Optimize Inner Loops

Optimize those code fragments where the algorithm spends most of its time.
Often these fragments
are inside loops.



Overall code organization: 

	lineMethod( )
	{
	    // 1. general set up
	    // 2. special case set up
	    while (notdone) {
	        // 3. inner loop
	    }
	}
  • remove unnecessary method invocations

    replace Math.round(m*x0 + b)
    with (int)(m*y0 + b + 0.5)
    Does this always work?

  • use incremental calculations

    Consider the expression
        y = (int)(m*x + b + 0.5)

    The value of y is known at x0 (i.e. it is y0 + 0.5)
    Future values of y can be expressed in terms of previous values with a difference equation:
    yi+1 = yi + m;
    or
    yi+1 = yi - m;
Lecture 5 Slide 8 6.837 Fall '98

Optimize Inner Loops

The most important code fragments to consider when optimizing are those where the algorithm spends most of its time. Often these areas are within loops.

Our improved algorithm can be partitioned into two major sections. The first section is basically set-up code that is used by the second section, the central pixel-drawing while-loop. Most of the pixels are drawn in this while loop.

First optimization: remove unnecessary method invocations. Notice the call to the round class-method of the Math object that is executed for each pixel drawn. You might think that proper rounding is so trival a task that it hardly warrants a special method to begin with. But, it is complicated when differences between postive and negative numbers, and proper handling of fractions with a value of exactly 0.5 are considered. However, in our line drawing code we are either not going to run into these cases (i.e. we only consider positive coordinate values) or we don't care (i.e. we don't expect to see values of 0.5 very often, and when we do we'd like to see them handled consistently). So for our purposes the call to Math.round(m*x0 + b) could be replaced with the following (int)(m*y0 + b + 0.5).

Second optimization: use incremental calculations. Incremental or iterative function calculations use previous function values to compute future values. Often expressions within loops depend on a value that is being incremented or decremented on each loop iteration. In these cases the actual function values might be more efficiently calculated by first computing an initial value of the function during the loop set up, and updating the subsequent function values using a discrete version of a differential equation called a difference equation.

Consider the expression,

m*x0 + b + 0.5,

that is computed inside the first loop. The We can initialize the first value of the function outside of the loop:

y[0] = m*x0 + b + 0.5

. Future values of y depend only on how the value of x changes within the loop, since all other values are constant. Thus subsequent values of y can be computed by one of the following equations:

y[i+1] = y[i] + m; // if x0 is incremented

y[i+1] = y[i] - m; // if x0 is decremented

We can use a single equation if we change the sign of m, based on whether we are incrementing or decrementing, outside of the loop.