計算機程序的構造和解釋 練習題2.71

#lang racket

(define (element-of-set? x set)
  (cond ((null? set) #f)
        ((equal? x (car set)) #t)
        (else (element-of-set? x (cdr set)))))

(define (make-leaf symbol weight)
  (list 'leaf symbol weight))        
(define (leaf? object)
  (eq? (car object) 'leaf))
(define (symbol-leaf x) (cadr x))
(define (weight-leaf x) (caddr x))
(define (make-code-tree left right)
  (list left 
        right
        (append (symbols left) (symbols right))
        (+ (weight left) (weight right))))
(define (left-branch tree) (car tree))
(define (right-branch tree) (cadr tree))
(define (symbols tree)
  (if (leaf? tree)
      (list (symbol-leaf tree))
      (caddr tree)))
(define (weight tree)
  (if (leaf? tree)
      (weight-leaf tree)
      (cadddr tree)))

(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 -- CHOSSE-BRANCH" bit))))

(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))))))

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

(define (encode-symbol message tree)
  (cond ((leaf? tree) null)
        ((element-of-set? message (symbols (left-branch tree))) 
         (cons 0 (encode-symbol message (left-branch tree))))
        ((element-of-set? message (symbols (right-branch tree))) 
         (cons 1 (encode-symbol message (right-branch tree))))
        (else (error "bad bit -- CHOSSE-BRANCH" message))))

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

(define (successive-merge set)
  (if (= (length set) 1) (car set)
      (successive-merge (adjoin-set (make-code-tree (car set) (cadr set)) (cddr set)))))


(define n5-pairs '((A 16) (B 8) (C 4) (D 2) (E 1)))
(define n10-pairs '((A 512) (B 256) (C 128) (D 64) (E 32) (F 16) (G 8) (H 4) (I 2) (J 1)))
(define n5-tree (generate-huffman-tree n5-pairs))
(define n10-tree (generate-huffman-tree n10-pairs))
(define n5-max '(A A A A A))
(define n5-min '(E E E E E))
(define n10-max '(A A A A A A A A A A))
(define n10-min '(J J J J J J J J J J))
(length (encode n5-max n5-tree))
(length (encode n5-min n5-tree))
(length (encode n10-max n10-tree))
(length (encode n10-min n10-tree))

運行結果

5
20
10
90

如果是有n個符號進行編碼，那麼，最小的編碼長度位n，最大的編碼長度爲n*(n-1)，因爲最低頻率出現的符號，在樹的最低端，樹一共有n-1層，所以他的編碼長度是n-1，所以一共就是n*(n-1)的長度