<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <em>the length of the shortest non-empty <strong>subarray</strong> of </em><code>nums</code><em> with a sum of at least </em><code>k</code>. If there is no such <strong>subarray</strong>, return <code>-1</code>.</p>

<p>A <strong>subarray</strong> is a <strong>contiguous</strong> part of an array.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> nums = [1], k = 1
<strong>Output:</strong> 1
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> nums = [1,2], k = 4
<strong>Output:</strong> -1
</pre><p><strong class="example">Example 3:</strong></p>
<pre><strong>Input:</strong> nums = [2,-1,2], k = 3
<strong>Output:</strong> 3
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>5</sup> &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= k &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution {
public:
    int shortestSubarray(vector<int>& nums, int k) {
        int shortest = 1000000;

        vector<long long> prefixes; prefixes.push_back(0);
        for (int num : nums) prefixes.push_back(num + prefixes.back());

        deque<tuple<long long, int>> dq; // <prefix sum, index>
        for (int right_idx = 0; right_idx < prefixes.size(); right_idx++) {
            long long right_prefix = prefixes[right_idx];
            while (!dq.empty()) {
                auto [back_prefix, back_idx] = dq.back();
                if (back_prefix < right_prefix) break; // still useful
                dq.pop_back();
            }
            dq.push_back({right_prefix, right_idx});
            while (true) {
                auto [left_prefix, left_idx] = dq.front();
                if (right_prefix - left_prefix < k) break; // invariant : candidates have no subarrays with sum of at least k within
                dq.pop_front();
                shortest = min(shortest, right_idx - left_idx);
            }
        }

        return (shortest == 1000000 ? -1 : shortest);
    }
};
129700