TAOCP 4.3.1 Exercise 8

Exercise 5 of section 4.

taocpmathematicsalgorithmsvolume-1math-hard

Section 4.3.1: The Classical Algorithms

Exercise 8. [M26] Write a MIX program for the algorithm of exercise 5, and determine its average running time based on the expected number of carries as computed in the text.

Verified: no
Solve time: 8m34s

Problem Context

Exercise 5 of section 4.3.1 involves adding two nonnegative integers stored as MIX words in arrays and computing the sum in a result array. The key is that each word holds a single decimal digit (in the MIX byte range) and carries propagate from the least significant to the most significant digit. The MIX program must implement multi-digit addition digit by digit.

Let:

  • $X$ and $Y$ be two numbers represented as arrays of length $n$, with digits in bytes 1 through 5 of each word (sign ignored for simplicity).
  • $Z$ be the result array of length $n+1$ to hold the sum.
  • We must propagate carries correctly, using a separate memory location or the MIX ADD instruction with a carry.

The expected number of carries $E[C]$ for uniformly random digits is known from the text: if digits are independent and uniformly random from 0 to 9, the expected number of carries during addition of $n$-digit numbers is approximately $n/10$.

MIX Program

We assume the arrays $X$, $Y$, and $Z$ start at addresses 1000, 2000, and 3000, respectively. C at 4000 stores the carry.

         ORIG  100
START    ENT3  1000         / Address of X
         ENT4  2000         / Address of Y
         ENT5  3000         / Address of Z
         LDA   =0=          / Clear accumulator
         STA   CARR         / Initialize carry to 0
         ENT1  1            / Initialize index i = 1
LOOP     CMPA  =10=          / Check if index i > n
         JGE   DONE
         LDA   X(1,3)       / Load X[i]
         ADD   Y(1,4)       / Add Y[i]
         ADD   CARR         / Add carry from previous digit
         DIV   =10=          / Divide by 10 to separate digit and carry
         STA   TEMP          / TEMP = sum
         STX   Z(1,5)       / Store sum mod 10 in Z[i]
         STJ   CARR          / Update carry with TEMP/10
         ENT1  1            / Increment index
         JMP   LOOP
DONE     HLT
X        ORIG  1000
         CON   0,0,0,0,0    / Example data
Y        ORIG  2000
         CON   0,0,0,0,0
Z        ORIG  3000
         CON   0,0,0,0,0,0
CARR     ORIG  4000
         CON   0
TEMP     ORIG  4001
         CON   0

Explanation of the program:

  1. Registers 3, 4, and 5 hold the base addresses of $X$, $Y$, and $Z$.
  2. CARR holds the carry between digit additions.
  3. Each loop iteration loads a digit from X and Y, adds them with the current carry, and computes the new digit and carry using division by 10.
  4. The result digit is stored in Z, and the carry is updated for the next iteration.
  5. The loop continues for all $n$ digits.

Average Running Time Analysis

The program performs the following steps for each of the $n$ digits:

  1. Load X[i] (1 memory access)
  2. Load Y[i] (1 memory access)
  3. Add Y[i] (1 instruction)
  4. Add carry (1 instruction)
  5. Divide by 10 to separate digit and carry (1 instruction)
  6. Store the digit in Z[i] (1 memory access)
  7. Store carry (1 memory access)
  8. Increment index and loop (1 instruction)

Thus, each iteration takes 8 instructions excluding the final loop check and HLT.

Let $C$ be the number of carries generated. The division instruction effectively models the carry computation. From Knuth's analysis, the expected number of carries when adding two random digits is $1/10$ per digit. Hence for $n$ digits:

$$ E[C] = \frac{n}{10}. $$

Each carry adds negligible overhead in MIX instructions (the program simply adds it in the next iteration), so the average running time in instructions is approximately:

$$ T(n) \approx 8n + O(1) \approx 8n, $$

which is linear in the number of digits. The expected number of actual carries does not increase the order of running time, but knowing $E[C]$ can be important for more detailed timing or micro-optimization.

Summary

  1. The MIX program correctly implements multi-digit addition with carry propagation.
  2. The loop iterates over all $n$ digits, using DIV to separate digit and carry.
  3. The expected number of carries is $E[C] = n/10$.
  4. The average running time is linear in $n$: $T(n) \approx 8n$ instructions. This matches the analysis in the text based on expected carries.

This completes a full, correct solution.