Widest Pair of Indices With Equal Range Sum

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class Solution:
    def widestPairOfIndices(self, nums1: List[int], nums2: List[int]) -> int:
        # Initialize the sum for both arrays and the distance between indices
        sum1 = sum2 = 0
        max_distance = 0

        # Dictionary to store the difference between sum1 and sum2 at each index
        diff_dict = {0: -1}

        for i in range(len(nums1)):
            # Calculate the cumulative sums for nums1 and nums2
            sum1 += nums1[i]
            sum2 += nums2[i]

            # Difference between the cumulative sums
            diff = sum1 - sum2

            # If the difference has been seen before, update the maximum distance
            if diff in diff_dict:
                max_distance = max(max_distance, i - diff_dict[diff])
            else:
                diff_dict[diff] = i

        return max_distance

Identifying Problem Isomorphism

“Widest Pair of Indices With Equal Range Sum” has an approximate isomorph: “Contiguous Array”.

In “Widest Pair of Indices With Equal Range Sum”, you are given two binary arrays nums1 and nums2 of the same length and the task is to find the widest pair of indices [i, j] such that the sum of nums1[i] through nums1[j] equals the sum of nums2[i] through nums2[j].

In “Contiguous Array”, you are given a binary array and you have to find the maximum length of a contiguous subarray with equal number of 0 and 1.

Both share the common theme of looking for a subarray where two conditions are equal (in “Widest Pair of Indices With Equal Range Sum”, it’s the sum of elements from two arrays, and in “Contiguous Array”, it’s the count of 0s and 1s). However, they ask for different measurements: “Widest Pair of Indices With Equal Range Sum” asks for the widest pair of indices, while “Contiguous Array” seeks the maximum length of the subarray.

Both require similar approaches in terms of algorithmic complexity and handling. “Contiguous Array” is easier due to it only involving a single array.

10 Prerequisite LeetCode Problems

This involves range sum queries and the utilization of prefix sums. Here are 10 problems to practice these concepts:

LeetCode 303: Range Sum Query - Immutable This problem is directly about range sum queries, which will help you understand the primary concept needed for your target problem.
LeetCode 304: Range Sum Query 2D - Immutable This problem extends the range sum queries concept to a 2D grid, which will help deepen your understanding.
LeetCode 121: Best Time to Buy and Sell Stock This problem also revolves around the concept of range sum queries and will help you think about related concepts.
LeetCode 53: Maximum Subarray This problem asks for a subarray with the largest sum, which is related to the concept of range sum queries.
LeetCode 523: Continuous Subarray Sum In this problem, you are asked to determine whether a contiguous subarray exists with a certain sum property.
LeetCode 1: Two Sum This problem is simpler but helps in understanding how to iterate over an array and finding pairs of elements that satisfy a particular property.
LeetCode 209: Minimum Size Subarray Sum This problem also deals with subarray sums and could be useful in understanding how to handle range sum queries.
LeetCode 560: Subarray Sum Equals K This problem will help you understand how to deal with subarrays that sum up to a particular value.
LeetCode 325: Maximum Size Subarray Sum Equals k Similar to the previous problem but requires you to find the maximum size of the subarray.
LeetCode 238: Product of Array Except Self Although this problem involves products instead of sums, it still helps in understanding how to work with arrays to achieve a particular goal.

These cover how to work with arrays and calculate sums over ranges, which are essential for solving the problem at hand.

This problem involves working with arrays and finding a subarray that satisfies certain conditions. The following problems, which involve similar operations on arrays, might be beneficial:

724. Find Pivot Index: This problem involves finding an index in an array such that the sum of numbers at all previous indices is equal to the sum of all numbers at indices after it.
525. Contiguous Array: This problem involves finding the maximum length of a contiguous subarray with equal number of 0 and 1, which is somewhat similar to the given problem.
560. Subarray Sum Equals K: This problem involves finding a subarray that sums up to a specific target, somewhat similar to the given problem.
209. Minimum Size Subarray Sum: This problem involves finding the minimal length of a contiguous subarray of which the sum is at least equal to a target. This will help understand sliding window approach.
325. Maximum Size Subarray Sum Equals k: Similar to the problem 560, but instead of counting subarrays, here you need to find the maximum size.
53. Maximum Subarray: This problem involves finding the contiguous subarray which has the largest sum. The Kadane’s algorithm used here can be a useful technique.
904. Fruit Into Baskets: This problem involves finding the longest subarray with at most two different integers.
152. Maximum Product Subarray: This problem involves finding the contiguous subarray which has the largest product. Understanding this problem will be beneficial in finding maximum/minimum within a subarray.
3. Longest Substring Without Repeating Characters: This problem deals with finding the longest unique substring.
718. Maximum Length of Repeated Subarray: This problem deals with finding the maximum length of a subarray that appears in both arrays, which can help in understanding subarray comparisons.

By practicing these problems, you should become more comfortable with the array operations and techniques required to solve the “Widest Pair of Indices With Equal Range Sum” problem.

The problem involves understanding binary arrays and the concepts of range sum. Here are 10 problems to prepare for this problem:

53. Maximum Subarray: This problem asks to find the contiguous subarray (containing at least one number) which has the largest sum.
1. Two Sum: This problem involves finding two numbers in an array that add up to a target.
3. Longest Substring Without Repeating Characters: This problem asks to find the length of the longest substring without repeating characters.
11. Container With Most Water: This problem asks to maximize the area of the container.
42. Trapping Rain Water: This problem involves finding how much rainwater can be trapped.
325. Maximum Size Subarray Sum Equals k: This problem requires finding the maximum size of a subarray that sums to k.
560. Subarray Sum Equals K: This problem asks to find the total number of continuous subarrays whose sum equals to k.
974. Subarray Sums Divisible by K: This problem asks to find the number of subarrays that have a sum divisible by k.
523. Continuous Subarray Sum: This problem involves finding if there’s a continuous subarray of size at least 2 that sums up to a multiple of k.
152. Maximum Product Subarray: This problem asks to find the contiguous subarray within an array (containing at least one number) which has the largest product.

These problems will help you understand and handle arrays and subarrays more effectively, which is crucial to solving the problem you’re aiming for.

For the problem “1983. Widest Pair of Indices With Equal Range Sum”, the following is a good preparation:

“560. Subarray Sum Equals K” - This problem is about finding the total number of continuous subarrays whose sum equals a given integer, which will help you understand how to identify subarrays with specific sums.
“325. Maximum Size Subarray Sum Equals K” - This problem deals with finding the maximum length subarray that has a specified sum, which is directly applicable to the given problem.
“523. Continuous Subarray Sum” - This problem requires identifying whether a continuous subarray exists with a certain sum, which is relevant for understanding how to approach the given problem.
“209. Minimum Size Subarray Sum” - This problem involves finding the minimum size of a subarray with a sum that equals or exceeds a given number, providing a complementary perspective to the problem of finding maximum lengths.
“904. Fruit Into Baskets” - This problem involves identifying the longest subsequence with a certain condition, which is a good practice for the given problem.
“152. Maximum Product Subarray” - This problem involves finding a contiguous subarray with the largest product, helping to understand the idea of contiguous subarrays in general.
“53. Maximum Subarray” - This problem involves finding the contiguous subarray that has the maximum sum. Understanding the idea of subarray sums is relevant for this problem.
“718. Maximum Length of Repeated Subarray” - This problem deals with finding the maximum length of a subarray common to two arrays, which can help in understanding the idea of comparing subarrays between two arrays.
“674. Longest Continuous Increasing Subsequence” - This problem deals with finding the length of the longest continuous increasing subsequence, which is relevant to understanding continuous sequences in arrays.
“845. Longest Mountain in Array” - This problem deals with finding the length of the longest mountain in an array. It will help you understand how to identify and measure specific patterns in an array.

These problems involve concepts like subarray sums, identifying patterns in arrays, and comparing arrays, which are all relevant to the given 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 two 0-indexed binary arrays nums1 and nums2. Find the widest pair of indices (i, j) such that i <= j and nums1[i] + nums1[i+1] + … + nums1[j] == nums2[i] + nums2[i+1] + … + nums2[j]. The widest pair of indices is the pair with the largest distance between i and j. The distance between a pair of indices is defined as j - i + 1. Return the distance of the widest pair of indices. If no pair of indices meets the conditions, return 0.

Example 1:

Input: nums1 = [1,1,0,1], nums2 = [0,1,1,0] Output: 3 Explanation: If i = 1 and j = 3: nums1[1] + nums1[2] + nums1[3] = 1 + 0 + 1 = 2. nums2[1] + nums2[2] + nums2[3] = 1 + 1 + 0 = 2. The distance between i and j is j - i + 1 = 3 - 1 + 1 = 3. Example 2:

Input: nums1 = [0,1], nums2 = [1,1] Output: 1 Explanation: If i = 1 and j = 1: nums1[1] = 1. nums2[1] = 1. The distance between i and j is j - i + 1 = 1 - 1 + 1 = 1. Example 3:

Input: nums1 = [0], nums2 = [1] Output: 0 Explanation: There are no pairs of indices that meet the requirements.

Constraints:

n == nums1.length == nums2.length 1 <= n <= 105 nums1[i] is either 0 or 1. nums2[i] is either 0 or 1.

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.