Formatted question description: https://leetcode.ca/all/673.html

673. Number of Longest Increasing Subsequence (Medium)

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.

Related Topics:
Dynamic Programming

Similar Questions:

Solution 1. DP

Consider the subproblem in range A[0..i] and the LIS must ends with A[i].

Let len[i] be the length of longest LIS ending with A[i].

Let cnt[i] be the count of longest LIS ending with A[i].

len[i] = 1 + max( len[j] | 0 <= j < i && A[j] < A[i] )
cnt[i] = sum( cnt[j] | 0 <= j < i && len[j] + 1 == len[i] )

The answer is

sum( cnt[i] | len[i] == maxLen )
    where maxLen = max( len[i] | 0 <= i < N )
// OJ: https://leetcode.com/problems/number-of-longest-increasing-subsequence/

// Time: O(N^2)
// Space: O(N)
class Solution {
public:
    int findNumberOfLIS(vector<int>& A) {
        if (A.empty()) return 0;
        int N = A.size(), ans = 0, maxLen = 0;
        vector<int> len(N, 1), cnt(N, 1);
        for (int i = 0; i < N; ++i) {
            for (int j = 0; j < i; ++j) {
                if (A[j] >= A[i]) continue;
                if (len[i] < 1 + len[j]) len[i] = 1 + len[j], cnt[i] = cnt[j];
                else if (len[i] == 1 + len[j]) cnt[i] += cnt[j];
            }
            if (len[i] > maxLen) maxLen = len[i], ans = cnt[i];
            else if (len[i] == maxLen) ans += cnt[i];
        }
        return ans;
    }
};

Java

class Solution {
    public int findNumberOfLIS(int[] nums) {
        if (nums == null || nums.length == 0)
            return 0;
        if (nums.length == 1)
            return 1;
        int maxLength = 1;
        int length = nums.length;
        int[] dp = new int[length];
        dp[0] = 1;
        for (int i = 1; i < length; i++) {
            dp[i] = 1;
            for (int j = 0; j < i; j++) {
                if (nums[j] < nums[i])
                    dp[i] = Math.max(dp[i], dp[j] + 1);
            }
            maxLength = Math.max(maxLength, dp[i]);
        }
        int[][] dp2d = new int[length][maxLength + 1];
        for (int i = 0; i < length; i++)
            dp2d[i][0] = 1;
        dp2d[0][1] = 1;
        for (int i = 1; i < length; i++) {
            dp2d[i][1] = 1;
            for (int j = 0; j < i; j++) {
                if (nums[j] < nums[i]) {
                    for (int k = 1; k < maxLength; k++)
                        dp2d[i][k + 1] += dp2d[j][k];
                }
            }
        }
        int count = 0;
        for (int i = 0; i < length; i++)
            count += dp2d[i][maxLength];
        return count;
    }
}

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