LeetCode #330 — HARD

Patching Array

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

Given a sorted integer array nums and an integer n, add/patch elements to the array such that any number in the range [1, n] inclusive can be formed by the sum of some elements in the array.

Return the minimum number of patches required.

Example 1:

Input: nums = [1,3], n = 6
Output: 1
Explanation:
Combinations of nums are [1], [3], [1,3], which form possible sums of: 1, 3, 4.
Now if we add/patch 2 to nums, the combinations are: [1], [2], [3], [1,3], [2,3], [1,2,3].
Possible sums are 1, 2, 3, 4, 5, 6, which now covers the range [1, 6].
So we only need 1 patch.

Example 2:

Input: nums = [1,5,10], n = 20
Output: 2
Explanation: The two patches can be [2, 4].

Example 3:

Input: nums = [1,2,2], n = 5
Output: 0

Constraints:

1 <= nums.length <= 1000
1 <= nums[i] <= 10⁴
nums is sorted in ascending order.
1 <= n <= 2³¹ - 1

Step 01

Brute Force Baseline

Problem summary: Given a sorted integer array nums and an integer n, add/patch elements to the array such that any number in the range [1, n] inclusive can be formed by the sum of some elements in the array. Return the minimum number of patches required.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Greedy

Example 1

[1,3]
6

Example 2

[1,5,10]
20

Example 3

[1,2,2]
5

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #330: Patching Array
class Solution {
    public int minPatches(int[] nums, int n) {
        long x = 1;
        int ans = 0;
        for (int i = 0; x <= n;) {
            if (i < nums.length && nums[i] <= x) {
                x += nums[i++];
            } else {
                ++ans;
                x <<= 1;
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #330: Patching Array
func minPatches(nums []int, n int) (ans int) {
	x := 1
	for i := 0; x <= n; {
		if i < len(nums) && nums[i] <= x {
			x += nums[i]
			i++
		} else {
			ans++
			x <<= 1
		}
	}
	return
}

# Accepted solution for LeetCode #330: Patching Array
class Solution:
    def minPatches(self, nums: List[int], n: int) -> int:
        x = 1
        ans = i = 0
        while x <= n:
            if i < len(nums) and nums[i] <= x:
                x += nums[i]
                i += 1
            else:
                ans += 1
                x <<= 1
        return ans

// Accepted solution for LeetCode #330: Patching Array
struct Solution;

impl Solution {
    fn min_patches(nums: Vec<i32>, n: i32) -> i32 {
        let mut res = 0;
        let mut miss: i64 = 1;
        let mut i = 0;
        while miss <= n as i64 {
            if i < nums.len() && nums[i] as i64 <= miss {
                miss += nums[i] as i64;
                i += 1;
            } else {
                miss += miss;
                res += 1;
            }
        }
        res
    }
}

#[test]
fn test() {
    let nums = vec![1, 3];
    let n = 6;
    let res = 1;
    assert_eq!(Solution::min_patches(nums, n), res);
    let nums = vec![1, 5, 10];
    let n = 20;
    let res = 2;
    assert_eq!(Solution::min_patches(nums, n), res);
    let nums = vec![1, 2, 2];
    let n = 5;
    let res = 0;
    assert_eq!(Solution::min_patches(nums, n), res);
}

// Accepted solution for LeetCode #330: Patching Array
function minPatches(nums: number[], n: number): number {
    let x = 1;
    let ans = 0;
    for (let i = 0; x <= n; ) {
        if (i < nums.length && nums[i] <= x) {
            x += nums[i++];
        } else {
            ++ans;
            x *= 2;
        }
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n log n)

Space

O(1)

Approach Breakdown

EXHAUSTIVE

O(2ⁿ) time

O(n) space

Try every possible combination of choices. With n items each having two states (include/exclude), the search space is 2ⁿ. Evaluating each combination takes O(n), giving O(n × 2ⁿ). The recursion stack or subset storage uses O(n) space.

GREEDY

O(n log n) time

O(1) space

Greedy algorithms typically sort the input (O(n log n)) then make a single pass (O(n)). The sort dominates. If the input is already sorted or the greedy choice can be computed without sorting, time drops to O(n). Proving greedy correctness (exchange argument) is harder than the implementation.

Shortcut: Sort + single pass → O(n log n). If no sort needed → O(n). The hard part is proving it works.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Using greedy without proof

Wrong move: Locally optimal choices may fail globally.

Usually fails on: Counterexamples appear on crafted input orderings.

Fix: Verify with exchange argument or monotonic objective before committing.