The solutions for this section are coded in Common Lisp.

Exercise 2.59

The union of two sets is the combination of elements from both sets, without duplicates. For sets set1 and set2, we will walk over the elements of set2, and keep only those that aren’t already members of set1. The remaining elements will be appended to set1.

(defun union-set (set1 set2)
  (append
    set1
    (remove-if
      (lambda (x)
        (element-of-set? x set1))
      set2)))

Exercise 2.60

Sets with duplicates are traditionally called multisets. element-of-set? is the same function. I’ll implement it using CL’s built in function member:

(defun element-of-multiset? (x set)
  (member x set :test #'equal))

Intersection is also similar:

(defun intersection-multiset (set1 set2)
  (cond ((or (null set1) (null set2)) '())
        ((element-of-multiset? (car set1) set2)
          (cons (car set1)
                (intersection-multiset (cdr set1) set2)))
        (t (intersection-multiset (cdr set1) set2))))

However, adjoin-set and union-set are much simpler. Since we no longer care about duplicates, we don’t have to check that the elements we’re adding to a set don’t already exist in it:

(defun adjoin-multiset (x set)
  (cons x set))

(defun union-multiset (set1 set2)
  (append set1 set2))

Efficiency: element-of-multiset? and intersection-multiset will perform roughly the same as their regular set counterparts¹. adjoin-multiset is O(1) (as opposed to O(1) for unordered-list sets) and union-multiset is O(n) (as opposed to O(n^2), assuming that both sets have roughly n elements).

This representation makes sense when most of the operations we do on sets are unions and adjoins, of course.

Exercise 2.61

(defun adjoin-set (x set)
  (cond ((null set) (cons x '()))
        ((< x (car set)) (cons x set))
        ((= x (car set)) set)
        (t (cons
            (car set)
            (adjoin-set x (cdr set))))))

Exercise 2.62

(defun union-set (set1 set2)
  (let ((x1 (car set1)) (x2 (car set2)))
    (cond ((null x1) set2)
          ((null x2) set1)
          ((= x1 x2)
            (cons x1 (union-set (cdr set1) (cdr set2))))
          ((< x1 x2)
            (cons x1 (union-set (cdr set1) set2)))
          (t
            (cons x2 (union-set set1 (cdr set2)))))))

Exercise 2.63

a. Yes, both procedures produce the same result for all trees—the in-order traversal.

b. There’s no difference in the order of growth, both require O(n) steps.

Exercise 2.64

a. partial-tree divides the first n elements from elts into two groups—left and right, such that the sizes of the groups are n/2 (the right group gets the extra element in case of an odd n). It recursively creates a right subtree and a left subtree from these two groups and then cons-es them into the current node, producing a larger tree. It’s a really beautiful example of recursive code, I must add.

The tree produced by it for the list (1 3 5 7 9 11) is (this should be printed out in fixed font):

     5
    / 
   /   
  1     9
      / 
    3 7  11

b. The order of growth is O(n), as eventually partial-tree traverses each element of the list once.

Exercise 2.65

Since we now know how to:

• Convert from an ordered binary tree to an ordered list² in O(n)
• Compute intersection and union of two ordered lists in O(n)
• Convert back from an ordered list to a balanced binary tree in O(n)

We can just combine the operations and end up with a O(n) procedure.

If we keep the name union-set for union computation on ordered lists, here is the union for binary trees:

(defun union-set-bintree (set1 set2)
  (let*  ((lset1 (tree->list-1 set1))
          (lset2 (tree->list-1 set2))
          (lunion (union-set lset1 lset2))
          (union (list->tree lunion)))
    union))

The intersection is very similar³:

(defun intersection-set-bintree (set1 set2)
  (let*  ((lset1 (tree->list-1 set1))
          (lset2 (tree->list-1 set2))
          (lintersect (intersection-set lset1 lset2))
          (intersect (list->tree lintersect)))
    intersect))

Exercise 2.66

The procedure is a simple conversion of the element-of-set? procedure:

(defun lookup (given-key set)
  (if (null set)
    nil
    (let*  ((cur-entry (entry set))
            (cur-key (key cur-entry)))
      (cond ((= cur-key given-key) cur-entry)
            ((< given-key cur-key)
              (lookup
                given-key
                (left-branch set)))
            ((> given-key cur-key)
              (lookup
                given-key
                (right-branch set)))))))

Note that I’ve defined the following simple abstraction for storing key-value pairs in the set instead of simple values:

(defun make-record (key data)
  (list key data))

(defun key (record)
  (car record))

(defun data (record)
  (cadr record))

¹ They will be a little slower on average, since multisets are longer than sets (because of all the duplicates). This depends on the actual data a lot.

² Since tree->list-1 (and tree->list-2) traverse the tree in-order, they will print out an ordered list for an ordered binary tree.

³ Naturally, in real code we’d find away to reuse the common code, perhaps abstracting the relations between the same operators on different representations in a dispatch table.

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