673. Number of Longest Increasing Subsequence

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.

Approach #1: C++. [DFS]

class Solution {
public:
    int findNumberOfLIS(vector<int>& nums) {
        int n = nums.size();
        if (n == 0) return 0;
        
        c_ = vector<int>(n, 0);
        l_ = vector<int>(n, 0);
        
        int max_len = 0;
        for (int i = 0; i < n; ++i)
            max_len = max(max_len, len(nums, i));
        
        int ans = 0;
        for (int i = 0; i < n; ++i)
            if (len(nums, i) == max_len)
                ans += count(nums, i);
        
        return ans;
    }
    
private:
    vector<int> c_;
    vector<int> l_;
    
    // find the total number of increasing subsequence from i to n of the index.
    int count(const vector<int>& nums, int n) {
        if (n == 0) return 1;
        if (c_[n] > 0) return c_[n];
        
        int total_count = 0;
        int l = len(nums, n);
        
        // find the number of increasing subsequence which is short than current subsquence.
        for (int i = 0; i < n; ++i) 
            if (nums[n] > nums[i] && len(nums, i) == l-1)
                total_count += count(nums, i);
        
        if (total_count == 0) 
            total_count = 1;
        
        return c_[n] = total_count;
    }
    
    // find the max length of increasing subsequence from i to n of the index.
    int len(const vector<int>& nums, int n) {
        if (n == 0) return 1;
        if (l_[n] > 0) return l_[n];
        
        int max_len = 1;
        
        for (int i = 0; i < n; ++i) 
            if (nums[n] > nums[i])
                max_len = max(max_len, len(nums, i) + 1);
        
        return l_[n] = max_len;
    }
    
};

　　

Appraoch #2: Interation. [Java]

class Solution {
    public int findNumberOfLIS(int[] nums) {
        int n = nums.length;
        if (n == 0) return 0;
        
        int[] c = new int[n];
        int[] l = new int[n];
        
        Arrays.fill(c, 1);
        Arrays.fill(l, 1);
        
        for (int i = 1; i < n; ++i) {
            for (int j = 0; j < i; ++j) {
                if (nums[i] > nums[j])
                    if (l[j] + 1 > l[i]) {
                        l[i] = l[j] + 1;
                        c[i] = c[j];
                    } else if (l[j] + 1 == l[i]){
                        c[i] += c[j];
                    }
            }
        }
        
        int max_len = 0;
        for (int i = 0; i < n; ++i)
            if (l[i] > max_len)
                max_len = l[i];
        
        int ans = 0;
        for (int i = 0; i < n; ++i) {
            if (l[i] == max_len)
                ans += c[i];
        }
        
        return ans;
    }
}

　　

Analysis:

The idea is to use two arrays l[n] ans c[n] to record the maximum length os Incresing Subsequence ans the coresponding number of there sequence which ends with nums[i], respectively. That is:

l[i]: the lenght of the Longest Increasing Subseuqence which ends with nums[i].

c[i]: the number of the Longest Increasing Subsequence which ends with nums[i].

Then, the result is the sum of each c[i] while its corresponding l[i] is the maximum length.

Reference:

https://leetcode.com/problems/number-of-longest-increasing-subsequence/discuss/107293/JavaC%2B%2B-Simple-dp-solution-with-explanation

http://zxi.mytechroad.com/blog/dynamic-programming/leetcode-673-number-of-longest-increasing-subsequence/