2.69

练习 2.69 下面过程以一个符号-频度对偶表为参数（其中没有任何符号出现在多于一个对偶中），并根据Huffman算法生成Huffman编码树。

(define (generate-huffman-tree pairs)
    (successive-merge (make-leaf-set pairs))
)

其中的make-leaf-set是前面给出的过程，它将对偶表变换为叶的有序集，successive-merge是需要你写的过程，它使用make-code-tree反复归并集合中具有最小权重的元素，直至集合里只剩下一个元素为止。这个元素就是我们所需要的Huffman树。（这一过程稍微有点技巧性，但并不很复杂。如果你正在设计的过程变得很复杂，那么几乎可以肯定是在什么地方搞错了。你应该尽可能地利用有序集合表示这一事实。）

make-leaf-set的代码如下：

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

adjoin-set 代码如下：

(define (adjoin-set x set)
  (cond
   ((null? set) (list x))
   ((<= (weight x) (weight (car set))) (cons x set))
   (else (cons (car set) (adjoin-set x (cdr set))))
   )
  )

上一练习中的代码：

(define (leaf? object) (eq? (car object) 'leaf))

(define (symbol-leaf x) (cadr x))
(define (weight-leaf x) (caddr x))

(define (left-branch tree) (car tree))
(define (right-branch tree) (cadr tree))

(define (encode-symbol-with-code char tree code)    
    (if (leaf? tree)
        (if (eq? (symbol-leaf tree) char)
            (list code)
            '()
        )

        (let
            (
                (left (encode-symbol-with-code char (left-branch tree) '0))
                (right (encode-symbol-with-code char (right-branch tree) '1))  
            )

            (let
                (
                    (res (append left right))
                )

                (if (null? res)
                    '()
                    (append (list code) res) 
                )
            )
        )
    )
)

(define (encode-symbol char tree)
    ; (encode-symbol-with-code char tree '0)

    (if (leaf? tree)
        (if (eq? (symbol-leaf tree) char)
            (list '0)
            '()
        )

        (let
            (
                (left-res (encode-symbol-with-code char (left-branch tree) '0))
                (right-res (encode-symbol-with-code char (right-branch tree) '1))
            )

            (append left-res right-res)
        )
    )
)

(encode-symbol 'A (list 'leaf 'A 1))

decode 的代码如下：

(define (decode bits tree)
    (define (decode-1 bits current-branch)
        (if (null? bits)
            '()
            (let 
                (
                    (next-branch
                        (choose-branch 
                            (car bits)
                            current-branch
                        )
                    )
                )

                (if (leaf? next-branch)
                    (cons 
                        (symbol-leaf next-branch) 
                        (decode-1 (cdr bits) tree)
                    )
                    (decode-1 (cdr bits) next-branch)
                )
            )
        )
    )

    (decode-1 bits tree)
)

(define (choose-branch bit branch)
    (cond 
        ((= bit 0) (left-branch branch))
        ((= bit 1) (right-branch branch))
        (else
            (error "bad bit -- CHOOSE-BRANCH" bit)
        )
    )
)

(define (make-leaf symbol weight)
    (list 'leaf symbol weight)
)

(define (make-code-tree left right)
    (list 
        left
        right
        (append (symbols left) (symbols right))
        (+ (weight left) (weight right))
    )
)

(define (symbols tree)
    (if (leaf? tree)
    (list (symbol-leaf tree))
    (caddr tree))
)

(define (weight tree)
    (if 
        (leaf? tree)
        (weight-leaf tree)
        (cadddr 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 '(0 1 1 0 0 1 0 1 0 1 1 1 0))

(decode sample-message sample-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)
            )
        )
    )
)


(encode-symbol 'A sample-tree)

测试一下 make-leaf-set：

(define sorted (make-leaf-set (list (list 'A 8) (list 'B 4) (list 'C 2) (list 'D 1))))

sorted

(define p (list
        (list 'A 2)
        (list 'NA 16)
        (list 'BOOM 1)
        (list 'SHA 3)
        (list 'GET 2)
        (list 'YIP 9)
        (list 'JOB 2)
        (list 'WAH 1)
    ))

(make-leaf-set 
    p
)

果然能将对偶表变成叶的有序集合！现在来写successive-merge：

(define (successive-merge sorted-leaf-set)
    (if (<= (length sorted-leaf-set) 1)
        sorted-leaf-set
        (let
            (
                (left (car sorted-leaf-set))
                (right (cadr sorted-leaf-set))
            )

            (let 
                (
                    (new-leaf (make-code-tree left right))
                    (rest (cddr sorted-leaf-set))
                )

                (if (null? rest) 
                    new-leaf 
                    (successive-merge 
                        (adjoin-set new-leaf rest)
                    )
                )
            )
        )
    )
)

(successive-merge sorted)

(generate-huffman-tree p)