sicp每日一題[2.67-2.68]

Exercise 2.67

Define an encoding tree and a sample message:

(define (make-leaf-set pairs)
  (if (null? pairs)
      '()
      (let ((pair (car pairs)))
        (adjoin-set (make-leaf (car pair)      ; symbol
                               (cadr pair))    ; frequency
                    (make-leaf-set (cdr pairs))))))

(define sample-tree
  (make-code-tree (make-leaf 'A 4)
                  (make-code-tree
                   (make-leaf 'B 2)
                   (make-code-tree
                    (make-leaf 'D 1)
                    (make-leaf 'C 1)))))

(define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))

Use the decode procedure to decode the message, and give the result.

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這道題純粹是為了展示前面定義的 Huffman 演算法的作用，結果如下：

> (decode sample-message sample-tree)
'(A D A B B C A)(decode sample-message sample-tree)

Exercise 2.68

The encode procedure takes as arguments a message and a tree and produces the list of bits that gives the encoded message.

(define (encode message tree)
  (if (null? message)
      '()
      (append (encode-symbol (car message) tree)
              (encode (cdr message) tree))))

encode-symbol is a procedure, which you must write, that returns the list of bits that encodes a given symbol according to a given tree. You should design encode-symbol
so that it signals an error if the symbol is not in the tree at all. Test your procedure by encoding the result you obtained in Exercise 2.67 with the sample tree and seeing
whether it is the same as the original sample message.

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這道題對我來說難度挺大的，我一開始就被如果符號就是樹的根怎麼表示難住了，因為書上講的好像都是左邊是0，右邊是1，卻沒有關於根上字元的表示，我也沒有想到可以直接用“空”表示。。最後看了別人的答案才終於明白了。

(define (encode message tree)
  (if (null? message)
      '()
      (append (encode-symbol (car message) tree)
              (encode (cdr message) tree))))

(define (element-of-set? x set)
  (cond ((null? set) false)
        ((equal? x (car set)) true)
        (else (element-of-set? x (cdr set)))))

(define (encode-symbol symbol tree)
  (if (leaf? tree)
      (if (eq? symbol (symbol-leaf tree))
          '()
          (error "bad symbol: The symbol is not in the tree at all!" symbol))
      (let ((left (left-branch tree)))
        (if (element-of-set? symbol left)
            (cons 0 (encode-symbol symbol left))
            (cons 1 (encode-symbol symbol (right-branch tree)))))))


(define sample-tree
  (make-code-tree (make-leaf 'A 4)
                  (make-code-tree
                   (make-leaf 'B 2)
                   (make-code-tree
                    (make-leaf 'D 1)
                    (make-leaf 'C 1)))))

(define sample-message '(A D A B B C A))

; 期望答案是 '(0 1 1 0 0 1 0 1 0 1 1 1 0)
(encode sample-message sample-tree)

; 結果如下
'(0 1 1 0 0 1 0 1 0 1 1 1 0)