题目链接: https://leetcode.com/problems/number-of-longest-increasing-subsequence/description/
Given an unsorted array of integers, find the number of longest increasing subsequence.
Example 1:
Input: [1,3,5,4,7]
Output: 2
Explanation: The two longest increasing subsequence are [1, 3, 4, 7] and [1, 3, 5, 7].
Example 2:
Input: [2,2,2,2,2]
Output: 5
Explanation: The length of longest continuous increasing subsequence is 1, and there are 5 subsequences' length is 1, so output 5.
Note: Length of the given array will be not exceed 2000 and the answer is guaranteed to be fit in 32-bit signed int.
动态规划求解,建立两个数组 len
、cnt
:
* len[k]
: 表示以 nums[k]
为末尾的最长子序列长度。
* cnt[k]
: 表示以 nums[k]
为末尾的最长子序列个数。
对于每一个 nums[i]
,只需要遍历其前面的所有数字 nums[j]
(0 <= j < i
) ,找到比它小且长度最长的 nums[k]
,就可得出以 nums[i]
为末尾的子序列的最大长度 len[i] = len[k] + 1
。同时,以 nums[i]
为末尾的最长子序列个数应该等于 nums[j]
(0 <= j < i
) 中,比它小且长度最长的所有 nums[k]
的最长子序列个数之和。
用两条公式来阐述上面一段难以理解的话
* len[k] = max(len[k], len[i] + 1), for all 0 <= i < k and nums[i] < nums[k]
* cnt[k] = sum(cnt[i]), for all 0 <= i < k and len[k] = len[i] + 1
举个例子
nums = {1, 3, 5, 4, 7}
初始状态,len = {1, 1, 1, 1, 1}
,cnt = {1, 1, 1, 1, 1}
开始遍历,
* 数字 3,比它小的只有 1 => len[1] = len[0] + 1 = 2
,cnt[1] = cnt[0] = 1
;
* 数字 5,比它小且长度最长的为 3 => len[2] = len[1] + 1 = 3
,cnt[2] = cnt[1] = 1
;
* 数字 4, 比它小且长度最长的为 3 => len[3] = len[1] + 1 = 3
,cnt[3] = cnt[1] = 1
;
* 数字 7,比它小且长度最长的为 5 和 4 => len[4] = len[2] + 1 = 4
,cnt[4] = cnt[2] + cnt[3] = 2
最终状态,len = {1, 2, 3, 3, 4}
,cnt = {1, 1, 1, 1, 2}
class Solution {
public:
int findNumberOfLIS(vector<int>& nums) {
int maxLen = 1;
int res = 0;
vector<int> len(nums.size(), 1);
vector<int> cnt(nums.size(), 1);
for (int i = 1; i < nums.size(); ++i) {
for (int j = 0; j < i; ++j) {
if (nums[j] < nums[i]) {
if (len[j] + 1 > len[i]) {
len[i] = len[j] + 1;
cnt[i] = cnt[j];
} else if (len[j] + 1 == len[i]) {
cnt[i] += cnt[j];
}
}
}
maxLen = max(maxLen, len[i]);
}
for (int i = 0; i < nums.size(); ++i) {
if (maxLen == len[i]) res += cnt[i];
}
return res;
}
};