2145

2145. Count the Hidden Sequences (Medium)

You are given a 0-indexed array of n integers differences, which describes the differences between each pair of consecutive integers of a hidden sequence of length (n + 1). More formally, call the hidden sequence hidden, then we have that differences[i] = hidden[i + 1] - hidden[i].

You are further given two integers lower and upper that describe the inclusive range of values [lower, upper] that the hidden sequence can contain.

• For example, given differences = [1, -3, 4], lower = 1, upper = 6, the hidden sequence is a sequence of length 4 whose elements are in between 1 and 6 (inclusive).

  <ul>
  	<li><code>[3, 4, 1, 5]</code> and <code>[4, 5, 2, 6]</code> are possible hidden sequences.</li>
  	<li><code>[5, 6, 3, 7]</code> is not possible since it contains an element greater than <code>6</code>.</li>
  	<li><code>[1, 2, 3, 4]</code> is not possible since the differences are not correct.</li>
  </ul>
  </li>
  

Return the number of possible hidden sequences there are. If there are no possible sequences, return 0.

 

Example 1:

Input: differences = [1,-3,4], lower = 1, upper = 6
Output: 2
Explanation: The possible hidden sequences are:
- [3, 4, 1, 5]
- [4, 5, 2, 6]
Thus, we return 2.

Example 2:

Input: differences = [3,-4,5,1,-2], lower = -4, upper = 5
Output: 4
Explanation: The possible hidden sequences are:
- [-3, 0, -4, 1, 2, 0]
- [-2, 1, -3, 2, 3, 1]
- [-1, 2, -2, 3, 4, 2]
- [0, 3, -1, 4, 5, 3]
Thus, we return 4.

Example 3:

Input: differences = [4,-7,2], lower = 3, upper = 6
Output: 0
Explanation: There are no possible hidden sequences. Thus, we return 0.

 

Constraints:

• n == differences.length
• 1 <= n <= 105
• -105 <= differences[i] <= 105
• -105 <= lower <= upper <= 105

Solution 1.

Assume hidden[0] = 0.

We can get all hidden[i+1] = hidden[i] + diff[i].

The hidden array forms a polyline. Assume the max/min values are max/min.

By changing hidden[0], we can shift this range up or down.

If we snap max to upper, we move up by upper - max steps. Then the number of possible of hidden sequences is min + (upper - max) - lower + 1.

Another way to think about it:

// OJ: https://leetcode.com/problems/count-the-hidden-sequences/
// Author: github.com/lzl124631x
// Time: O(N)
// Space: O(1)
class Solution {
public:
    int numberOfArrays(vector<int>& A, int lower, int upper) {
        long sum = 0, mn = 0, mx = 0;
        for (int n : A) {
            sum += n;
            mn = min(mn, sum);
            mx = max(mx, sum);
        }
        return max(0L, mn - mx + upper - lower + 1);
    }
};

Discuss

https://leetcode.com/problems/count-the-hidden-sequences/discuss/1709710/C%2B%2B-One-pass-O(N)-time