數據結構python實現（二）遞歸

（二）遞歸

1. 什麼樣的問題可以用遞歸來解決

遞歸須要知足三個條件：

（1）一個問題的解能夠分解成幾個子問題的解

（2）這個問題與分解以後的子問題，除了數據規模不一樣，求解思路徹底同樣	

（3）存在遞歸終止條件

寫遞歸代碼的關鍵也是，找到遞推公式和終止條件

2. 案例

案例1 階乘

#循環實現
#時間O(n)
#空間O(1)
def factorial(n):
    result = 1
    for i in range(1, n+1):
        result *= i
    return result
  
#遞歸實現
#時間O(n)
#空間O(n)
ef factorial_recursive(n):
    if n == 1:
        return 1
    return n * factorial_recursive(n - 1)

案例2 斐波那契數列

#遞歸實現
#時間O(2^n)
def fibonacci1(n):
    assert(n>0)
    if (n <= 2): 
        return 1
    return fibonacci2(n-1) + fibonacci2(n-2)
  
  #循環實現
  #時間O(n)
  def fibonacci2(n):
    assert(n>0)
    a,b=0,1
    for i in range(1,n+1):
      a,b=b,a+b
    return a
  
  def fibonacci3(n):
    assert(n>0)
    if (n <= 1): 
        return (n,0)
    (a, b) = fibonacci3(n-1)
    return (a+b, a)

案例3 打印尺子

（熱身）打印一下內容：

1

1 2 1

1 2 1 3 1 2 1

1 2 1 3 1 2 1 4 1 2 1 3 1 2 1

#時間O(2^n)
def ruler_bad(n):
    assert(n>=0)
    if (n==1):
        return "1"
    return ruler(n-1) + " " + str(n) + " " + ruler(n-1)

#時間O(n)
def ruler(n):
    assert(n>=0)
    if (n==1):
        return "1"
    t = ruler(n-1)
    return t + " " + str(n) + " " + t

#循環實現
def ruler2(n):
    result = ""
    for i in range(1, n+1):
        result = result + str(i) + " " + result
    return result

畫尺子：

#畫橫線
def draw_line(tick_length, tick_label=''):
    line = '-' * tick_length
    if tick_label:
        line += ' ' + tick_label
    print(line)
#畫每個大格子    
def draw_interval(center_length):
    if center_length > 0:
        draw_interval(center_length - 1)
        draw_line(center_length)
        draw_interval(center_length - 1)
#畫多個格子    
def draw_rule(num_inches, major_length):
    draw_line(major_length, '0')
    for j in range(1, 1 + num_inches):
        draw_interval(major_length - 1)
        draw_line(major_length, str(j))

案例4 數字表達式

Given two integers a ≤ b, write a program that transforms a into b by a minimum sequence of increment (add 1) and unfolding (multiply by 2) operations.

For example,

23 = ((5 * 2 + 1) * 2 + 1)

113 = ((((11 + 1) + 1) + 1) * 2 * 2 * 2 + 1)

def intSeq(a, b):
    if (a == b):
        return str(a)
    
    if (b % 2 == 1):
        return "(" + intSeq(a, b-1) + " + 1)"
    
    if (b < a * 2):
        return "(" + intSeq(a, b-1) + " + 1)"
        
    return intSeq(a, b/2) + " * 2";

案例5 漢諾塔

def hanoi(n, start, end, by):
    if (n==1):
        print("Move from " + start + " to " + end)
    else:
        hanoi(n-1, start, by, end)
        hanoi(1, start, end, by)
        hanoi(n-1, by, end, start)