Reading

These notes and closely review Unit 3 Section 4 and the end of Section 3.

Quiz questions

No quiz questions for this weekend, but you are required to read the project description, download the file (I fixed the broken links), run the sample code to generate input problems and the sample code to produce solutions, and at least take a look at the sample code for producing solutions, since you're supposed to analyze the algorithm they follow.

Reviewing Old Quiz Questions

We went over the quiz problems for today.

XGCD

The Unit notes and lecture notes showed the Extended Euclidean GCD Algorithm as a recursive algorithm. Here's an iterative version:
Algorithm: XGCD(x,y)
Input : integers x,y ≥ 0
Output: g,s,t such that g is the gcd of x and y, and g = s*a + t*b
------------------------
  (a,b)   = (x,y)
  (sa,ta) = (1,0)
  (sb,tb) = (0,1)

  while b != 0 do
    (q,r) = divide(a,b) ← i.e. a = q*b + r, 0 ≤ 0 < b
    sbp = sa - q*sb
    tbp = ta - q*tb
    (sa,ta) = (sb,tb)
    (sb,tb) = (sbp,tbp)
    (a,b) = (b,r)

  return (a, sa, ta);
}
Great, huh? But do you actually have any reason to believe it works? What would it take to convince you that it works? This is actually quite a simple algorithm in the grand scheme of things, but already it's hard to be confident it is correct. Let me try to convince you! Here's a loop invariant for you: \[ a = s_a x + t_a y \text{ and } b = s_b x + t_b y \] So let's do this:
  1. Initialization: As we encounter the loop for the first time, we have \[ a = x = 1\cdot x + 0\cdot y = s_a x + t_a y \text{ ... check!} \text{ and we have } b = y = 0\cdot x + 1\cdot y = s_b x + t_b y \text{ ... check!} \] So the invariant holds at the beginning
  2. Matainance: Assuming the invariant holds, we have \[ a = qb + r \Rightarrow r = a - qb = (s_a x + t_a y) - q(s_b x + t_b y) = (s_a - q s_b) x + (t_a - q t_b) y = s'_b x + t'_b y \] Then we set $(s_a,t_a) = (s_b,t_b)$ so that after the assignment $a = b$ we have the first part of the invariant. Then we set $(s_b,t_b) = (s'_b,t'_b)$ so that after the assignment $b = r$ we have the second part of the invariant.
  3. Termination: When the loop exits, $a$ is the gcd of $x$ and $y$, as required. The loop invariant tells us that $a = s_a x + t_a y$, so returning $(g,s,t) = (a,s_a,t_a)$ meets the algorithm's specification.
Some important notes: The loop invariant is not the loop continuation condition. It is something new that is not in the code, but tells us something extra about the code. Second, the loop invariant references variables from the program, referring to their value at a point in time. We don't write something like "$a_i$", intending to indicate the value of $a$ "at the $i$th iteration". That's not how it works. The invariant is true each and every time we get to the point where we are about to check the loop continuation condition. Finally, the invariant is strong enough that, we can easily use it to prove the program correct.

Reviewing Problem Set 1

I thought it would be good to go over some of the important some of the Problem Set problems.