What is the time complexity of Patching Array?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Patching Array?

This problem uses Array, Greedy. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Patching Array?

Insight 1: The core idea is to maintain a 'miss' variable, representing the smallest number that cannot be formed by the current set of numbers. If the next number in the array `nums` is less than or equal to `miss`, we can add it to the existing numbers to extend the range of representable numbers. Otherwise, we need to 'patch' `miss` into the array, as we cannot form `miss` from the numbers we have seen so far. Insight 2: Each time we can add a number from nums to the current coverage, the coverage extends. When we cannot add from nums, we add (patch) 'miss' to the coverage. Adding 'miss' to the current coverage extends the range of representable numbers up to the current 'miss' value plus the current coverage value.

Which companies ask LeetCode 330. Patching Array in interviews?

This problem is frequently asked at Flipkart, Snowflake.

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Patching Array - LeetCode 330 Solution

Patching Array - Complete Solution Guide

Patching Array is LeetCode problem 330, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

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

Detailed Explanation

The problem asks us to determine the minimum number of 'patches' needed to add to a given sorted array `nums` such that any number in the range `[1, n]` can be formed by the sum of a subset of elements from the array (including the patches we add). We are given a sorted array `nums` and an integer `n`. We need to return the minimum number of elements we must 'patch' (add) into `nums` so that every number from 1 to `n` can be represented as the sum of some subset of the elements in the array (including the patches).

Solution Approach

The solution uses a greedy approach to find the minimum number of patches. It maintains a variable `miss` which represents the smallest positive integer that is not yet representable as the sum of some subset of the elements in `nums` and the patches we've added so far. The algorithm iterates while `miss` is less than or equal to `n`. In each iteration, if there is an element in `nums` that is less than or equal to `miss`, we add it to the range of representable numbers by incrementing `miss` by that element. Otherwise, we need to add `miss` as a patch to the array. We increment the `patches` count and also increment `miss` by itself, as this extends our range of representable numbers up to the new `miss` value.

Step-by-Step Algorithm

Step 1: Initialize `patches` to 0 and `miss` to 1. Initialize index `i` to 0 to iterate through the `nums` array.
Step 2: Iterate while `miss` is less than or equal to `n`.
Step 3: Inside the loop, check if `i` is within the bounds of the `nums` array and if `nums[i]` is less than or equal to `miss`.
Step 4: If the condition in Step 3 is true, then we can add `nums[i]` to our coverage. So, increment `miss` by `nums[i]` and increment `i` to move to the next element in `nums`.
Step 5: If the condition in Step 3 is false, it means `nums[i]` is greater than `miss` or we've reached the end of the `nums` array. In this case, we must add a patch equal to `miss`. Increment `patches` by 1 and increment `miss` by itself.
Step 6: After the loop finishes, return the value of `patches`.

Key Insights

Insight 1: The core idea is to maintain a 'miss' variable, representing the smallest number that cannot be formed by the current set of numbers. If the next number in the array `nums` is less than or equal to `miss`, we can add it to the existing numbers to extend the range of representable numbers. Otherwise, we need to 'patch' `miss` into the array, as we cannot form `miss` from the numbers we have seen so far.
Insight 2: Each time we can add a number from nums to the current coverage, the coverage extends. When we cannot add from nums, we add (patch) 'miss' to the coverage. Adding 'miss' to the current coverage extends the range of representable numbers up to the current 'miss' value plus the current coverage value.
Insight 3: Using a greedy approach by always patching the smallest missing number is optimal. This guarantees that we cover the widest possible range with each patch.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Greedy.

Companies

Asked at: Flipkart, Snowflake.

Patching Array | LeetCode 330

Hard

Array Greedy

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int minPatches(int[] nums, int n) {
        int patches = 0;
        long miss = 1;
        int i = 0;
        
        while (miss <= n) {
            if (i < nums.length && nums[i] <= miss) {
                miss += nums[i];
                i++;
            } else {
                miss += miss;
                patches++;
            }
        }
        
        return patches;
    }
}

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Problem Info

Description

Problem 330: Patching Array - Detailed explanation and solution approaches.

Topics

Array Greedy

Companies

FlipkartSnowflake

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:330

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

6 steps

1
Step 1: Initialize `patches` to 0 and `miss` to 1. Initialize index `i` to 0 to iterate through the `nums` array.
2
Step 2: Iterate while `miss` is less than or equal to `n`.
3
Step 3: Inside the loop, check if `i` is within the bounds of the `nums` array and if `nums[i]` is less than or equal to `miss`.
4
Step 4: If the condition in Step 3 is true, then we can add `nums[i]` to our coverage. So, increment `miss` by `nums[i]` and increment `i` to move to the next element in `nums`.
5
Step 5: If the condition in Step 3 is false, it means `nums[i]` is greater than `miss` or we've reached the end of the `nums` array. In this case, we must add a patch equal to `miss`. Increment `patches` by 1 and increment `miss` by itself.
6
Step 6: After the loop finishes, return the value of `patches`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is determined by the number of iterations of the while loop. In the worst case, we add patches greedily until we reach `n`, which takes logarithmic time with base 2. The space complexity is constant because we only use a fixed number of variables, regardless of the input size.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Dynamic Programming

An alternative approach could involve dynamic programming, where dp[i] represents whether the sum `i` can be formed. However, given the constraint on `n` (up to 2^31 - 1), this approach would be impractical due to the huge memory requirements.

Time: O(n * m), where m is the sum of numbers in the nums array (including patches)Space: O(n)

Trade-offs: The DP approach would be extremely slow and require massive memory, making it a poor choice for this problem, especially given the constraints.

✨

Key Insights

1Insight 1: The core idea is to maintain a 'miss' variable, representing the smallest number that cannot be formed by the current set of numbers. If the next number in the array `nums` is less than or equal to `miss`, we can add it to the existing numbers to extend the range of representable numbers. Otherwise, we need to 'patch' `miss` into the array, as we cannot form `miss` from the numbers we have seen so far.
2Insight 2: Each time we can add a number from nums to the current coverage, the coverage extends. When we cannot add from nums, we add (patch) 'miss' to the coverage. Adding 'miss' to the current coverage extends the range of representable numbers up to the current 'miss' value plus the current coverage value.
3Insight 3: Using a greedy approach by always patching the smallest missing number is optimal. This guarantees that we cover the widest possible range with each patch.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Greedy Algorithm

Making the best choice at each step without looking ahead.

Edge Cases

Edge case 1: `nums` is empty. The algorithm correctly handles this because the `while` loop continues as long as `miss <= n`, and the patch count is incremented as needed.
Edge case 2: The first element of `nums` is greater than 1. In this case, the algorithm correctly identifies that `1` needs to be patched, and the `patches` counter is incremented.
Edge case 3: `n` is very large (close to the maximum integer value). Using `long` for `miss` avoids integer overflow issues.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer overflow. Since `miss` can become very large, it's essential to use a `long` data type in Java and C++, and take advantage of Python's auto-conversion to arbitrarily long integers, to prevent overflow issues.

✗Common mistake 2: Not handling the case where `nums` is empty or the first element is greater than 1. The algorithm should correctly patch the necessary numbers in these scenarios.

✗Common mistake 3: Incorrectly incrementing `miss` after adding a patch. It's crucial to increment `miss` by itself (`miss += miss`) to effectively double the range of representable numbers.

✗Common mistake 4: Incorrectly assuming you have to add every number from `nums` to the representable sums. You only add the number if it's less than or equal to the missing number.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Resource allocation. Imagine you have a set of resources with different sizes. The problem is similar to finding the minimum number of additional resource pieces required to cover all needs up to a certain capacity.

🏦Real-world scenario 2: Network coverage. If you have a set of base stations covering different ranges, the patching array problem can be adapted to find the minimum number of new base stations to add to ensure complete network coverage up to a certain distance.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Hard problems often combine multiple techniques. Break the problem into sub-problems.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

👨‍💻 From Experience

I encountered a variation of this problem in a system design interview. The key was recognizing the underlying pattern.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 29, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 14

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I verify if a greedy approach is correct?

Prove the greedy choice property: the locally optimal choice leads to globally optimal solution. Use exchange argument: show that swapping any element with the greedy choice doesn't improve the solution. If you can't prove it, test extensively or consider DP.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Patching Array"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether greedy techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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