Recursive Data Structures

Read this page. In the previous unit of our course we studied recursive algorithms. Recursion is a concept that also applies to data. Here we look at recursive data structures - lists, trees, and sets. A list is a structure that consists of elements linked together. If an element is linked to more than one element, the structure is a tree. If each element is linked to two (sub) elements, it is called a binary tree. Trees can be implemented using lists, as shown in the resource for this unit. Several examples of the wide applicability of lists are presented. A link points to all the remaining links, i.e. the rest of the list or the rest of the tree; thus, a link points to a list or to a tree - this is data recursion.

The efficiency of the programming process includes both running time and size of data. This page discusses the latter for recursive lists and trees.

Lastly, why read the last section on sets? Sets are another recursive data structure and the last section 2.7.6, indicates their connection with trees, namely, a set data type can be implemented in several different ways using a list or a tree data type. Thus, the programming process includes implementation decisions, in addition, to design or algorithm decisions. Each of these types of decisions is constrained by the features of the programming language used. The decision choices, such as which data structure to use, will impact efficiency and effectiveness of the program's satisfaction of the program's requirements.

Note: You will notice an unusual use of C++ here. What the author is doing is showing how to pass a fixed-value data-structure as a calling argument.

4. Accidental complexity

Rotating a square in this recursive manner is intellectually stimulating, but our code is encumbered with some accidental complexity. Here's a flashing strobe-and-neon hint of what it is:

const firstHalf = (array) => array.slice(0, array.length / 2);
const secondHalf = (array) => array.slice(array.length / 2);

const divideSquareIntoRegions = (square) => {
  const upperHalf = firstHalf(square);
  const lowerHalf = secondHalf(square);

  const upperLeft = upperHalf.map(firstHalf);
  const upperRight = upperHalf.map(secondHalf);
  const lowerRight = lowerHalf.map(secondHalf);
  const lowerLeft= lowerHalf.map(firstHalf);

  return [upperLeft, upperRight, lowerRight, lowerLeft];
};

divideSquareIntoRegions is all about extracting region squares from a bigger square, and while we've done our best to make it readable, it is rather busy. Likewise, here's the same thing in rotateAndCombineArrays:

const rotateAndCombineArrays = ([upperLeft, upperRight, lowerRight, lowerLeft]) => {
  // rotate
  [upperLeft, upperRight, lowerRight, lowerLeft] =
    [lowerLeft, upperLeft, upperRight, lowerRight];

  // recombine
  const upperHalf = [...zipWith(concat, upperLeft, upperRight)];
  const lowerHalf = [...zipWith(concat, lowerLeft, lowerRight)];

  return concat(upperHalf, lowerHalf);
};

rotateAndCombineArrays is a very busy function. The core thing we want to talk about is actually the rotation: Having divided things up into four regions, we want to rotate the regions. The zipping and concatenating is all about the implementation of regions as arrays.

We can argue that this is necessary complexity, because squares are arrays, and that's just what we programmers do for a living, write code that manipulates basic data structures to do our bidding.

But what if our implementation wasn't an array of arrays? Maybe divide and combine could be simpler? Maybe that complexity would turn out to be unnecessary after all?