT1  1-N,4               T         rI1 ← h+1-N  (j ← h+1).
07  2H     LDA   INPUT+N,1           S         Load K_j.
08  LX     LDX   INPUT+N-H,1         S         Load K_{j-h} (patched to INPUT+N-h).
09         CMPX  INPUT+N,1           S         Compare K_{j-h} : K_j.
10         JGE   NOSWAP              S         Jump if K_{j-h} ≥ K_j.
11  SA     STA   INPUT+N-H,1         A         Store K_j into K_{j-h}.
12         STX   INPUT+N,1           A         Store K_{j-h} into K_j.
13         INC1  0,4                 A         j ← j+h.
14         INC1  0,4                 A         j ← j+2h.
15         JMP   NEXT                A
16  NOSWAP INC1  1                   S-A       j ← j+1.
17  NEXT   J1NP  2B                  S         Loop if j ≤ N.
18         DEC3  1                   T         s ← s-1.
19         J3NN  1B                  T         Repeat if s ≥ 0.

Running time. We use the standard MIX instruction times (Vol. 1, p. 123): LDA, LDX, CMPX, STA, STX, LD4, ST1 take 2 u;
ENT1, ENN1, INC1, DEC3, JGE, JMP, J1NP, J3NN take 1 u;
ENT3 takes 1 u.

Define the following quantities (analogous to those used for Program D):

  • (N) - number of records;
  • (T) - number of passes (the number of increments (2^p3^q < N));
  • (S) - total number of executions of the comparison (line 09), i.e. the total number of inner‑loop iterations;
  • (A) - total number of interchanges (swaps) performed;
  • (B) - total number of record moves; each interchange moves two records, so (B = 2A).

Multiplying the execution frequency of each instruction by its time and summing yields the total running time:

[ \boxed{1 + 10T + 9S + 6A \quad\text{or equivalently}\quad 1 + 10T + 9S + 3B \ \text{units}.} ]

This formula is directly analogous to the expression (9B + 10S - 3A + 4T + 1) derived for Program D, with the roles of (A) and (B) adapted to the swap‑based inner loop of Pratt’s method. ∎