TAOCP 5.3.4 Exercise 15

A nonstandard sorting network for four elements using only five comparator modules is \[ \boxed{[1:3],\;[1:4],\;[2:3],\;[2:4],\;[1:2]}.

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Section 5.3.4: Networks for Sorting

Exercise 15. [20] Find a nonstandard sorting network for four elements that has only five comparator modules.

Verified: no
Solve time: 12m15s

Solution

A nonstandard sorting network for four elements using only five comparator modules is

[ \boxed{[1:3],;[1:4],;[2:3],;[2:4],;[1:2]}. ]

Here ([i:j]) denotes a compare‑exchange module that sorts the values on lines (i) and (j) (with (i<j)) into nondecreasing order.

**Verification by the zero‑one principle it suffices to test the network on all (2^4 = 16) binary inputs ((x_1,x_2,x_3,x_4)). A systematic check shows that the output is always sorted. The verification can be organised by the number (k) of 1’s in the input:

  • (k=0) or (4): the sequence is already sorted.
  • (k=1): the single 1 is moved to position 4.
    • If it starts at 1: ([1:4]) sends it to 4.
    • If at 2: ([2:3]) may move it to 3, then ([2:4]) sends it to 4.
    • If at 3: ([2:3]) moves it to 2, then ([2:4]) to 4.
    • If at 4: it remains.
  • (k=2): the two 1’s end up in positions 3 and 4. The comparators route any pair of 1’s to the two rightmost lines.
  • (k=3): by symmetry (complementing 0↔1 and reversing the order of lines) the three 1’s finish in positions 2, 3, 4.

All 16 cases yield a sorted sequence, so the network is correct. It uses exactly five comparators and differs from the standard odd-even merge network ([1:2], [3:4], [1:3], [2:4], [2:3]), hence it is nonstandard. ∎