Asked in: Amazon
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Solution
#include <iostream>
#include <vector>
#include <algorithm>
// ... rest of solution available after purchaseExplanation
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To approach the problem of finding the lexicographically maximal resulting array sorted in non-increasing order by performing specific operations on a given array of package weights, you need to carefully analyze the problem constraints and how the operations affect the outcome.
Let's break down the problem and build the right mental model:
1. **Understanding the Operations and Constraints:**
- Operation 1 lets you discard the first package from the current array. This essentially means you can skip a package without adding it to the result.
- Operation 2 involves taking the first package, adding it to the resulting array, and then removing it and the next k elements from the current array.
- If fewer than k elements remain after the chosen element, you remove all the remaining elements after adding the chosen one.
- The resulting array must be sorted in non-increasing order (weights are non-increasing).
- Your goal is to produce the lexicographically maximal array from these rules.
2. **Lexicographically Maximal and Non-Increasing Order:**
The phrase "lexicographically maximal" means that when comparing two valid resulting arrays, the one that is "greater" in dictionary order is preferred. For arrays, lexicographic order compares elements from left to right; the first position at which they differ determines which array is larger.
Additionally, the resulting array must be sorted in non-increasing order. This means the weights you add to the result must be in order such that each next element is not greater than the previous.
3. **Challenges:**
- You can't reorder the original array; you only operate on it sequentially, deciding whether to skip or to pick a package plus skipping k subsequent elements.
- You must maintain the non-increasing order in the resulting array.
- You want to maximize the lexicographical order, which pushes you to pick heavier packages earlier if possible.
4. **Initial Thoughts on the Approach:**
Since the problem involves sequential decisions — to pick or discard elements — think about it as a decision process where each step decides whether to:
- Discard the current package (operation 1), moving forward.
- Pick the current package and remove it plus k elements after it (operation 2).
Each decision influences the future available elements and the resulting array.
5. **Key Insight:**
The choice to pick a package is constrained by the non-increasing order requirement. You can't add a package heavier than the last picked package because that would violate the order.
Therefore, at each step, you need to consider the largest possible package you can pick that maintains the non-increasing order relative to the previous chosen package.
6. **Greedy or Recursive Thinking:**
To get the lexicographically maximal array, a greedy approach makes sense: always try to pick the largest valid package you can at the current position that won't violate the non-increasing order.
However, since the operation 2 removes k elements ahead, a naive greedy approach might fail, because skipping or picking affects future choices drastically.
7. **Lookahead and Subproblems:**
Because of the effect of removing k elements after picking, the problem can be broken down into subproblems: for each position, what's the best resulting array if you start there, considering the constraints?
This suggests formulating the problem using recursion or dynamic programming where:
- The state includes the current index in the weight array.
- Also keep track of the last chosen weight to ensure the non-increasing order.
- From the current index, try two options:
- Skip the current package.
- Pick the current package if it doesn't violate the order, then skip k elements ahead.
- Choose the option that leads to the lexicographically maximal resulting array.
8. **Handling Lexicographical Comparison:**
When deciding which option leads to a better result, you need to be able to compare arrays lexicographically.
This requires either:
- Storing the resulting arrays explicitly in the recursion and comparing them, which can be memory-heavy.
- Or storing partial results and using memoization to avoid recalculations.
Since you only need to pick packages, the solution can build resulting arrays incrementally and backtrack efficiently.
9. **Optimizing the Search:**
The complexity can be high if you explore every possibility.
- To optimize, you might prune suboptimal choices early.
- Use memoization keyed by the current index and last chosen weight.
- Compare partial solutions and prune those that cannot yield a better lexicographical result.
10. **Result Assembly:**
After evaluating all possible paths with pruning and memoization, the solution would return the lexicographically maximal array of picked packages sorted non-increasingly.
11. **Edge Cases and Final Checks:**
- When the array is small or k is large enough to remove all following elements, check how the approach handles these edge cases.
- Make sure to handle cases where no packages are picked (empty result).
- Confirm that the resulting array is indeed sorted non-increasingly.
**Summary:**
- The problem is a sequence decision-making problem with constraints on ordering and removal of elements.
- Model it as a recursive or dynamic programming problem with memoization.
- At each step, decide whether to discard the package or pick it (and remove k subsequent packages).
- Ensure that the picked packages maintain non-increasing order.
- Use lexicographical comparison to choose between possible resulting arrays.
- Optimize using memoization and pruning to manage complexity.
- Finally, return the lexicographically maximal array that can be formed under these constraints.
This approach lets you systematically explore valid sequences, enforcing the ordering constraint and maximizing lexicographical order, resulting in the desired maximal array of packages.
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