# SICP读书笔记-huffman编码的实现

huffman 编码是一种变长前缀式编码，通过利用被编码消息中符号的出现频率（频率出现越高的用越少的码），可以有效的节约空间。在 SICP 的2.3.4节通过实现一个huffman编码树来阐述通过表和数据抽象去操作集合和数的例子。

#### 构建 huffman 编码树

• huffman 树以表的方式来表示，将树分为 叶子节点*和 *非叶子节点

('leaf symbol weight) : 叶子节点，包含标示叶子的符号'leaf， 实际字符 symbol，权重 weight
(left right symbols weight): 非叶子节点, 包含左子树 left, 右子树 right, 实际字符表（它孩子节点的符号汇总表）, 权重 weight (它孩子节点的权重之和).

• 构建过程

```(define (generate-huffman-tree pairs)
(define (successive-merge entry-set)
(cond ((null? entry-set) '())
((null? (cdr entry-set)) (car entry-set))
(else
(make-code-tree (car entry-set) (cadr entry-set))
(cddr entry-set))))))
(successive-merge (make-leaf-set pairs)))```

#### 通过huffman编码和解码

• 编码通过针对消息中的每个字符，遍历 huffman 数，如果往左则增加一个0，往右为1，到达叶子节点时得到的2进制序列就是该字符的编码。以下是针对单个符号的编码算法：
```;编码单个字符
(define (encode-symbol symbol tree)
(if (leaf? tree)
'()
(let ((code-br-pair (encode-branch symbol tree)))
(cons
(car code-br-pair)
(encode-symbol symbol (cadr code-br-pair))))))

;根据字符是在左树还是右树进行编码
(define (encode-branch symbol tree)
(let (
(left (left-branch tree))
(right (right-branch tree))
)
(cond ((member? symbol (symbols left)) (list 0 left))
((member? symbol (symbols right)) (list 1 right))
(else (error "symbol not int left or right branch - " symbol)))))

(define (decode bits tree)
(define (decode-l 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-l (cdr bits) tree))
(decode-l (cdr bits) next-branch)))))
(decode-l 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))))```

#### 完整的代码如下

• 树的构建 `huffman-tree.scm`
```;构建如((A 4) (B 2) (C 1) (D 1))的符号和权重的序对列表，构建huffman树
(define (generate-huffman-tree pairs)
(define (successive-merge entry-set)
(cond ((null? entry-set) '())
((null? (cdr entry-set)) (car entry-set))
(else
(make-code-tree (car entry-set) (cadr entry-set))
(cddr entry-set))))))
(successive-merge (make-leaf-set pairs)))

;定义树叶子的表示法
(define (make-leaf symbol weight)
(list 'leaf symbol weight))
;判断是否是叶子节点
(define (leaf? object)
(eq? (car object) 'leaf))
;获取叶子节点的符号
(define (symbol-leaf x) (cadr x))
;获取叶子节点的权重
(define (weight-leaf x) (caddr x))
;获取树的符号表
(define (symbols tree)
(if (leaf? tree)
(list (symbol-leaf tree))
;获取树的权重
(define (weight tree)
(if (leaf? tree)
(weight-leaf tree)
;获取树的左子树
(define (left-branch tree) (car tree))
;获取树的右子树
(define (right-branch tree) (cadr tree))
;树表示为1个具有4个元素的表：左节点，右节点，符号列表，权重
(define (make-code-tree left right)
(list left
right
(append (symbols left) (symbols right))
(+ (weight left) (weight right))))
;根据权重，构建叶子和树的有序标，方便归并一对最小项
(define (adjoin-set x set)
(cond ((null? set) (list x))
((> (weight x) (weight (car set))) (cons (car set) (adjoin-set x (cdr set))))
(else (cons x set))))
;构造叶子的初始排序集合
(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))))))```

• 编码和解码 `huffman-code.scm`
```(load "huffman-tree.scm")
;编码消息
(define (encode message tree)
(if (null? message)
'()
(append
(encode-symbol (car message) tree)
(encode (cdr message) tree))))

;编码单个字符
(define (encode-symbol symbol tree)
(if (leaf? tree)
'()
(let ((code-br-pair (encode-branch symbol tree)))
(cons
(car code-br-pair)
(encode-symbol symbol (cadr code-br-pair))))))

;根据字符是在左树还是右树进行编码
(define (encode-branch symbol tree)
(let (
(left (left-branch tree))
(right (right-branch tree))
)
(cond ((member? symbol (symbols left)) (list 0 left))
((member? symbol (symbols right)) (list 1 right))
(else (error "symbol not int left or right branch - " symbol)))))
;包含关系判断
(define (member? item set)
(not (equal? (member item set) false)))
;解码消息
(define (decode bits tree)
(define (decode-l 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-l (cdr bits) tree))
(decode-l (cdr bits) next-branch)))))
(decode-l 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))))```

• 测试代码 `huffman-use.scm`
```(load "huffman-code.scm")
;定义一个特定的文本串
(define message '(a a b a c a c b b d d d d d d))
;定义初始的叶子节点
(define leaf-set (list '(a 4) '(b 3) '(c 2) '(d 6)))
;根据叶子节点，生成对应的huffman树
(define huffman (generate-huffman-tree leaf-set))
;encode
(define bits (encode message huffman))
(display bits)
;decode
(define msg (decode bits huffman))
(newline)
(display msg)```

• 测试结果如下
```(1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0)
(a a b a c a c b b d d d d d d)```

SICP读书笔记-huffman编码的实现

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