Maximum Subarray Sum After One Operation

This problem can be approached using dynamic programming. The idea is to keep track of the maximum subarray sum with and without squaring an element. For each position, we calculate:

The maximum subarray sum without squaring an element. This is either the current element itself or the current element added to the maximum subarray sum up to the previous element (without squaring).
The maximum subarray sum with squaring an element. This is either the squared current element or the current element squared and added to the maximum subarray sum up to the previous element (either with or without squaring, whichever is larger).

Finally, we return the maximum value from these two arrays.

Here is the code:

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class Solution:
    def maxSumAfterOperation(self, nums: List[int]) -> int:
        n = len(nums)
        no_square = [0]*n
        with_square = [0]*n
        no_square[0] = nums[0]
        with_square[0] = nums[0]*nums[0]
        for i in range(1, n):
            no_square[i] = max(nums[i], no_square[i-1] + nums[i])
            with_square[i] = max(nums[i]*nums[i], no_square[i-1] + nums[i]*nums[i], with_square[i-1] + nums[i])
        return max(max(no_square), max(with_square))

In the code:

no_square and with_square are arrays to store the maximum subarray sum without squaring and with squaring respectively.
The maximum subarray sum without squaring at position i is the maximum of the current element nums[i] and the sum of the current element and the maximum subarray sum up to the previous element without squaring (no_square[i-1] + nums[i]).
The maximum subarray sum with squaring at position i is the maximum of the squared current element nums[i]*nums[i], the sum of the squared current element and the maximum subarray sum up to the previous element without squaring (no_square[i-1] + nums[i]*nums[i]), and the sum of the current element and the maximum subarray sum up to the previous element with squaring (with_square[i-1] + nums[i]).
Finally, we return the maximum value from the two arrays.

Identifying Problem Isomorphism

“Maximum Subarray Sum After One Operation” is isomorphic to “Optimal Energy Boost”.

In “Optimal Energy Boost”, consider an array of different energy bars, each with a certain energy level (positive or negative). You can eat one bar and double its energy level to get an extra boost. The goal is to maximize the total energy you can obtain from a contiguous selection of energy bars after exactly one boost operation.

The isomorphism is evident in the similarity of operations and objective. Both problems require performing exactly one operation on an array element that maximizes a sum of a subarray. This operation includes squaring a number in “Maximum Subarray Sum After One Operation” and doubling the energy level in “Optimal Energy Boost”.

The differences between the problems are more contextual than functional. “Maximum Subarray Sum After One Operation” deals with mathematical numbers and their operations, while the “Optimal Energy Boost” problem translates this concept into a real-world scenario of consuming energy bars for maximum energy.

Both problems share an equal level of complexity as they both necessitate understanding of array manipulation and optimization techniques. The choice of which problem might be easier to comprehend could depend on whether a learner prefers abstract mathematical problems or tangible, real-world scenarios.

10 Prerequisite LeetCode Problems

“1746. Maximum Subarray Sum After One Operation” requires understanding of the concept of subarray sum and the impact of a single operation on it.

Here are 10 simpler problems to prepare for problem:

LeetCode 53: Maximum Subarray This problem introduces the concept of finding the maximum subarray sum.
LeetCode 121: Best Time to Buy and Sell Stock This problem involves a similar concept of maximizing profit after a single operation.
LeetCode 122: Best Time to Buy and Sell Stock II This problem extends the previous one with the possibility of multiple transactions.
LeetCode 152: Maximum Product Subarray This problem introduces the concept of finding the maximum product of a subarray.
LeetCode 238: Product of Array Except Self This problem introduces the concept of performing an operation on every element except one.
LeetCode 283: Move Zeroes This problem focuses on array manipulation after a certain operation, in this case, moving zeroes.
LeetCode 448: Find All Numbers Disappeared in an Array This problem involves identifying missing elements after certain operations on an array.
LeetCode 560: Subarray Sum Equals K This problem deals with finding a subarray with a specific sum.
LeetCode 724: Find Pivot Index This problem involves identifying a specific index after performing a cumulative sum operation on the array.
LeetCode 747: Largest Number At Least Twice of Others This problem deals with identifying a number that is at least twice as big as all others in the array.

These cover array manipulation, subarray operations, and associated concepts, which are needed to solve the “Maximum Subarray Sum After One Operation” problem.

Clarification Questions

What are the clarification questions we can ask about this problem?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement:You are given an integer array nums. You must perform exactly one operation where you can replace one element nums[i] with nums[i] * nums[i].

Return the maximum possible subarray sum after exactly one operation. The subarray must be non-empty.

Example 1:

Input: nums = [2,-1,-4,-3] Output: 17 Explanation: You can perform the operation on index 2 (0-indexed) to make nums = [2,-1,16,-3]. Now, the maximum subarray sum is 2 + -1 + 16 = 17.

Example 2:

Input: nums = [1,-1,1,1,-1,-1,1] Output: 4 Explanation: You can perform the operation on index 1 (0-indexed) to make nums = [1,1,1,1,-1,-1,1]. Now, the maximum subarray sum is 1 + 1 + 1 + 1 = 4.

Constraints:

1 <= nums.length <= 105 -104 <= nums[i] <= 104

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Why do we need to sort the events by its startTime to get the correct answer?

Sorting the events by their start time is crucial to this problem because it helps in structuring the decision-making process. We’re trying to attend the maximum number of non-overlapping events, and the ordering of the events can drastically impact which ones we’re able to attend.

By sorting the events by start time, we ensure that we’re considering the events in the order they occur. This is helpful because it allows us to adopt a “greedy” strategy, where we make the best decision we can at each step, assuming that this will lead to the best overall outcome.

Furthermore, this sorting is crucial for efficiently finding the next non-overlapping event using binary search. When the events are sorted by start time, we can use binary search to find the next event that starts after the current one ends.

Finally, sorting the events by their start time provides a clear order for our dynamic programming solution, as it relies on using previously computed results of earlier (in time) subproblems to solve later ones.

So, sorting the events by start time not only helps us get the correct answer, but it also allows us to solve the problem more efficiently.

Why did you sort the events by end time? It lead to wrong answer.

If sorting by end time led to a wrong answer, it might be due to the nature of the problem. In some scheduling or interval-based problems, sorting by end time is a common strategy as it allows us to choose the next non-overlapping event that finishes the earliest, thereby leaving the maximum amount of time for subsequent events. This approach is typically employed in problems where the goal is to maximize the number of events attended.

However, in this problem, each event also has a value associated with it, so simply choosing the event that finishes the earliest might not always lead to the maximum total value. Sorting by start time might be more appropriate here because it aligns with the chronological order of events, allowing us to make optimal choices at each step by considering the start time, end time, and the value of the events.

The actual sorting criterion can vary depending on the specific details and requirements of the problem. It’s always crucial to understand the problem context and constraints before deciding on an approach.

What should be the result for: I believe it should be 1500 not 1400 but accepted is 1400. Can anyone tell what I am missing here? [[11,22,50],[33,44,500],[22,33,1000],[55,66,400]] 2

This problem is a type of interval scheduling problem where each event has a start time, end time, and a value. The task is to select up to k events such that they do not overlap and the total value is maximized.

Let’s examine your specific case:

[[11,22,50], [33,44,500], [22,33,1000], [55,66,400]], k=2

We sort the events by end time:

[[11,22,50], [22,33,1000], [33,44,500], [55,66,400]]

Now we select up to k=2 events:

Choosing event [11,22,50] first would leave us with only one viable subsequent event to choose, [33,44,500] or [55,66,400], as [22,33,1000] overlaps with the first event. This would give a total value of 50 + 500 = 550 or 50 + 400 = 450, which are less than 1400.
Choosing event [22,33,1000] first allows us to choose [55,66,400] as the second event, leading to a total value of 1000 + 400 = 1400.

Therefore, the maximum total value we can achieve by selecting k=2 events is 1400.

In your assumption of the maximum total value being 1500, it appears you’ve selected [33,44,500] and [22,33,1000]. However, these two events overlap (since the second event starts before the first one ends), and thus can’t both be selected according to the problem’s constraints. Therefore, 1500 would not be a correct answer for this case given the problem’s conditions.

Not able to memoize it. Can anyone help?

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class Solution {
    int ans = 0;
public:
    void helper(vector<vector<int>>& events, int value, int preEnd, int k, int pos) {
        if (k == 0 || pos == events.size()) {
            ans = max(ans, value);
            return;
        }
        if (events[pos][0] > preEnd) {
            helper(events, value + events[pos][2], events[pos][1], k-1, pos+1);
        }
        helper(events, value, preEnd, k, pos+1);
    }
    
    int maxValue(vector<vector<int>>& events, int k) {
        sort(events.begin(), events.end(), [](const vector<int>& a, const vector<int>& b) {
          return a[1] < b[1];
       });
        
        helper(events, 0, 0, k, 0);
        return ans;
    }
};

You are attempting to use a depth-first search (DFS) to explore all possibilities, and your base cases are correct. However, due to the overlapping subproblems in your DFS, you are doing a lot of repeated computation. This is a classic scenario where memoization (a form of dynamic programming) can be beneficial to avoid repeated calculation and improve time efficiency.

Here’s how you could add memoization to your solution:

Create a memoization matrix/array where dp[i][j] represents the maximum value that can be obtained by picking from the first i events with j picks.
During the recursion, before starting any new recursion, check if the subproblem has already been solved i.e., check if dp[pos][k] has been set before.
If it has, there’s no need to recompute it, you can simply use the saved result.
If it hasn’t been set before, then compute it and save the result in the dp table before returning it.

Here’s your updated code:

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class Solution {
public:
    int dp[1001][51] = {};
    
    int helper(vector<vector<int>>& events, int k, int pos) {
        if (k == 0 || pos >= events.size()) {
            return 0;
        }
        if (dp[pos][k] != 0) { // check if this subproblem has been solved before
            return dp[pos][k]; // if so, return the saved result
        }
        int taken = 0;
        // find next event that does not overlap with the current one
        int next_pos = pos+1;
        while (next_pos < events.size() && events[next_pos][0] <= events[pos][1]) {
            next_pos++;
        }
        if (events[pos][0] > events[pos][1]) {
            taken = events[pos][2] + helper(events, k-1, next_pos);
        }
        int not_taken = helper(events, k, pos+1);
        dp[pos][k] = max(taken, not_taken); // save the result before returning it
        return dp[pos][k];
    }
    
    int maxValue(vector<vector<int>>& events, int k) {
        sort(events.begin(), events.end(), [](const vector<int>& a, const vector<int>& b) {
          return a[1] < b[1];
       });
        
        return helper(events, k, 0);
    }
};

Note that dp[1001][51] assumes the maximum length of events is 1000 and the maximum k is 50. Adjust these values as per the problem constraints.

In the helper function, the idea is to maximize the result for each subproblem by choosing between two options: take the current event and move to the next non-overlapping event (taken) or skip the current event (not_taken). The memoization table dp is updated with the maximum value for each subproblem, which is then returned.

Finally, we initialize the recursion by calling helper from maxValue, and return the result.

Please note that memoization adds a space complexity of O(n*k) where n is the length of events and k is the number of picks allowed. It greatly improves time complexity by preventing recalculating results for the same subproblems.

Can we say this is 0/1 knapsack?

In some respects, this problem is similar to the 0/1 Knapsack problem. Both problems involve making a series of decisions to maximize a certain value under specific constraints:

In the 0/1 Knapsack problem, you’re deciding whether or not to include each item in the knapsack to maximize the total value, without exceeding the knapsack’s capacity.
In this problem, you’re deciding whether or not to include each event to maximize the total happiness, without exceeding the maximum number of events you can attend.

However, there are also key differences:

In the 0/1 Knapsack problem, the decision to include an item is only constrained by the total weight of the items included so far.
In this problem, the decision to include an event is constrained not just by the number of events included so far, but also by the time overlap between events. You can’t attend two events if their times overlap.

Because of this additional constraint, this problem requires a more complex solution than the 0/1 Knapsack problem. You need to sort the events by end time and use binary search to find the next non-overlapping event, which is not required in the 0/1 Knapsack problem.

So while the problem does bear some resemblance to the 0/1 Knapsack problem, it is not a direct application of it and requires additional techniques to solve.

can you please tell me if sorting by start time and sorting by end time are equivalent？

Sorting by start time and sorting by end time are not always equivalent, and the choice between them depends on the specific problem.

In this problem of choosing events to attend, it is not equivalent to sort the events by start time or end time.

Sort by start time: If you sort events by start time, you might start attending events early, but you also might run into a situation where an event starts early and ends late, leaving you unable to attend shorter events happening within this time frame.
Sort by end time: If you sort events by end time, you’re prioritizing events that finish earlier. This approach generally allows you to attend more events, as you free up your schedule earlier. In the context of this problem where we want to attend the maximum number of non-overlapping events, this is the more suitable approach.

So, while both sorts organize events chronologically, they prioritize different aspects of the events (start time vs end time), and the results can lead to different solutions. Always consider the specific requirements and constraints of your problem when deciding which sorting strategy to use.