Iteration in PictureWorld -- Row Example

We've seen how to create "rows" of pictures in PictureWorld. The recursive definition for such a method is given below:

// This method returns a picture with p arranged in numberItems
// equally spaced columns (ie a row with numberItems elements).
public Picture row (Picture p, int numberItems) {
  if (numberItems<=0) { // base case: a row with no elements is empty
    return empty();
  } else { // general case: a row is a picture with a row of one
           // fewer number of elements to its right
    return beside(p, row(p, numberItems-1), 1.0/numberItems);
  }
}

To create an iterative version, we need to eliminate the pending operation above which is the beside that is left to do once the recursive call finishes. In order to do that, we will need to introduce a new state variable which will hold our partial results at each step of the iteration. For this particular problem, the following state table can be derived for the execution of row(patch(Color.red),4)

numberItems	numberDone	p	new_p
4	0
4	1
4	2
4	3
4	4

In the table above, the state variable numberDone indicates the number of pictures we have added to that row and the state variable new_p is the partial result thus far. Looking at the state table, we can make the following observations on how to get from one row of the table to the next row of the table:

the numberItems does not change
the numberDone increases by one
the picture to be added, p, does not change
the new_p for each row can be derived from the previous row by doing
beside(p,new_p,1.0/(numberDone+1))
If we already have three pictures and we want to add a fourth picture, that fourth picture will take up 1/4 of the space which is 1.0/(3+1).

The code for the tail-recursive, while loop, and for loop version of row are below:

// iterative (tail-recursive) version of row
public Picture rowIter (Picture p, int numberItems) {
  return rowTail(numberItems,0,p,empty());
}
	
public Picture rowTail (int numberItems, int numberDone, Picture p, Picture new_p) {
  if (numberDone==numberItems) { // we're finished
    return new_p;
  } else { // create next row of state table
    return rowTail(numberItems, numberDone+1, p,
                   beside(p,new_p,1.0/(numberDone+1)));
  }
}
	
// while loop version of row
public Picture rowWhile (Picture p, int numberItems) {
  Picture new_p = empty(); // initialize state variable
  int numberDone = 0; // initialize counter
  while (numberDone<numberItems) { // continuation condition
    new_p = beside(p,new_p,1.0/(numberDone+1)); // update new_p
    numberDone = numberDone + 1; // update counter
  }
  return new_p; // we're done so return our picture
}
	
// for loop version of row
public Picture rowFor (Picture p, int numberItems) {
  Picture new_p = empty(); // initialize state variable
  for (int numberDone=0; numberDone<numberItems; numberDone=numberDone+1) {
    new_p = beside(p,new_p,1.0/(numberDone+1));
  }
  return new_p; // we're done so return our picture
}

Notice the similarity between the while and for loop versions above. A for loop is just syntactic sugar (an easier way of writing) for a while loop. The blue bold lines in the while loop get placed all on one line in the for loop.