Count the Number of K-Big Indices

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class Solution:
    def kBigIndices(self, nums: List[int], k: int) -> int:
        prefix = [False] * len(nums)
        pq = []
        for i, x in enumerate(nums): 
            if len(pq) == k and -pq[0] < x: 
                prefix[i] = True
            heappush(pq, -x)
            if len(pq) > k: heappop(pq)
        ans = 0 
        pq = []
        for i, x in reversed(list(enumerate(nums))): 
            if len(pq) == k and -pq[0] < x and prefix[i]: 
                ans += 1
            heappush(pq, -x)
            if len(pq) > k: 
                heappop(pq)
        return ans 

Identifying Problem Isomorphism

“Count the Number of K-Big Indices” can be approximately mapped to “Kth Largest Element in an Array”.

Reasoning:

Both are related to the concept of identifying elements in an array based on their order or rank (Kth largest or smallest). In “Count the Number of K-Big Indices”, it likely involves identifying indices based on some ‘K-Big’ criteria. In “Kth Largest Element in an Array”, the goal is to find the Kth largest element, which involves ordering or rank.

The tasks are not identical, as “Count the Number of K-Big Indices” probably involves counting a certain number of indices meeting some criteria, while “Kth Largest Element in an Array” involves finding a single specific element based on its rank.

“Kth Largest Element in an Array” is simpler as it involves a direct task of finding the Kth largest element, whereas “Count the Number of K-Big Indices” may involve more intricate logic to determine the ‘K-Big’ indices.

The “2519. Count the Number of K-Big Indices” involves concepts like array manipulation, sorting, and binary search. Here are 10 problems to prepare:

LeetCode 704. Binary Search
- This is a straightforward implementation of the binary search algorithm. Mastering this algorithm will help you search through sorted arrays quickly and efficiently.
LeetCode 278. First Bad Version
- This problem is another application of binary search where you’ll need to minimize the number of checks to find the first bad version.
LeetCode 35. Search Insert Position
- This problem requires using binary search to determine where a new element should be inserted in a sorted array.
LeetCode 88. Merge Sorted Array
- This problem involves merging two sorted arrays. It’s a useful skill for manipulating sorted arrays.
LeetCode 215. Kth Largest Element in an Array
- This problem can be solved using a variety of techniques including sorting, heap, and quickselect.
LeetCode 75. Sort Colors
- This problem will help you understand in-place sorting and Dutch national flag problem.
LeetCode 442. Find All Duplicates in an Array
- This problem will help you get comfortable with manipulating arrays and handling their elements.
LeetCode 349. Intersection of Two Arrays
- In this problem, you need to return the intersection of two arrays. It’ll help you understand set operations and working with arrays.
LeetCode 283. Move Zeroes
- This problem helps you learn about in-place operations in arrays, which is an important skill when dealing with array manipulation.
LeetCode 167. Two Sum II - Input array is sorted
- This problem helps you understand the advantages of having a sorted array and how you can use that to optimize your solution.

Another set:

560. Subarray Sum Equals K: This problem requires you to find the number of continuous subarrays whose sum equals to K.
974. Subarray Sums Divisible by K: Here, you have to count all the subarrays with a sum divisible by K.
523. Continuous Subarray Sum: Similar to the above, but here you’re looking for subarrays with sum that is a multiple of K.
209. Minimum Size Subarray Sum: This problem involves finding a continuous subarray which has a sum greater than or equal to a number K.
325. Maximum Size Subarray Sum Equals k: This problem involves finding a subarray with a sum equal to a given value K.
1679. Max Number of K-Sum Pairs: In this problem, you’re tasked with finding the maximum number of pairs that sum up to a certain value K in an array.
1498. Number of Subsequences That Satisfy the Given Sum Condition: Here, you need to find the number of subsequences in an array that satisfy a sum condition related to K.
454. 4Sum II: This problem asks you to count the number of tuples in four arrays that sum up to a value K.
713. Subarray Product Less Than K: This problem requires you to count the number of contiguous subarrays where the product of all the elements in the subarray is less than K.
152. Maximum Product Subarray: This problem asks you to find the contiguous subarray within an array (containing at least one number) which has the largest product, which can help understanding array manipulation.

Problem Classification

Problem Statement:You are given a 0-indexed integer array nums and a positive integer k.

We call an index i k-big if the following conditions are satisfied:

There exist at least k different indices idx1 such that idx1 < i and nums[idx1] < nums[i]. There exist at least k different indices idx2 such that idx2 > i and nums[idx2] < nums[i]. Return the number of k-big indices.

Example 1:

Input: nums = [2,3,6,5,2,3], k = 2 Output: 2 Explanation: There are only two 2-big indices in nums:

i = 2 –> There are two valid idx1: 0 and 1. There are three valid idx2: 2, 3, and 4.
i = 3 –> There are two valid idx1: 0 and 1. There are two valid idx2: 3 and 4. Example 2:

Input: nums = [1,1,1], k = 3 Output: 0 Explanation: There are no 3-big indices in nums.

Constraints:

1 <= nums.length <= 105 1 <= nums[i], k <= nums.length

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.

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.

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?

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

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

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

Thought Process

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.

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?

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.