#3181

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Total Reward Using Operations II

Maximize your total reward using dynamic programming with state transitions in this challenging problem involving array manipulation.

array dynamic programming bit manipulation

Problem Statement

You are given an integer array rewardValues representing the reward values at each index. Your initial total reward is 0, and all indices are unmarked. You are allowed to perform the following operation any number of times:

You can select indices to mark in a way that maximizes your total reward, following the rule that the total reward can only increase when marking certain indices in a specific order. Return the maximum reward achievable after performing these operations optimally.

Examples

Example 1

Input: rewardValues = [1,1,3,3]

Output: 4

During the operations, we can choose to mark the indices 0 and 2 in order, and the total reward will be 4, which is the maximum.

Example 2

Input: rewardValues = [1,6,4,3,2]

Output: 11

Mark the indices 0, 2, and 1 in order. The total reward will then be 11, which is the maximum.

Constraints

1 <= rewardValues.length <= 5 * 104
1 <= rewardValues[i] <= 5 * 104

Solution Approach

Sort the Reward Array

Start by sorting the rewardValues array. Sorting helps ensure that we prioritize larger reward values, which leads to maximizing the total reward during the marking process. Sorting the array simplifies the selection of optimal indices.

Use State Transition Dynamic Programming

Implement a dynamic programming solution where the state transitions represent the choices of marked indices. The idea is to maintain a running total and track how the reward can be accumulated through optimal transitions across the array indices.

Track Maximum Total Reward

While processing the sorted array, track the current total reward by considering different transition states for each index. After considering all possible transitions, return the maximum reward possible.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexities depend on the sorting step and dynamic programming approach. Sorting takes O(n log n), and dynamic programming requires O(n) space and time for state transitions.

What Interviewers Usually Probe

Looks for an understanding of dynamic programming with state transitions.
Evaluates ability to optimize solutions through sorting and state-based accumulation.
Assesses proficiency in handling large arrays with efficient algorithms.

Common Pitfalls or Variants

Common pitfalls

Failing to sort the reward values first, which could lead to suboptimal choices during the selection of indices.
Not properly managing state transitions, resulting in incorrect calculations of the maximum reward.
Overcomplicating the problem by failing to leverage dynamic programming efficiently.

Follow-up variants

A variant could involve adding constraints where only certain subsets of indices are allowed to be marked.
Consider allowing a limited number of operations to be performed, requiring optimization of each move.
Modify the problem to allow for non-integer reward values, changing the optimization dynamics.

FAQ

What is the main pattern used to solve the "Maximum Total Reward Using Operations II" problem?

The problem primarily uses state transition dynamic programming to determine the best way to accumulate rewards by marking indices optimally.

How do I optimize my solution for the "Maximum Total Reward Using Operations II" problem?

Sorting the reward values array first is key to ensuring you always pick the largest rewards, followed by using dynamic programming to track state transitions for optimal reward accumulation.

What is the time complexity of the "Maximum Total Reward Using Operations II" problem?

The time complexity is dominated by the sorting step, which takes O(n log n), followed by the dynamic programming calculation that takes O(n).

What are common pitfalls when solving the "Maximum Total Reward Using Operations II" problem?

Common pitfalls include forgetting to sort the reward array, mishandling state transitions, and inefficiently managing dynamic programming states.

What is the optimal approach for solving the "Maximum Total Reward Using Operations II" problem?

The optimal approach involves sorting the reward array and then using dynamic programming to track and maximize the reward through optimal state transitions.

terminal

Solution

Solution 1: Dynamic Programming + Bit Manipulation

We define $f[i][j]$ as whether it is possible to obtain a total reward of $j$ using the first $i$ reward values. Initially, $f[0][0] = \textit{True}$, and all other values are $\textit{False}$.

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class Solution:
    def maxTotalReward(self, rewardValues: List[int]) -> int:
        nums = sorted(set(rewardValues))
        f = 1
        for v in nums:
            f |= (f & ((1 << v) - 1)) << v
        return f.bit_length() - 1

Continue Practicing

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