Leetcode-673. Number of Longest Increasing Subsequence 最长递增子序列的个数 （DP） -python

题目

给定一个未排序的整数数组，找到最长递增子序列的个数。
链接：https://leetcode.com/problems/number-of-longest-increasing-subsequence/

Given an unsorted array of integers, find the number of longest increasing subsequence.

Example:

Input: [1,3,5,4,7]
Output: 2
Explanation: The two longest increasing subsequence are [1, 3, 4, 7] and [1, 3, 5, 7].

思路及代码

DP

• dp[i]：包含第i个元素在内的最长子序列的长度
• count[i]：包含第i个元素在内的最长子序列的个数
• dp[i] = max(dp[j]) + 1，其中nums[j] < nums[i]
• count[i] = sum(count[j])，其中j代表了所有max(dp)的index，j<i且nums[j] < nums[i]
• 返回值sum(count[i])，其中i为max(dp)的所有index

class Solution:
    def findNumberOfLIS(self, nums: List[int]) -> int:
        if not nums:
            return 0
        dp = [1] * len(nums)
        count = [1] * len(nums)
        for i in range(len(nums)):
            prev = 0
            for j in range(i-1, -1, -1):
                if nums[i] > nums[j]:
                    if dp[j] > prev:
                        prev = dp[j]
                        count[i] = count[j]
                    elif dp[j] == prev:
                        count[i] += count[j]
            dp[i] = prev + 1
        lis = max(dp)
        cnt = 0
        for i in range(len(dp)):
            if dp[i] == lis:
                cnt += count[i]
        return cnt

复杂度

T = $O(n^2)$
S = $O(n)$