查找最长双调子序列的Python程序

给定一个包含 n 个正整数的数组 arr[0 … n-1]，如果 arr[] 的子序列先递增，然后递减，则称为双调。编写一个函数数组为参数并返回最长双音子序列长度的函数。
一个按升序排序的序列被认为是双调的，降序部分为空。类似地，降序序列被认为是双调，增加的部分为空。
例子：

Input arr[] = {1, 11, 2, 10, 4, 5, 2, 1};
Output: 6 (A Longest Bitonic Subsequence of length 6 is 1, 2, 10, 4, 2, 1)

Input arr[] = {12, 11, 40, 5, 3, 1}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 12, 11, 5, 3, 1)

Input arr[] = {80, 60, 30, 40, 20, 10}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 80, 60, 30, 20, 10)

来源：微软面试题

解决方案
此问题是标准最长递增子序列 (LIS) 问题的变体。令输入数组为长度为 n 的 arr[]。我们需要使用 LIS 问题的动态规划解决方案来构造两个数组 lis[] 和 lds[]。 lis[i] 存储以 arr[i] 结尾的最长递增子序列的长度。 lds[i] 存储从 arr[i] 开始的最长递减子序列的长度。最后，我们需要返回 lis[i] + lds[i] – 1 的最大值，其中 i 从 0 到 n-1。
以下是上述动态规划解决方案的实现。

Python

# Dynamic Programming implementation of longest bitonic subsequence problem
"""
lbs() returns the length of the Longest Bitonic Subsequence in
arr[] of size n. The function mainly creates two temporary arrays
lis[] and lds[] and returns the maximum lis[i] + lds[i] - 1.
  
lis[i] ==> Longest Increasing subsequence ending with arr[i]
lds[i] ==> Longest decreasing subsequence starting with arr[i]
"""
  
def lbs(arr):
    n = len(arr)
  
  
    # allocate memory for LIS[] and initialize LIS values as 1
    # for all indexes
    lis = [1 for i in range(n+1)]
  
    # Compute LIS values from left to right
    for i in range(1 , n):
        for j in range(0 , i):
            if ((arr[i] > arr[j]) and (lis[i] < lis[j] +1)):
                lis[i] = lis[j] + 1
  
    # allocate memory for LDS and initialize LDS values for
    # all indexes
    lds = [1 for i in range(n+1)]
      
    # Compute LDS values from right to left
    for i in reversed(range(n-1)): #loop from n-2 downto 0
        for j in reversed(range(i-1 ,n)): #loop from n-1 downto i-1
            if(arr[i] > arr[j] and lds[i] < lds[j] + 1):
                lds[i] = lds[j] + 1 
  
  
    # Return the maximum value of (lis[i] + lds[i] - 1)
    maximum = lis[0] + lds[0] - 1
    for i in range(1 , n):
        maximum = max((lis[i] + lds[i]-1), maximum)
      
    return maximum
  
# Driver program to test the above function
arr =  [0 , 8 , 4, 12, 2, 10 , 6 , 14 , 1 , 9 , 5 , 13,
        3, 11 , 7 , 15]
print "Length of LBS is",lbs(arr)
  
# This code is contributed by Nikhil Kumar Singh(nickzuck_007)

输出：

Length of LBS is 7

时间复杂度：O(n^2)
辅助空间：O(n)

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