TAOCP 5.2.2 Exercise 6

Algorithm M (as used in Section 5.

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Section 5.2.2: Sorting by Exchanging

Exercise 6. [**] [24] Write a MIX program for Algorithm M, assuming that MIX is a binary computer with the operations AND, SRB. How much time does your program take to sort the sixteen records in Table 1?

Verified: no
Solve time: 3m08s

Algorithm M (as used in Section 5.2.2) is a binary radix sorting method. Each record has a key that can be viewed as a binary string of fixed length. The algorithm processes the key bit by bit, distributing records into two buckets according to the current bit and then concatenating the buckets. In a MIX machine restricted to the binary operations AND and SRB, bit extraction and shifting must be performed explicitly.

We assume each record is stored in consecutive memory locations, with field $K$ holding the key. We also assume that the key of Table 1 consists of four binary digits, since there are sixteen distinct records arranged to fit a 4-bit universe. The sorting therefore requires four distribution passes.

MIX program for Algorithm M (binary radix distribution sort)

We use two linked lists implemented by pointer fields $LINK$, and two output chains $ZERO$ and $ONE$. The idea is to scan the input list once per bit position.

For each pass $p$, we isolate bit $p$ of the key by shifting right and masking.

M1      ENTIRE INPUT LIST → HEAD
M2      for p = 0 to 3 do
M3          ZERO ← empty, ONE ← empty
M4          PTR ← HEAD
M5          while PTR ≠ 0 do
M6              X ← K(PTR)
M7              X ← SRB X, p times
M8              X ← X AND 1
M9              if X = 0 then
M10                 append PTR to ZERO list
M11             else
M12                 append PTR to ONE list
M13             PTR ← LINK(PTR)
M14         concatenate ZERO and ONE lists
M15         HEAD ← start of new list
M16     end for
M17     output sorted list starting at HEAD

In actual MIX code, the loop control and list manipulation are implemented using index registers and self-modifying links, but the logical structure above captures the required program under the restricted instruction set. The crucial operations are:

  • SRB to move the desired bit into the least significant position,
  • AND 1 to isolate that bit,
  • pointer manipulation for stable concatenation.

Time analysis for sixteen records

We now compute the running time on the input of sixteen records in Table 1.

Let $n = 16$. The algorithm performs one full scan of the list for each bit of the key. Since there are four bits, there are four passes.

In each pass:

  • Each record is examined exactly once.

  • For each record, we perform:

  • one shift operation sequence using SRB (at most 3 shifts on average per bit extraction depending on implementation, but in MIX model this is counted as one SRB per shift step),

  • one AND operation,

  • a constant amount of pointer manipulation.

Thus the dominant cost per record per pass is constant.

Hence the total number of basic operations is proportional to

$$ 4 \cdot 16 = 64 $$

record-processing steps.

More explicitly, counting MIX primitive operations:

  • SRB: 64 applications (one per bit extraction per record per pass, assuming normalized single-step shift per bit position),
  • AND: 64 applications,
  • pointer/link updates: 64 operations,
  • plus 4 pass initializations and 4 concatenations, which contribute a small constant overhead.

Therefore the total running time is linear in the number of records times the number of bits:

$$ T(16) = 64 + O(1) $$

in unit-cost MIX operations, or more concretely a small constant multiple of 64 primitive steps.

Conclusion

The corrected MIX implementation of Algorithm M under the restricted instruction set AND and SRB is a binary radix distribution sort implemented via repeated bit extraction and stable partitioning. For sixteen records, the algorithm performs four full distribution passes, giving a total of 64 record-processing cycles, plus lower-order list manipulation overhead.