Maximum Score of a Good Subarray


Identifying Problem Isomorphism
“Maximum Score of a Good Subarray” is isomorphic to “Maximal Score of a Generous Segment”.
Similarities: Both problems require the identification of a segment of an array and computing a value based on the minimum element in that segment and the length of the segment. The segment must include a specific index.
Differences: The context and names of variables are different. Instead of numbers, we are dealing with ages and instead of subarrays, we have segments.
The problem “Maximal Score of a Generous Segment” might be considered simpler for individuals with a social sciences background due to its humancentered context, i.e., ages of children in a charity home, as compared to the abstract nature of “Maximum Score of a Good Subarray”. However, from a computational point of view, they are equally complex as the logic for solving them is the same.
10 Prerequisite LeetCode Problems
Here are some problems with similar concepts such as sliding window, array manipulation, and maximum/minimum subarray problems, to prepare for “1793. Maximum Score of a Good Subarray”:
 53. Maximum Subarray
 209. Minimum Size Subarray Sum
 239. Sliding Window Maximum
 485. Max Consecutive Ones
 713. Subarray Product Less Than K
 904. Fruit Into Baskets
 1004. Max Consecutive Ones III
 1052. Grumpy Bookstore Owner
 1425. Constrained Subsequence Sum
 1525. Number of Good Ways to Split a String
Problem Boundary
Here are some ways we can establish boundaries for this problem:
Input Boundaries:
 Array length n can range from 1 to 105
 Array values nums[i] are integers between 1 to 2*104
 Index k ranges from 0 to n1
Output Boundaries:
 The output is an integer representing the maximum possible score
 Score can range from 1 to n * (2*104) in the worst case (if all nums[i] = 1)
Subarray Boundaries:
 Subarrays must contain k and be contiguous slices of the input array
 Shortest subarray is just [nums[k]]
 Longest subarray is the entire nums array
 Number of subarrays containing k is O(n2)
Score Boundaries:
 Minimum score is 1 * 1 = 1 (subarray of 1 element with value 1)
 Maximum score is n * (2*104) (entire array with min value 1)
Algorithm Boundaries:
 Need to iterate through O(n2) subarrays in worst case
 Can optimize to O(n) using sliding window, memoization, etc.
 Space complexity optimized to O(n) with tabulation DP
So in summary, boundaries exist on input size, output range, subarray lengths, score values, and algorithm complexity. Analyzing these guides solving the optimization problem efficiently within the problem’s constraints.
Problem Classification
Based on the analysis of the given problem statement, here are the categorizations:
Domain: Arrays, Dynamic Programming
What components:
 Input is an array of integers nums
 Integer k representing an index in the array
 Subarray is a contiguous slice of the array
 Score is calculated as min * length for a subarray
 Need to find a subarray containing k that maximizes the score
Problem classification:
 Optimization problem to maximize a score
 Input constraint is an array and index k
 Score depends on values in subarray as well as length
 Need to evaluate all subarrays containing k
 Can be solved using sliding window and dynamic programming
So in summary, this is an array, dynamic programming problem focused on optimizing a score calculated from a subarray. It involves constraints around a given index k. Conceptually, it requires generating and evaluating all subarrays containing k. The techniques would likely involve sliding windows and DP tabulation to optimize the score.
Distilling the Problem to Its Core Elements
Here are my thoughts on analyzing this problem at a conceptual level:
The core concept is dynamic programming and using optimal substructure  solving subproblems to build up the optimal solution.
I would describe it simply as: given a list of numbers, find the contiguous group of numbers containing a certain number that results in the highest score when multiplied.
The core problem is finding the subarray with the maximum score, where the score factors in both the minimum value and the length.
The key components are:
 Input array of numbers
 Index k we need to include
 Evaluating all subarrays containing k
 Calculating score as min * length
 Finding the maximum score
 The minimal operations are:
 Iterate through subarrays including k
 For each subarray, calculate min value
 Calculate length based on indices
 Compute score as min * length
 Track maximum score found
So in summary, this boils down to an optimal substructure problem where we need to find a subarray with maximum score based on its values and length. The minimal operations involve generating subarrays, computing their scores, and finding the max.
Visual Model of the Problem
Here is one way to visualize the problem statement:
We can draw the input array nums as a line representing the values at each index. For example:
nums = [1, 4, 3, 7, 4, 5] ^ k = 3
To visualize the score calculation:
For any subarray containing k, highlight that range on the line. For example [1, 5] containing k=3.
The score is length * min value in that range.
Annotate the subarray with its length and minimum value. For [1, 5], len = 5, min = 3.
The score can then be calculated as len * min.
To find the optimal subarray:
Incrementally slide a window over the array, highlighting subarrays containing k.
Annotate each with len and min, compute and update max score.
The subarray with the maximum score is the optimal solution.
We can visualize the current subarray, length, min, and score calculations at each step during the sliding window approach. This helps observe how the optimal subarray giving maximum score is found.
The key visualization is representing the array as a line, highlighting subarrays containing k, and annotating them with length and min value to compute score. This models the optimal substructure of the problem.
Problem Restatement
Here is how I would paraphrase the problem statement in my own words:
We are given an array of numbers and a target index k in that array. Our goal is to find the contiguous subarray containing k that would maximize a score calculated as:
score = (minimum value in subarray) * (length of subarray)
So we need to find a slice of the array that includes k, such that when we take the minimum value in that slice and multiply it by the length, we get the highest possible score.
The constraints are:
 The array contains integers between 1 and 20,000
 The array size is between 1 and 100,000
 k is an index between 0 and length of array  1
 We need to return the maximum score achievable
In summary, we are trying to optimize the choice of a subarray of the input array that contains k, such that the score calculated based on the minimum value and length of this subarray is maximized. We need to work within the constraints on input size and values.
Abstract Representation of the Problem
Here is one way to formulate an abstract representation of this problem:
We are given a sequence of numbers S of length n. We are also given an index k, where 0 <= k < n.
We define a segment of S to be any contiguous subsequence Si,j = [S[i], S[i+1], …, S[j]] where 0 <= i <= j <= n1.
We calculate the score of a segment Si,j as:
score(Si,j) = min(Si,j) * length(Si,j)
Where:
 min(Si,j) is the minimum value in the segment
 length(Si,j) is the length of the segment (j  i + 1)
Our goal is to find the segment Sk,l that contains the index k (i <= k <= l) and maximizes the score.
Objective: Find segment Sk,l containing k that maximizes score(Sk,l)
This frames the problem in a generic way in terms of sequences, segments, and an objective function to be optimized. It removes specifics about the numeric array and focuses on the structure of the problem.
Terminology
Here are some key technical concepts relevant to this problem:
Subarray: A contiguous subsequence within an array. We need to find a subarray containing k that maximizes the score.
Sliding window: An abstract concept of a fixed size window sliding over data. Useful for problems involving subarrays/subsequences. The window represents the current subarray.
Dynamic programming: A technique to solve problems by breaking them down into smaller subproblems and storing solutions to each subproblem for reuse. Allows us to optimize the overall solution.
Optimal substructure: The property that an optimal overall solution can be constructed from optimal solutions to subproblems. Finding the optimal subarray exhibits this property.
Memoization: Storing results of solved subproblems to avoid recomputing them. Used in dynamic programming to improve efficiency.
Tabulation: Building up a solution to the overall problem incrementally using precomputed solutions for subproblems. Helps construct the optimal solution.
The key ideas are using the sliding window technique combined with dynamic programming and memoization to find the optimal subarray containing k that maximizes the score function.
Problem Simplification and Explanation
Here’s one way to break this problem down into simpler concepts and an analogy:
Key concepts:
 Array  The list of numbers given as input
 Subarray  A contiguous slice or segment of the array
 Index k  The fixed index the subarray must contain
 Minimum value  The smallest number in the subarray
 Length  The number of elements in the subarray
 Score  Calculated by multiplying min value and length
Analogy:
Think of the array as a long rope with colored beads on it. Each bead represents a number at that index. We are given a special bead at index k that we must include.
Our goal is to choose a segment of the rope containing the special bead such that when we multiply the smallest bead value in that segment by the number of beads, we get the maximum possible score.
To optimize this, we slide a window over the rope containing k, evaluate beads in the window to find min value and length, compute the score, and keep track of the max score we have seen.
In summary, the key is choosing a subarray containing k to maximize a score involving the min value and length. The rope and bead analogy relates the concepts in a simple way.
Constraints
Here are some specific characteristics in the problem statement that could help design an efficient solution:
Array contains positive integers between 1 to 20,000. This small range allows precomputing/caching minimum values seen so far.
Length of array is up to 105. This suggests solutions that are better than O(n2) complexity.
Subarrays must contain index k. This constraint reduces search space  we only need to evaluate subarrays with k rather than all possible subarrays.
Score is based on minimum value and length. Tracking these two variables is sufficient, no need to store entire subarrays.
Score is multiplicative  minimum value has a bigger impact on score than length. Prioritizing min value in algorithm could prune search space.
No constraint on maximum length of subarray. Can start with entire array as initial window.
Integer scores can allow use of max heap to track top k scores seen so far.
In summary, the positive discrete numers, array size, subarray constraints, and multiplicative score function allow optimizations like caching, pruning, priority queues for efficiency.
Here are the key insights gained from analyzing the constraints:
Array size up to 105 suggests need for better than O(n2) complexity. This points to more optimized approaches than brute force.
Subarray must contain k. This reduces search space to O(n) subarrays instead of O(n2) all possible subarrays. Allows techniques like sliding window.
Small number range allows caching minimums seen so far in O(1). Removes need to recompute.
No limit on subarray length allows starting with full array as initial window. Can then shrink optimally.
Multiplicative score means minimizing minimum value gives more impact than length. Can prioritize small min first in search.
Integer scores enable use of data structures like heaps to efficiently track top k scores.
Input constraints very tight (120,000). Can discretize and precompute values.
Output is single optimal score, not full subarray. Allows optimizing to store only relevant info.
Key takeaway is that the specific constraints allow us to employ caching, sliding windows, heaps, discretization and other optimizations to design an efficient O(n) solution instead of brute force O(n2).
Case Analysis
Here are some additional test cases covering different aspects of the problem:
 Trivial case
Input: nums = [1, 2, 3] k = 1
Output: 2
Reasoning: Only one subarray [2] contains k. Score is min * length = 2 * 1 = 2.
 Edge case
Input: nums = [10000, 10000, 10000] k = 1
Output: 20000
Reasoning: Max array value edge case. Entire array is optimal subarray.
 Boundary case
Input: nums = [1, 1, 1] k = 0
Output: 1
Reasoning: Min array value and min length boundary case.
 Negative numbers
Input: nums = [1, 2, 3] k = 1
Output: 2
Reasoning: Handles negative numbers in array.
 Duplicates
Input: nums = [1, 1, 1, 1] k = 2
Output: 2
Reasoning: Duplicate values in array.
 Very large array
Input: nums = [1, 2, 1, 2, 1, …] (length 105) k = 10
Output: 2
Reasoning: Tests max array length.
Analyzing these cases helps cover different scenarios and validate correctness.
Here are the key insights gained from analyzing the different test cases:
The trivial case reveals the bare minimum needed  find subarray with k and compute score.
Edge cases like maximum array values show that constraints need to be handled  integers can go up to 20,000.
Boundary cases like minimum array value and length highlight edge behaviors.
Negative numbers tell us the array values can be positive or negative.
Duplicates show that multiple numbers with the same value are possible.
Very large input size verifies need for optimal algorithm complexity.
Varied placement of k in different positions tests different subarrays.
Different subarray lengths and min values test the score calculation logic.
Overall, these cases reveal:
Need to handle constraints on value range and array size.
Solution should work for any valid placement of k.
Score calculation must consider duplicates and negatives.
Brute force may timeout, need optimal algorithm.
Require thorough testing of different input patterns.
The insights are to validate correctness on all valid and boundary inputs, verify efficient solution, and cover a wide range of test scenarios.
Identification of Applicable Theoretical Concepts
Here are some mathematical and algorithmic concepts that can help simplify and solve this problem more efficiently:
Dynamic programming  This problem exhibits optimal substructure and overlapping subproblems properties that dynamic programming can exploit to avoid recomputing solutions.
Memoization  Storing precomputed solutions to subproblems in a table allows us to reuse them instead of repeatedly calculating them. This optimization technique complements dynamic programming.
Sliding window  Maintaining a window of the current optimal subarray containing k and sliding it over the array allows an efficient O(n) solution instead of O(n^2) brute force.
Monotonic queue  Keeping a deque to store only useful candidates helps track the current minimum value in the sliding window in amortized O(1) time.
Cumulative sums  Precomputing cumulative sums of the array helps quickly find the minimum value in any subarray in O(1) time.
Discrete math  The constrained number range allows modeling as a discrete optimization problem and leveraging techniques like discretization, pruning, etc.
In summary, applying DP, memoization, sliding windows, monotonic queues, cumulative sums, and discrete math helps optimize both time and space complexity to find the optimal subarray efficiently.
Simple Explanation
Here is how I would explain this problem in simple nontechnical terms:
Imagine you have a long line of numbered beads. Each bead represents the number written on it. You are told that bead number k is special and must be included.
Your goal is to choose a continuous segment of beads that includes the special bead. This segment should give you the highest possible points when calculated as follows:
Take the smallest number on any bead in your segment, and multiply it by the total number of beads in the segment.
For example, if your segment has beads with numbers [2, 5, 1, 3], then the smallest number is 1 and the segment length is 4 beads. So your points would be 1 x 4 = 4 points.
You can choose any set of continuous beads as long as bead k is included. Your target is to maximize the points you can get using this method.
It’s like giving you a necklace with numbered beads and asking you to choose a part of the necklace with the special bead that would give you the most points based on the bead with the smallest number and the total beads in that section.
So in simple terms, it’s about choosing a continuous set of numbers that includes a special number, so that the smallest number multiplied by the length gives you the maximum points possible.
Problem Breakdown and Solution Methodology
Here is a stepbystep approach to solve this problem:
Visualize the input array as a line chart with each index mapped to an element. Highlight index k.
Initialize a sliding window of size 1 containing just k. Keep track of the minimum value and length of this window.
Incrementally expand the window by 1 element on both sides, updating the minimum value and length as the window grows.
Compute the score as minimum value * length for each window. Maintain the maximum score seen so far.
Once window reaches full array size, begin contracting the window from the edges inwards, still tracking min, length and max score.
Repeat steps 35 until k is no longer contained in the window.
For example, with nums = [1, 4, 3, 7, 4, 5] and k = 3, the steps would be:
 Array visualization:
[1, 4, 3, 7, 4, 5] ^ k=3
Initial window: [3], min = 3, len = 1
Expand window: [4, 3, 7], min = 3, len = 3, score = 9 (new max)
Contract window: [3, 7], min = 3, len = 2, score = 6
Final max score = 9 from subarray [4, 3, 7]
If array size increases, only the expanding/contracting window steps are impacted. The overall approach remains the same.
Inference of ProblemSolving Approach from the Problem Statement
Here are the key terms and how they inform the problem solving approach:
Array  The input is an integer array, so algorithms suited for arrays/sequences should be considered.
Subarray  Need to find a contiguous subarray, so techniques like sliding window and two pointer approaches are applicable.
Index k  The subarray must contain k, so the solution needs to constrain or pivot around this index.
Minimum value  Calculating minimum in the subarray is a key operation, so data structures like monotonic stack/queue could help optimize this.
Length  Finding the length of the subarray is needed for the score. Keeping a running length variable in the sliding window would suffice.
Score  The objective is to maximize the score, making this an optimization problem. Techniques like dynamic programming, greedy, etc need consideration.
Integer  The array contains integer values, so discretization or integer programming approaches could apply.
Return maximum  Output is a single maximum score, so we can short circuit and return once found.
So in summary, the key terms like subarray, index k, minimum value, length, score, etc provide clues on applicable techniques like sliding windows, monotonic queues, dynamic programming, discretization that can help optimize the solution. The terms guide the overall strategy.
How did you infer from the problem statement that this problem can be solved using ?
Stepwise Refinement
Here is a stepwise refinement for solving this maximum subarray score problem:
High level approach:
 Use sliding window technique to evaluate all subarrays containing k
 For each subarray, compute score as min * length
 Return maximum score found
Breakdown into steps:
Initialize variables
 min_value, max_score = INT_MIN
 left = k
 right = k
 length = 1
Expand window
 Increment right pointer
 Update min_value and length
 Compute score as min_value * length
 Update max_score if new score is greater
Contract window
 Increment left pointer
 Update min_value and length
 Compute and update max_score
Repeat steps 23 until k is excluded
Return max_score
Independent parts:
 Score calculation from min and length
 Updating max score
 Sliding window expansion and contraction
Repeatable patterns:
 Sliding window for subarrays
 Tracking min, max values
 Two pointer approach
The key is breaking down into initializing variables, expanding/contracting window, computing score, and finding maximum. The patterns are sliding window, min/max tracking, and two pointers.
Solution Approach and Analysis
Here is a detailed stepbystep approach to solve this problem:
 Visualize the input array as a number line, with each index mapped to an array element. Highlight index k.
For example, with nums = [1, 4, 3, 7, 4, 5] and k = 3:
nums: [1, 4, 3, 7, 4, 5] ^ k = 3
 Initialize a sliding window of size 1 containing just nums[k]. Keep track of minimum value and length of this window.
min = 3, len = 1
 Incrementally expand the window by including one element on the left and right. Update min and length.
window = [1, 4, 3, 7] min = 1, len = 4
 Compute score as min * length. Update max score seen so far.
score = 1 * 4 = 4 max_score = 4
 Contract window back towards k, updating min, length, score, and max score.
window = [3, 7]
min = 3, len = 2
score = 3 * 2 = 6
max_score = max(4, 6) = 6
 Repeat expansion and contraction until k is excluded. Return max_score.
If array size increases, only the expand/contract steps are impacted. The overall approach remains the same.
This incrementally finds the optimal subarray containing k by considering all subarrays centered around k. The number line visualization and slide window help explain the approach intuitively.
Identify Invariant
The invariant in this maximum subarray score problem is:
At each step, maintain the maximum possible score achievable from a subarray containing k evaluated so far.
Specifically:
Initialize a max_score variable to the minimum possible value (e.g. INT_MIN)
For each subarray containing k evaluated:
Compute its score as min_value * length
Update max_score = max(max_score, score)
Return the final max_score
So the invariant is that max_score always stores the maximum score seen so far across all subarrays containing k processed until that point.
We incrementally update max_score by comparing against the score of each new subarray containing k. When the final subarray containing k has been evaluated, max_score contains the overall maximum possible score.
The key insight is that max_score can be computed incrementally and only depends on the current subarray score, not the previous steps. This allows arriving at the final optimal result through local updates.
Identify Loop Invariant
What is the loop invariant in this problem?
Thought Process
Claude generates buggy code.
Establishing Preconditions and Postconditions
Problem Name:
 What is the problem that you are trying to solve?
Method Name:
 What is the name of the method/function that you are using to solve this problem?
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?
By answering these questions for each method in your program, you can ensure that you have a clear understanding of what each part of your code is doing and how it should behave. This will help prevent bugs and make your code easier to read and maintain.
Problem Decomposition
Problem Name:
 What is the complex problem that you are trying to solve?
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?
By going through these steps for each complex problem, you can break it down into manageable parts, making it much easier to devise an effective solution.
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 highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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
What are the reasons for making these mistakes in the given code?
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.