Minimum Swaps To Make Sequences Increasing

Identifying Problem Isomorphism

“Minimum Swaps to Make Sequences Increasing” has an approximate isomorphism: “Minimum Number of Swaps to Make the String Balanced”.

In “Minimum Number of Swaps to Make the String Balanced”, you’re given a string containing parentheses, and your task is to perform the minimum number of swaps to balance the string. A string is balanced if the number of opening and closing parentheses is equal and for every prefix of the string, the number of opening parentheses is not less than the number of closing parentheses.

Similarly, “Minimum Swaps to Make Sequences Increasing” involves performing the minimum number of swaps to make two given sequences strictly increasing.

In both problems, the core computational task involves finding the minimum number of swaps to satisfy a certain condition, be it balancing parentheses or making sequences strictly increasing. Both problems thus require similar strategic considerations, although the specifics of the condition and the structure of the input are different.

“Minimum Swaps to Make Sequences Increasing” is simpler because you only need to compare numerical values and their order. “Minimum Number of Swaps to Make the String Balanced” is a bit more complex as it involves manipulating a string to meet certain conditions of balance.

10 Prerequisite LeetCode Problems

For “801. Minimum Swaps To Make Sequences Increasing”, the following are good preparation:

“746. Min Cost Climbing Stairs” - This problem is about finding a path with minimum cost which is similar to our main problem where we need to find minimum swaps.
“198. House Robber” - This problem is about dynamic programming and making choices at each step which is the basis of our main problem.
“55. Jump Game” - This problem is about finding a path in an array which is similar to finding minimum swaps in our main problem.
“70. Climbing Stairs” - This problem is about recursion and how to reach the final step with minimum efforts which is similar to our main problem.
“322. Coin Change” - This problem is about making decisions at each step to minimize the total number of coins which is similar to our main problem.
“300. Longest Increasing Subsequence” - This problem is about finding an increasing subsequence in an array which directly relates to our main problem.
“1143. Longest Common Subsequence” - This problem is about finding a common subsequence between two arrays which is similar to finding minimum swaps between two arrays in our main problem.
“64. Minimum Path Sum” - This problem is about finding a path with minimum sum in a grid, which can be seen as a similar problem to finding a sequence with minimum swaps.
“72. Edit Distance” - This problem is about converting one string to another with minimum operations. It’s similar because swapping elements in arrays to make them increasing is akin to transforming one sequence into another.
“120. Triangle” - This problem is about finding a path with the minimum total from top to bottom in a triangle. The dynamic programming approach used in this problem can be useful to solve the main problem.

These deal with dynamic programming, making choices at each step, recursion, or finding paths in arrays/grids. Understanding how to solve these problems can provide a solid foundation for tackling the main problem.

Problem Classification

Problem Statement: You are given two integer arrays of the same length nums1 and nums2. In one operation, you are allowed to swap nums1[i] with nums2[i].

For example, if nums1 = [1,2,3,8], and nums2 = [5,6,7,4], you can swap the element at i = 3 to obtain nums1 = [1,2,3,4] and nums2 = [5,6,7,8]. Return the minimum number of needed operations to make nums1 and nums2 strictly increasing. The test cases are generated so that the given input always makes it possible.

An array arr is strictly increasing if and only if arr[0] < arr[1] < arr[2] < … < arr[arr.length - 1].

Example 1:

Input: nums1 = [1,3,5,4], nums2 = [1,2,3,7] Output: 1 Explanation: Swap nums1[3] and nums2[3]. Then the sequences are: nums1 = [1, 3, 5, 7] and nums2 = [1, 2, 3, 4] which are both strictly increasing.

Example 2:

Input: nums1 = [0,3,5,8,9], nums2 = [2,1,4,6,9] Output: 1

Constraints:

2 <= nums1.length <= 105 nums2.length == nums1.length 0 <= nums1[i], nums2[i] <= 2 * 105

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.

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?

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?

Is invariant and loop invariant the same for 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 code for the solution of this problem.

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

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving 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. The response text is of the following format:

Here are 10 problems that use similar underlying concepts:

for i in range(1, n):
    if nums1[i] > nums1[i-1] and nums2[i] > nums2[i-1]:
        keep[i] = keep[i-1]
        swap[i] = swap[i-1] + 1
    if nums1[i] > nums2[i-1] and nums2[i] > nums1[i-1]:
        swap[i] = min(swap[i], keep[i-1] + 1)
        keep[i] = min(keep[i], swap[i-1])

return min(keep[-1], swap[-1])


This code implements the `minSwap` function to find the minimum number of swaps required to make both `nums1` and `nums2` strictly increasing. Note that the function signature indicates that both `nums1` and `nums2` are lists of integers and the function will return an integer. The actual algorithm is a dynamic programming approach where we track the minimum number of swaps needed at each step when we decide to swap or keep the numbers.

## Problem Classification

The problem can be classified into the following categories:

Domain: 
- Algorithms: This problem falls under the category of algorithms because it requires us to devise a process or a sequence of steps to reach a solution.

What components: 
- Input: Two equal length integer arrays, nums1 and nums2.
- Output: The minimum number of operations (swaps between the elements of nums1 and nums2) needed to make both arrays strictly increasing.
- Constraints: The lengths of nums1 and nums2 are in the range 2 <= nums1.length, nums2.length <= 10^5. Each element in the arrays nums1[i], nums2[i] falls in the range 0 <= nums1[i], nums2[i] <= 2 * 10^5.

Classification:
- Dynamic Programming: The problem seems to have the property of overlapping subproblems and optimal substructure which are key hallmarks of dynamic programming problems. This is because each operation might affect future decisions (i.e., whether or not to swap in future operations), suggesting that we might want to keep track of previous states to avoid recalculation.
- Array Manipulation: The problem involves manipulating the elements of two arrays, making it an array manipulation problem. 

Based on these categorizations, it seems like a dynamic programming problem that requires careful array manipulation. The primary task is to find the minimum number of operations (swaps) required to make both input arrays strictly increasing.

## Thought Process

Let's go over a few more examples and explain how the "minimum" swaps is determined.

**Example 1:**

Let's take the example with `nums1 = [1,3,5,4]` and `nums2 = [1,2,3,7]` from the problem statement. Here, by swapping the last two numbers (one swap operation), both the arrays become strictly increasing, so the answer is 1.

**Example 2:**

Consider `nums1 = [1,5,3]` and `nums2 = [2,3,4]`. Here, by swapping the second numbers of the two arrays (one swap operation), both the arrays become strictly increasing (`nums1 = [1,3,3]`, `nums2 = [2,5,4]`). But `nums1` is not strictly increasing. So we need to swap the last numbers too. So the total number of swaps is 2.

**Example 3:**

Consider `nums1 = [1,5,6,4]` and `nums2 = [2,3,7,8]`. Here, by swapping the second and fourth numbers of the two arrays (two swap operations), both the arrays become strictly increasing, so the answer is 2.

In the given problem, we aim to make both the arrays strictly increasing with the "minimum" number of swaps. The algorithm for this uses dynamic programming where two states are maintained - `keep` and `swap`. `keep` tracks the minimum number of swaps to make the arrays increasing assuming the current element is NOT swapped, and `swap` tracks the minimum number of swaps assuming the current element is swapped.

At each step, we update `keep` and `swap` based on the comparison between the current element and the previous element in both arrays. This ensures that the number of swaps is minimized at each step, thereby giving us the minimum number of swaps required to make both arrays strictly increasing.

This problem involves modifying two arrays to be strictly increasing by swapping elements, and doing so with a minimal number of swaps. This is an optimization problem with sequence dependencies - the decision to swap or not at one position depends on the previous positions.

A key insight here is that we're going to have two potential states at each position `i` - one where we do a swap and one where we don't do a swap. We can track the minimum number of swaps needed to reach each state.

To solve this problem:

1. Initialize two lists, `keep` and `swap`, of the same length as `nums1`. `keep[i]` represents the minimum number of swaps to make `nums1[:i+1]` and `nums2[:i+1]` increasing if we don't swap `nums1[i]` and `nums2[i]`, and `swap[i]` represents the minimum number of swaps if we do.

2. We iterate over each position `i`. There are two possibilities - if `nums1[i] > nums1[i-1]` and `nums2[i] > nums2[i-1]`, we can keep the arrays increasing either by not swapping at `i-1` and `i`, or by swapping at both. Similarly, if `nums1[i] > nums2[i-1]` and `nums2[i] > nums1[i-1]`, we can keep the arrays increasing either by swapping at `i-1` and not swapping at `i`, or swapping at both.

3. By comparing these cases, we can update `keep[i]` and `swap[i]` accordingly.

4. The minimum number of swaps to make the arrays increasing is the minimum between `keep[-1]` and `swap[-1]` because the last element is the end of the array.

Now let's write this algorithm in Python.

```python
class Solution:
    def minSwap(self, nums1: List[int], nums2: List[int]) -> int:
        n = len(nums1)
        # Initialization
        keep = [float('inf')] * n
        swap = [float('inf')] * n
        keep[0] = 0
        swap[0] = 1
        
        for i in range(1, n):
            if nums1[i] > nums1[i - 1] and nums2[i] > nums2[i - 1]:
                # 'keep' situation: keep the current element, keep the previous element
                keep[i] = keep[i - 1]
                # 'swap' situation: swap the current element, swap the previous element
                swap[i] = swap[i - 1] + 1
                
            if nums1[i] > nums2[i - 1] and nums2[i] > nums1[i - 1]:
                # 'keep' situation: keep the current element, swap the previous element
                keep[i] = min(keep[i], swap[i - 1])
                # 'swap' situation: swap the current element, keep the previous element
                swap[i] = min(swap[i], keep[i - 1] + 1)
                
        return min(keep[-1], swap[-1])

This Python code implements the strategy explained above, and it is a solution to the problem.

This problem is actually more complicated than it seems at first glance. Your confusion seems to be coming from the fact that you’re trying to sort the individual arrays instead of making them both strictly increasing.

The task here is to make both nums1 and nums2 strictly increasing arrays, not just one of them. This means that you can’t just swap elements to make one of the arrays sorted. You have to consider both arrays at the same time.

So let’s look at your example:

nums1 = [0, 7, 8, 10, 10, 11, 12, 13, 19, 18]
nums2 = [4, 4, 5, 7, 11, 14, 15, 16, 17, 20]

The indices where the sequence is not strictly increasing are: 1, 3, 4 and 9 (0-based index). However, swapping at indices 1, 3, and 4 alone would not make nums1 and nums2 both strictly increasing, because nums2[4] (after swapping) would still be 10, which is not less than nums1[5] = 11. Hence, one more swap is needed at index 5.

Therefore, the minimum number of operations to make both nums1 and nums2 strictly increasing would indeed be 4. You need to swap the elements at indices 1, 3, 4 and 5. Here’s how the arrays look like after each swap:

After swapping at index 1:
nums1 = [0, 4, 8, 10, 10, 11, 12, 13, 19, 18]
nums2 = [4, 7, 5, 7, 11, 14, 15, 16, 17, 20]

After swapping at index 3:
nums1 = [0, 4, 8, 7, 10, 11, 12, 13, 19, 18]
nums2 = [4, 7, 5, 10, 11, 14, 15, 16, 17, 20]

After swapping at index 4:
nums1 = [0, 4, 8, 7, 11, 11, 12, 13, 19, 18]
nums2 = [4, 7, 5, 10, 10, 14, 15, 16, 17, 20]

After swapping at index 5:
nums1 = [0, 4, 8, 7, 11, 14, 12, 13, 19, 18]
nums2 = [4, 7, 5, 10, 10, 11, 15, 16, 17, 20]

After these swaps, both nums1 and nums2 are strictly increasing. The minSwap function will return the correct minimum number of swaps which is 4 in this case.

The memoization could be done using a 2D memoization array, where the first dimension is the index idx, and the second dimension represents the state of the array, which could either be swapped or not swapped. Here’s how you can incorporate memoization:

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class Solution {
public:
    vector<vector<int>> memo;

    int go(int idx, vector<int> &v1, vector<int> &v2, int swapped)
    {
        if(idx == v1.size())
        {
            bool flag = true;
            for(int i=1;i<v1.size();i++)
            {
                flag &= (v1[i]>v1[i-1]);
                flag &= (v2[i]>v2[i-1]);
            }
            if(flag) return 0;
            return 100000000;
        }

        if(memo[idx][swapped] != -1)
        {
            return memo[idx][swapped];
        }

        int result = 100000000;

        if(swapped)
        {
            swap(v1[idx], v2[idx]);
            result = min(result, 1 + go(idx + 1, v1, v2, 0));
            swap(v1[idx], v2[idx]);
        }

        result = min(result, go(idx + 1, v1, v2, 1));
        
        return memo[idx][swapped] = result;
    }

    int minSwap(vector<int>& nums1, vector<int>& nums2) 
    {
        memo = vector<vector<int>>(nums1.size(), vector<int>(2, -1));
        return go(0, nums1, nums2, 0);
    }
};

In this code, I initialized memo as a 2D array with dimensions nums1.size() x 2 and filled it with -1. This represents an unvisited state for each position in nums1 and nums2 and for each state (swapped or not swapped).

In the go() function, I first checked if the current state was already computed (i.e., if memo[idx][swapped] != -1). If so, I simply returned the precomputed value. If not, I computed the minimum number of swaps as before, but stored the result in memo[idx][swapped] before returning it.

Finally, I returned the minimum number of swaps required to make both arrays strictly increasing by calling go(0, nums1, nums2, 0).

Language Agnostic Coding Drills

1. Dissect the code and identify each distinct concept it contains

Here are the distinct coding concepts found in the above code:

Variable Initialization: Initialization of variables and lists are seen in the lines where n, keep, and swap are defined.
Looping/Iteration: The for loop is used to iterate over the lists nums1 and nums2.
Conditions: The if conditions inside the loop are used to decide when to swap elements and when to keep them as they are.
List Indexing: Indexing is used to access and modify the elements of keep and swap lists.
Min function: The min function is used to keep track of the minimum number of swaps required.

2. List out the coding concepts or drills in order of increasing difficulty

Ordered from least difficult to most difficult:

Variable Initialization: This is a basic concept, where we declare and initialize variables.
List Indexing: This concept involves accessing or modifying elements of a list based on their position or index.
Looping/Iteration: This concept involves using loops to iterate over elements. In this code, a for loop is used to traverse the lists.
Conditions: Conditional statements are used to decide the course of action. In this code, conditions are used to decide whether to swap elements or not.
Min function: It’s a bit tricky as it’s used here not only to find minimum values but also to update the state of the keep and swap lists during the iteration.

3. Describe the problem-solving approach

The problem involves deciding when to swap elements between two lists in order to make both lists strictly increasing.

The key strategy in this code is to maintain two lists - keep and swap.

keep[i] represents the minimum number of swaps needed to fix the lists up to index i when we did NOT swap nums1[i] and nums2[i]. Similarly, swap[i] represents the minimum swaps needed when we DID swap the i-th elements.

In each iteration of the loop, we evaluate both the scenarios - swapping and not swapping elements, and update the keep and swap lists accordingly. In the end, we return the minimum value from the last element of both lists, which represents the minimum number of swaps required to make both lists strictly increasing.

Each of these steps or ‘drills’ builds upon the previous ones and are necessary for the final solution. Thus, understanding and implementing these steps individually and then integrating them would lead us to the final solution.

Targeted Drills in Python

1. Variable Initialization

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# Here we are initializing an integer variable and a list
n = 5
my_list = [float('inf')] * n

2. List Indexing

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# Here we are accessing and modifying an element of the list using indexing
my_list = [1, 2, 3, 4, 5]
my_list[2] = 10  # The list now becomes [1, 2, 10, 4, 5]

3. Looping/Iteration

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# Here we are iterating over a list of integers
for i in range(1, len(my_list)):
    print(my_list[i])

4. Conditions

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# Here we are using a conditional statement inside a loop
for i in range(1, len(my_list)):
    if my_list[i] > my_list[i - 1]:
        print(f"{my_list[i]} is greater than {my_list[i - 1]}")

5. Min function

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# Here we are using the min function to find the minimum between two numbers
a = 10
b = 20
minimum = min(a, b)

Now, let’s discuss the problem-specific concepts:

In the given problem, we are dealing with two lists and we need to decide whether to swap elements or keep them as they are at each index. This requires an understanding of the conditions for when a swap is beneficial. We need to compare the elements at the same indices in the two lists, and the elements at the current index with the previous index.

Problem-specific drill:

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# Assuming nums1 and nums2 are two lists
if nums1[i] > nums1[i-1] and nums2[i] > nums2[i-1]:
    # Update the 'keep' and 'swap' lists

Now, how do these pieces integrate together?

The integration is pretty straightforward. We start by initializing our variables and lists (keep and swap). Then, we enter a loop that goes from the second element to the end of our lists. Inside the loop, we apply the conditions to determine whether to swap or keep the elements and update our keep and swap lists accordingly.

At the end of the loop, we return the minimum value between the last elements in keep and swap, which gives us the minimum number of swaps required to make both lists strictly increasing.

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int solve(vector<int>&nums,vector<int>&nums2,int idx,bool swapped){
        if(idx == nums.size()){
            return 0;
        }
        // case1 not strictly inc. so perform swap
        int prev1 = nums[idx-1];
        int prev2 = nums2[idx-1];
        int ans = 0;
        if(swapped){
            swap(prev1,prev2);
        }

        if(nums[idx]<=prev1 || nums2[idx]<=prev2){
            ans +=  1 + solve(nums,nums2,idx+1,1);
        }
        else{
            ans += min(solve(nums,nums2,idx+1,0),1 + solve(nums,nums2,idx+1,1));
        }
        return ans;
    }

It seems that you’re trying to solve this problem recursively, checking at each index whether the numbers at the current index in the two arrays are less than or equal to their previous numbers. If they are, then you consider that a swap is needed and increment the ans by 1. Then you proceed to the next index (idx+1) with swapped set to 1 (true). If the numbers at the current index in the two arrays are both greater than their previous numbers, then you have two options: either don’t perform a swap, or perform a swap if it leads to a lesser number of total swaps. You then take the minimum of these two options.

One potential issue in this code is the lack of a base case for the recursion when idx is 0 (i.e., the first call of the function when idx equals 0). This may cause an out-of-bounds error when trying to access nums[idx-1] and nums2[idx-1]. A base case that returns 0 when idx equals 0 might solve this issue.

Additionally, it seems that this code does not handle the case where a swap is not possible (i.e., when both nums[idx] and nums2[idx] are less than or equal to their respective previous numbers even after a swap). This could result in an incorrect answer.

Finally, the function currently recalculates the same subproblems multiple times because it lacks memoization, which could cause efficiency issues for larger inputs. Consider using a memoization technique to store already computed results of subproblems.

for test case with input [0,7,8,10,10,11,12,13,19,18] [4,4,5,7,11,14,15,16,17,20] expected output is 4 but swapping 2,3,4 elements would give strictly increasing sequence. Anyone please explain where I am wrong or why the expected output is 4.

In the given input,

nums1: [0,7,8,10,10,11,12,13,19,18] nums2: [4,4,5,7,11,14,15,16,17,20]

If you only swap the 2nd, 3rd and 4th elements, you get:

nums1: [0,4,5,7,10,11,12,13,19,18] nums2: [4,7,8,10,11,14,15,16,17,20]

Which seems strictly increasing. But the main point of the problem statement is that both arrays should be strictly increasing after swapping. Here, the array nums2 is not strictly increasing since the first and the second element are the same (4, 4), hence it violates the condition of strict increase.

So, we need to perform one more swap, the first elements of both arrays.

After swapping, you get:

nums1: [4,4,5,7,10,11,12,13,19,18] nums2: [0,7,8,10,11,14,15,16,17,20]

Now, nums1 still isn’t strictly increasing because the first and second elements are the same. So, the second element also needs to be swapped.

Finally, after swapping, you get:

nums1: [4,7,5,7,10,11,12,13,19,18] nums2: [0,4,8,10,11,14,15,16,17,20]

Now, nums1 isn’t strictly increasing after the 2nd and 3rd elements. Hence, the third element also needs to be swapped.

After swapping, you get:

nums1: [4,7,8,7,10,11,12,13,19,18] nums2: [0,4,5,10,11,14,15,16,17,20]

Now, nums1 isn’t strictly increasing after the 3rd and 4th elements. Hence, the fourth element also needs to be swapped.

After the final swapping, you get:

nums1: [4,7,8,10,10,11,12,13,19,18] nums2: [0,4,5,7,11,14,15,16,17,20]

Now, both arrays are strictly increasing, and we performed a total of 4 swaps which is the expected output. Hence, 4 swaps are necessary to make both arrays strictly increasing.

I think people have over_complicate\over_engineered the solution.

In simple term , we can think of problem as normal knapsack problem. let f(idx,_) := minimum number of swap required to make both array strictly increasing when we are allowed to use A[0..idx] and B[0..idx] array only.

Now, at idx we have the following choice: (a) swap the array (b) Do not swap the array

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    int findAns(int idx, bool swapped,vector<int>& A, vector<int>&B)
    {
        if(idx>1  && A[idx-2] >= A[idx-1]) return INF; 
        if(idx>1 && B[idx-2] >= B[idx-1]) return INF;
        
        if(idx>= A.size()) return 0;

        int &mans = mem[idx][swapped];
        if(mans != -1) return mans;
        

       //number of ways to generate f(idx,_)
        int val1 = 0; //no not swap
        int val2 = 0; //swap it
        
        val1 = findAns(idx+1,0,A,B);
        
        swap(A[idx],B[idx]);
        val2 = 1+ findAns(idx+1,1,A,B);
        swap(A[idx],B[idx]);
        
        return mans = min(val1,val2);
    }

For memoization, if we do not use swapped variable, then we would have to use prevA and prevB in memoization. (where prevA=A[idx-1] and prevB = B[idx-1])

To simplify the code, we swapped the original array itself && encoded this swapping operation in an independent variable swapped. This way for memoization we can use this swapped variable itself instead of using preA and preB, which greatly simplify the code.

This is indeed a simplified and clear explanation of how to approach this problem. This dynamic programming approach uses a top-down memoization technique to solve the problem, making the solution efficient.

Here’s a further explanation of your code for better understanding:

The function findAns returns the minimum number of swaps needed to make both the A and B arrays strictly increasing when only considering elements up to the index idx.
The base case checks whether the arrays are strictly increasing up to the current index. If not, it returns INF to indicate that this is not a valid configuration.
If idx is greater than or equal to the size of the arrays, it means all elements have been considered and 0 swaps are returned.
val1 and val2 represent the minimum number of swaps needed if the current index is not swapped or swapped, respectively. It is determined by recursively calling the findAns function on the next index and adding 1 to val2 because a swap operation is performed in that case.
The minimum of val1 and val2 is stored in mans and returned. This is the minimum number of swaps needed to make the arrays strictly increasing up to the current index.
The mem 2D array is used for memoization to avoid re-computation of the same subproblems.
The ‘swapped’ variable keeps track of whether the current index is swapped or not, which simplifies the state space for memoization.

Remember, this is a recursive solution that uses dynamic programming and memoization, making it efficient in terms of time complexity. The primary idea here is to decide for each index, whether to swap or not, and then take the minimum of the two cases.

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