The problem requires finding the maximum sum divisible by three from a given integer array. The solution involves dynamic programming where we track states based on the sum's remainder when divided by three. Using state transition DP, we can efficiently determine the greatest sum divisible by three.
Problem Statement
Given an integer array nums, return the maximum possible sum of the elements such that the sum is divisible by three. If no such sum exists, return 0.
You must maximize the sum while ensuring that the sum modulo three equals zero. Use dynamic programming with a state transition approach to solve the problem efficiently.
Examples
Example 1
Input: nums = [3,6,5,1,8]
Output: 18
Pick numbers 3, 6, 1 and 8 their sum is 18 (maximum sum divisible by 3).
Example 2
Input: nums = [4]
Output: 0
Since 4 is not divisible by 3, do not pick any number.
Example 3
Input: nums = [1,2,3,4,4]
Output: 12
Pick numbers 1, 3, 4 and 4 their sum is 12 (maximum sum divisible by 3).
Constraints
- 1 <= nums.length <= 4 * 104
- 1 <= nums[i] <= 104
Solution Approach
State Transition Dynamic Programming
Represent the state as DP[pos][mod], where pos refers to the current index in the array and mod is the current sum modulo 3. For each number in the array, update the states and track the maximum sum that is divisible by three.
Greedy and Sorting Optimization
In certain cases, sorting the array and greedily selecting numbers based on their modulo 3 values can provide quick insights into which numbers contribute to the maximum sum divisible by three.
Iterative State Updates
Iterate through the array, updating possible states at each step, ensuring that the sum modulo 3 is adjusted based on the current number. This ensures that the dynamic programming state transitions are efficiently managed.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the chosen dynamic programming approach, typically O(n) or O(n * 3) for state transitions, where n is the length of the input array. Space complexity can be O(3) or O(n * 3), depending on the representation of states.
What Interviewers Usually Probe
- Ability to recognize state transition patterns in dynamic programming.
- Understanding of greedy techniques for optimizing the sum.
- Comfort with handling arrays and ensuring the sum is divisible by three.
Common Pitfalls or Variants
Common pitfalls
- Not properly managing the modulo 3 states during dynamic programming updates.
- Confusing the maximum sum with the largest number divisible by three.
- Incorrectly handling edge cases where no sum is divisible by three.
Follow-up variants
- Modify the problem to find the minimum sum divisible by three.
- Extend the problem to include negative integers in the array.
- Apply this approach to a multidimensional array for greater complexity.
How GhostInterview Helps
- GhostInterview provides optimized solutions using state transition dynamic programming for similar problems.
- It helps break down the problem into manageable steps, showing you how to apply dynamic programming effectively.
- Our solver assists in recognizing the right approach for handling modulo arithmetic within dynamic programming.
Topic Pages
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FAQ
What is the approach for solving the Greatest Sum Divisible by Three?
The solution uses dynamic programming with state transitions, represented as DP[pos][mod], to track the maximum possible sum divisible by three.
How can I optimize this problem using greedy algorithms?
Sorting the array based on modulo 3 values and selecting numbers greedily can optimize the solution, though dynamic programming is the most efficient method.
What is the time complexity of the solution?
The time complexity depends on the approach, usually O(n) or O(n * 3), where n is the length of the array.
Can this approach be extended to multidimensional arrays?
Yes, the state transition dynamic programming approach can be adapted for multidimensional arrays, increasing the problem's complexity.
What should I consider when implementing the solution?
Ensure that you correctly manage the modulo 3 states and handle edge cases where no sum is divisible by three.
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Find the largest number divisible by three by selecting and ordering digits optimally using state transition dynamic programming techniques.
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