Maximum Total Reward Using Operations I

Maximum Total Reward Using Operations I

1. What is this problem about?

The Maximum Total Reward Using Operations I problem involves an array of reward values. You start with a total reward of 0 and you can perform operations. In each operation, you pick a reward value v from the array that you haven't picked before. However, you can only pick v if the value v is strictly greater than your current total reward. Your goal is to find the maximum total reward you can accumulate by picking elements in the best possible order.

2. Why is this asked in interviews?

This "Maximum Total Reward Using Operations I interview question" is a clever variation of the Knapsack problem. It is used by companies like Mitsogo to test a candidate's ability to adapt Dynamic Programming to a problem with unique constraints. The condition "v > current_reward" is the key—it implies that you should process the rewards in a sorted order and that the maximum possible reward is bounded by twice the maximum value in the array.

3. Algorithmic pattern used

The "Array, Dynamic Programming interview pattern" is the way to solve this. First, sort the rewards array. Sorting is crucial because if you can pick a set of rewards, you can always pick them in increasing order to satisfy the v > current condition. The problem then becomes: "Can we achieve a sum X?" This can be solved with a boolean DP array (or a bitset for optimization) where dp[i] is true if a reward sum of i is possible.

4. Example explanation

Rewards: [1, 3, 2]

• Sorted: [1, 2, 3]
• Current Sum 0: Can pick 1. New sum 1.
• Current Sum 1: Can pick 2 (since 2 > 1). New sum 3.
• Current Sum 3: Cannot pick 3 (since 3 is not > 3).
• Alternatively: Pick 2 first? Sum becomes 2. Now you can only pick 3 (3 > 2). Sum becomes 5.
• Alternatively: Pick 3 first? Sum becomes 3. Now you can't pick 1 or 2. The maximum is 5.

5. Common mistakes candidates make

In the "Maximum Total Reward Using Operations I coding problem," a common mistake is not sorting the array, which makes the problem feel like it requires a complex permutation search. Another error is not realizing the upper bound of the DP array; since you can only pick v if your current sum is < v, and the largest v is max_val, the highest possible sum you can reach is max_val + (max_val - 1) = 2 * max_val - 1. Failing to handle this bound can lead to memory or logic issues.

6. Interview preparation tip

Practice "Subset Sum" variations. This problem is essentially a Subset Sum problem where the elements you include must satisfy a specific order-based condition. For "Operations I" (with smaller constraints), a boolean array or std::bitset in C++ is usually enough. For "Operations II" (with larger constraints), bitset optimizations become mandatory to pass the time limits. Mentioning the bitset optimization to your interviewer shows you are aware of performance bottlenecks in DP.