What is the time complexity of Maximum Total Reward Using Operations I?

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

What data structures are used to solve Maximum Total Reward Using Operations I?

This problem uses Array, Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum Total Reward Using Operations I?

Insight 1: Sorting the unique reward values is crucial. By processing rewards in ascending order, we guarantee that once a reward `r` is considered, we know all possible total rewards less than `r` that can be achieved with previously considered rewards. Insight 2: The core idea is to use dynamic programming, represented by a bitmask (or array in the C implementation), to track which total reward sums are achievable at each step. If a sum `x` is achievable and `x < r`, then `x + r` is also achievable.

Which companies ask LeetCode 3180. Maximum Total Reward Using Operations I in interviews?

This problem is frequently asked at Mitsogo.

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Maximum Total Reward Using Operations I - LeetCode 3180 Solution

Q: How to solve Maximum Total Reward Using Operations I?

The solution uses a dynamic programming approach with a bitmask (represented as a long integer or an array of integers). The bitmask represents the set of achievable total rewards at each step. The algorithm iterates through the unique reward values in sorted order. For each reward value `r`, it identifies the achievable total rewards that are less than `r`. Then, it adds `r` to each of those achievable total rewards, effectively marking new achievable total rewards in the bitmask. Finally, the algorithm finds the largest achievable total reward, which is the index of the most significant bit in the final bitmask.

Q: Which companies ask LeetCode 3180. Maximum Total Reward Using Operations I in interviews?

This problem is frequently asked at Mitsogo.

Maximum Total Reward Using Operations I - Complete Solution Guide

Maximum Total Reward Using Operations I is LeetCode problem 3180, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an integer array rewardValues of length n , representing the values of rewards. Initially, your total reward x is 0, and all indices are unmarked . You are allowed to perform the following operation any number of times: Choose an unmarked index i from the range [0, n - 1] . If rewardValues[i] is greater than your current total reward x , then add rewardValues[i] to x (i.e., x = x + rewardValues[i] ), and mark the index i . Return an integer denoting the maximum total reward you can

Detailed Explanation

The problem asks us to find the maximum possible total reward we can obtain by strategically selecting elements from the input array `rewardValues`. We start with a total reward of 0 and can only add a reward value at a particular index if that reward value is greater than the current total reward. The goal is to maximize the final total reward by choosing the indices in an optimal order. We are allowed to choose any index any number of times, but we can only add it to the total reward if the reward is greater than our current total reward, and the index is unmarked. Once an index is used, it's marked, and we can't use it again.

Solution Approach

The solution uses a dynamic programming approach with a bitmask (represented as a long integer or an array of integers). The bitmask represents the set of achievable total rewards at each step. The algorithm iterates through the unique reward values in sorted order. For each reward value `r`, it identifies the achievable total rewards that are less than `r`. Then, it adds `r` to each of those achievable total rewards, effectively marking new achievable total rewards in the bitmask. Finally, the algorithm finds the largest achievable total reward, which is the index of the most significant bit in the final bitmask.

Step-by-Step Algorithm

Step 1: Extract unique reward values from the input array `rewardValues` and sort them in ascending order. This ensures optimal reward selection during the dynamic programming phase.
Step 2: Initialize a bitmask (or array `dp`) to represent achievable rewards. Initially, only a total reward of 0 is achievable, so set the 0th bit (or `dp[0]`) to 1.
Step 3: Iterate through the sorted unique reward values. For each reward `r`, perform the following:
Step 4: Create a mask to identify the achievable rewards less than `r`. This is done by setting the bits from 0 to `r-1` to 1.
Step 5: Using bitwise AND, determine the achievable rewards that are less than `r` from the current bitmask.
Step 6: Left-shift the bitmask containing achievable rewards less than `r` by `r` positions. This represents adding `r` to each of those rewards, forming new achievable rewards.
Step 7: Update the main bitmask by performing a bitwise OR with the shifted bitmask. This marks the new achievable rewards in the main bitmask.
Step 8: After iterating through all unique reward values, find the maximum total reward by identifying the index of the most significant bit in the final bitmask. This index represents the largest achievable sum.
Step 9: Return the maximum total reward.

Key Insights

Insight 1: Sorting the unique reward values is crucial. By processing rewards in ascending order, we guarantee that once a reward `r` is considered, we know all possible total rewards less than `r` that can be achieved with previously considered rewards.
Insight 2: The core idea is to use dynamic programming, represented by a bitmask (or array in the C implementation), to track which total reward sums are achievable at each step. If a sum `x` is achievable and `x < r`, then `x + r` is also achievable.
Insight 3: By representing achievable sums as a bitmask, we can efficiently determine if sums are reachable. Bitwise operations (AND, OR, left shift) are used for concise and fast updates to the bitmask. The size of array needed will be the sum of all unique reward values.

Complexity Analysis

Time Complexity: O(n log n)

Space Complexity: O(n)

Topics

This problem involves: Array, Dynamic Programming.

Companies

Asked at: Mitsogo.

Maximum Total Reward Using Operations I | LeetCode 3180

Medium

Array Dynamic Programming

Time: O(n log n)Space: O(n)

Solution Code

Available in:

import java.util.Arrays;
import java.util.HashSet;
import java.util.List;
import java.util.Set;
import java.util.stream.Collectors;

class Solution {
    public int maxTotalReward(int[] rewardValues) {
        Set<Integer> uniqueRewardsSet = new HashSet<>();
        for (int reward : rewardValues) {
            uniqueRewardsSet.add(reward);
        }

        List<Integer> uniqueRewardsList = uniqueRewardsSet.stream().collect(Collectors.toList());
        Integer[] uniqueRewardsArray = uniqueRewardsList.toArray(new Integer[0]);
        Arrays.sort(uniqueRewardsArray);
        List<Integer> unique_rewards = Arrays.asList(uniqueRewardsArray);
        

        long dp = 1;

        for (int r : unique_rewards) {
            long mask = (1L << r) - 1;
            long achievable_less_than_r = dp & mask;
            long newly_achievable = achievable_less_than_r << r;
            dp |= newly_achievable;
        }

        return Long.toBinaryString(dp).length() - 1;
    }
}

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Problem Info

Description

Problem 3180: Maximum Total Reward Using Operations I - Detailed explanation and solution approaches.

Topics

Array Dynamic Programming

Companies

Mitsogo

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:3180

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

9 steps

1
Step 1: Extract unique reward values from the input array `rewardValues` and sort them in ascending order. This ensures optimal reward selection during the dynamic programming phase.
2
Step 2: Initialize a bitmask (or array `dp`) to represent achievable rewards. Initially, only a total reward of 0 is achievable, so set the 0th bit (or `dp[0]`) to 1.
3
Step 3: Iterate through the sorted unique reward values. For each reward `r`, perform the following:
4
Step 4: Create a mask to identify the achievable rewards less than `r`. This is done by setting the bits from 0 to `r-1` to 1.
5
Step 5: Using bitwise AND, determine the achievable rewards that are less than `r` from the current bitmask.
6
Step 6: Left-shift the bitmask containing achievable rewards less than `r` by `r` positions. This represents adding `r` to each of those rewards, forming new achievable rewards.
7
Step 7: Update the main bitmask by performing a bitwise OR with the shifted bitmask. This marks the new achievable rewards in the main bitmask.
8
Step 8: After iterating through all unique reward values, find the maximum total reward by identifying the index of the most significant bit in the final bitmask. This index represents the largest achievable sum.
9
Step 9: Return the maximum total reward.

💡

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 log n)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is dominated by the loop through the unique rewards and operations on the `dp` bitmask. The space complexity is determined by the size of the `dp` bitmask, which must be large enough to represent all possible sums up to S. Finding the unique rewards takes O(n) time in the worst case and sorting them takes O(n log n) time. Finding unique rewards has space complexity of O(n) and the DP bitmask uses O(S) space. O(S) will dominate overall space complexity since S >= n.

📖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

Recursive approach with Memoization

A recursive approach can be used to explore different combinations of reward selections. Memoization can be used to store the results of subproblems to avoid redundant calculations. This can be conceptualized as a knapsack problem where we decide whether to include or exclude each reward value.

Time: O(n * S), where n is the number of reward values and S is the sum of reward values.Space: O(n * S) due to the memoization table.

Trade-offs: The recursive approach with memoization can be easier to understand for some, but it generally has higher overhead than the bitmask-based dynamic programming solution. The space complexity is also higher, which can be a concern for large input sizes.

Greedy Approach (Incorrect)

A seemingly intuitive approach is to sort the rewards in descending order and pick the largest reward at each step if it exceeds the current total reward. However, this approach is not guaranteed to produce the optimal solution. For example, consider the input [1, 6, 4, 3, 2]. A greedy approach will pick 6, which only gives the sum of 6, when 11 can be achieved.

Time: O(n log n) for sorting.Space: O(1) if sorting in-place, otherwise O(n).

Trade-offs: The greedy approach is simpler to implement, but it does not guarantee the optimal result. It is not suitable for this problem.

✨

Key Insights

1Insight 1: Sorting the unique reward values is crucial. By processing rewards in ascending order, we guarantee that once a reward `r` is considered, we know all possible total rewards less than `r` that can be achieved with previously considered rewards.
2Insight 2: The core idea is to use dynamic programming, represented by a bitmask (or array in the C implementation), to track which total reward sums are achievable at each step. If a sum `x` is achievable and `x < r`, then `x + r` is also achievable.
3Insight 3: By representing achievable sums as a bitmask, we can efficiently determine if sums are reachable. Bitwise operations (AND, OR, left shift) are used for concise and fast updates to the bitmask. The size of array needed will be the sum of all unique reward values.

📚

Key Terms Explained

For Students

Array

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

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Empty input array. The provided code does not explicitly handle an empty array. In that case, the expected output is 0. It would be better to add a check at the beginning of the function to return 0 if the input array is empty.
Edge case 2: Array containing only a single element. The code correctly handles this case. The unique rewards will contain only one element, and the bitmask will be updated accordingly. The final result will be the value of that single element.
Edge case 3: Array containing identical elements. The `set` or the unique function filters out duplicate values. The code correctly handles cases where `rewardValues` contains many identical values. The program should handle cases with very large rewards within constraint limits. If any value becomes too large, operations performed during bit masking such as left shift operations on very large integers could result in errors.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not sorting the unique rewards. Sorting the unique rewards is essential for the dynamic programming approach to work correctly. If the rewards are not sorted, the algorithm may not be able to accurately track the achievable sums.

✗Common mistake 2: Using the original rewardValues array instead of unique values. This will cause errors, because the amount of memory needed grows according to the total amount, not the range of values.

✗Common mistake 3: Not handling the edge case of an empty input array. Failing to handle this edge case can lead to unexpected behavior or errors.

✗Common mistake 4: Integer Overflow. When reward values become too high, the `dp` variable could potentially lead to memory allocation errors or overflow. Therefore, it must be ensured that the type being used to store the `dp` will be sufficient to store all rewards. (e.g. use `long long` instead of `int`). Similar with the shift and mask operations.

✅

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 projects with varying costs and returns. You can only invest in a project if your current capital is less than the project's cost. This problem can be used to determine the maximum total return you can achieve with a limited budget.

🏦Real-world scenario 2: Knapsack problem variants. This problem shares similarities with the knapsack problem, where you want to maximize the value of items you can fit into a knapsack with a limited weight capacity. The constraint that the reward must be greater than the current total reward introduces a unique twist, but the underlying optimization principles are similar.

💡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

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

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

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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: September 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 identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

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 "Maximum Total Reward Using Operations I"?

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

🤖 Need help with this problem?