Equal Row and Column Pairs

To find the number of pairs (ri, cj) such that row ri and column cj are equal in a given matrix grid, we can follow these steps:

  1. Iterate through all possible row and column pairs.
  2. For each row and column pair, compare the elements to see if they are equal.
  3. If they are equal, increment a counter.
  4. Return the counter.

Here’s the code:

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class Solution:
    def equalPairs(self, grid: List[List[int]]) -> int:
        n = len(grid)
        count = 0

        for i in range(n):
            for j in range(n):
                if grid[i] == [grid[row][j] for row in range(n)]:
                    count += 1

        return count

Explanation:

  • We iterate through all possible rows (i) and columns (j).
  • For each row and column pair, we compare the elements of row i and column j by forming a list containing the elements of column j.
  • If they are equal, we increment the count.
  • Finally, we return the count, which gives the number of equal row and column pairs.

This code will return the correct result for the given problem, taking into account the constraints.

Identifying Problem Isomorphism

“Equal Row and Column Pairs” can be mapped to “Find All Duplicates in an Array”.

Both involve exploring arrays to find specific pairs. In “Equal Row and Column Pairs”, you are tasked to find pairs where the sum of elements in a row equals the sum of elements in a column. In the problem “Find All Duplicates in an Array”, you have to find all elements that appear twice in an array.

Both problems require traversing an array and applying certain conditions to find desired pairs. “Equal Row and Column Pairs” is more complex because it involves a 2D matrix and you need to calculate sums for each row and column. “Find All Duplicates in an Array” is simpler as it only involves a 1D array and straightforward condition checking.

10 Prerequisite LeetCode Problems

This deals with matrix manipulations and requires a good understanding of how to traverse and compare elements in a 2D grid. Here are some simpler problems to prepare:

  1. 867. Transpose Matrix: You need to understand how to transpose a matrix, as transposing can help to compare rows with columns.

  2. 766. Toeplitz Matrix: This problem asks you to compare elements diagonally in a matrix, which will be helpful for understanding how to traverse and compare elements in a grid.

  3. 832. Flipping an Image: Here you are asked to perform manipulations on a 2D grid, which is a basic requirement for the main problem.

  4. 48. Rotate Image: This problem involves complex manipulations of 2D grid and would be a good practice for understanding grid transformations.

  5. 485. Max Consecutive Ones: Although this problem deals with a 1D array, it will help you understand how to identify equal consecutive elements, which is a subproblem of the main problem.

  6. 561. Array Partition I: This problem asks you to pair up elements from an array, which is a simple version of pairing rows and columns in the main problem.

  7. 118. Pascal’s Triangle: Understanding how to construct Pascal’s triangle will improve your skill at dealing with nested arrays, which is necessary for handling a 2D grid.

  8. 169. Majority Element: This problem requires you to find an element that appears more than n/2 times in an array. It can help you understand how to count element occurrences, a subproblem of the main problem.

  9. 88. Merge Sorted Array: This problem requires you to merge two sorted arrays into one. Understanding this will help with comparing rows and columns in the main problem.

  10. 26. Remove Duplicates from Sorted Array: This problem asks you to remove duplicates from a sorted array and helps you understand how to process arrays for comparisons, which is a part of the main problem.

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class Solution:
    def equalPairs(self, grid: List[List[int]]) -> int:
        m = defaultdict(int)
        cnt = 0

        for row in grid:
            m[str(row)] += 1

        for i in range(len(grid[0])):
            col = []
            for j in range(len(grid)):
                col.append(grid[j][i])
            cnt += m[str(col)]
        return cnt

Problem Understanding

The problem asks us to find the number of pairs of rows and columns in a given square matrix where the row and column have the same elements in the same order.

Initial Breakdown

  1. Iterate through each row.
  2. Iterate through each column.
  3. Compare the row and column.
  4. Count pairs that are equal.

Subproblem Refinement

  1. Store each row in a data structure.
  2. Store each column in a data structure.
  3. Compare rows and columns using their stored representations.
  4. Maintain a counter for identical pairs.

Task Identification

  1. Row Extraction: Take out a row from the matrix.
  2. Column Extraction: Take out a column from the matrix.
  3. Comparison: Compare a row and a column.
  4. Counter Management: Update the count of identical pairs.

Task Abstraction

  1. Row Extraction: Collect each element in a specific row and store it in a list.
  2. Column Extraction: Collect each element in a specific column and store it in another list.
  3. Comparison: Check if two lists are equal.
  4. Counter Management: Increment a counter if the comparison is true.

Method Naming

  1. extract_row(): Extracts a row from the matrix.
  2. extract_column(): Extracts a column from the matrix.
  3. compare_arrays(): Compares two arrays for equality.
  4. update_count(): Updates the counter.

Subproblem Interactions

  1. Extract a row using extract_row().
  2. Extract a column using extract_column().
  3. Compare them using compare_arrays().
  4. Update the count using update_count() if they are equal.

Brute-force Solution

  1. Use two nested loops to iterate over rows and columns.
  2. Use extract_row() and extract_column() to get row and column as lists.
  3. Use compare_arrays() to see if they are equal.
  4. If yes, increment the count using update_count().

The time complexity for this approach would be (O(n^3)) as we have two nested loops, and array comparison would take (O(n)).

Optimizing the Solution

We can optimize by hashing each row and column in a dictionary where the key is the string representation of the row or column, and the value is the frequency. Then we can sum the frequencies of identical rows and columns.

This reduces the time complexity to (O(n^2)).

Example

Let’s consider the first example.

  1. For each row and column, hash it and count frequencies. Row 2 and Column 1 will have the same hash value ‘2,7,7’.
  2. The total count of ‘2,7,7’ in both rows and columns is 1.
  3. The answer is 1.

Summary

  1. Hash rows and columns.
  2. Count frequencies.
  3. Sum frequencies for identical rows and columns.
  4. Return the total count.

Language Agnostic Coding Drills

  1. Defining a Class and Function: The basic unit of encapsulation in object-oriented programming.
  2. Creating a List: A list is a collection which is ordered and changeable.
  3. Working with a Dictionary: A dictionary is a collection of key-value pairs. Here, we’re using defaultdict which is a subclass of the built-in dict class that overrides one method and adds one writable instance variable.
  4. Looping Through a List: Both for and while loops are used in this code.
  5. Accessing Elements in a 2D Array: The grid is a two-dimensional array and this code accesses elements row-wise and column-wise.
  6. Using List Methods: The append() function is used to add an element to the end of the list.
  7. Dictionary Manipulation: Elements are being added to the dictionary and values are being incremented.
  8. Working with Strings: Lists are being converted to strings so they can be used as dictionary keys.
  9. Accumulator Pattern: A common pattern in programming where a variable (in this case cnt) accumulates some result in a loop.
  10. Returning a Value from a Function: The function returns the final count.

In terms of increasing difficulty:

  1. Defining a Class and Function
  2. Creating a List
  3. Looping Through a List
  4. Accessing Elements in a 2D Array
  5. Working with a Dictionary
  6. Using List Methods
  7. Dictionary Manipulation
  8. Working with Strings
  9. Accumulator Pattern
  10. Returning a Value from a Function.

Targeted Drills in Python

Drill 1: Defining a Class and Function

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class TestClass:
    def test_function(self):
        pass

Drill 2: Creating a List

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list_example = [1, 2, 3, 4, 5]

Drill 3: Looping Through a List

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list_example = [1, 2, 3, 4, 5]
for i in list_example:
    print(i)

Drill 4: Accessing Elements in a 2D Array

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array_2d = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(array_2d[1][2])  # Outputs 6

Drill 5: Working with a Dictionary

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dictionary_example = {'key1': 'value1', 'key2': 'value2'}
print(dictionary_example['key1'])  # Outputs 'value1'

Drill 6: Using List Methods (append)

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list_example = [1, 2, 3, 4, 5]
list_example.append(6)
print(list_example)  # Outputs [1, 2, 3, 4, 5, 6]

Drill 7: Dictionary Manipulation

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dictionary_example = {'key1': 1, 'key2': 2}
dictionary_example['key3'] = 3
dictionary_example['key1'] += 1
print(dictionary_example)  # Outputs {'key1': 2, 'key2': 2, 'key3': 3}

Drill 8: Working with Strings

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string_example = "hello"
print(string_example.capitalize())  # Outputs 'Hello'

Drill 9: Accumulator Pattern

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accumulator = 0
for i in range(5):
    accumulator += i
print(accumulator)  # Outputs 10

Drill 10: Returning a Value from a Function

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class TestClass:
    def return_function(self):
        return "Returned value"

test_class = TestClass()
print(test_class.return_function())  # Outputs 'Returned value'

Clarification Questions

  1. Can the elements in the rows or columns be negative?
  2. Are duplicate numbers allowed within a single row or column?
  3. Is it possible for the matrix to contain empty rows or columns?
  4. What should be returned if no matching row and column pairs are found? Is it zero?
  5. Is it guaranteed that the matrix will always be square (number of rows equals the number of columns)?
  6. Can the same row or column be counted multiple times if it matches with multiple other rows or columns?
  7. Do we need to consider the order of elements in the row and column, or just the elements themselves?
  8. Is the matrix guaranteed to be non-empty?
  9. Is there a memory constraint we should be aware of for the solution?
  10. Is the goal to optimize for time complexity, space complexity, or both?

Problem Analysis and Key Insights

  1. Square Matrix: The problem deals with an n x n square matrix, meaning the number of rows equals the number of columns. This simplifies the range over which we have to look for matches.

  2. Order Matters: The elements in the matching row and column must be in the same order. This means we can’t simply use a set or sort the array; the original order must be maintained for comparison.

  3. Element Range: The elements in the matrix range from 1 to 10^5, so we don’t have to worry about negative numbers or zero.

  4. Count Pairs: The problem asks for the number of pairs of matching rows and columns, not the pairs themselves. This suggests that we only need to keep track of a count, not the actual indices of matching rows and columns.

  5. Matrix Size: The matrix size is up to 200x200, so a solution with time complexity O(n^2) is likely acceptable, but anything more might be too slow.

  6. No Duplicates Within a Pair: The problem doesn’t specify whether a row or column can participate in multiple pairs, but the use of the word “pair” suggests that each row and each column can be part of only one matching pair.

  7. Matching Criterion: A row and a column are considered a pair only if they contain the same elements in the same order. This indicates that straightforward array comparison is required.

These insights guide the problem-solving approach, particularly focusing on how to efficiently find and count matching row-column pairs.

Problem Boundary

The scope of the problem is confined to the following aspects:

  1. Input: A square matrix of integers, grid, with dimensions n x n. The size of n can range from 1 to 200, and the individual matrix elements can range from 1 to 10^5.

  2. Objective: The task is to find and count the number of pairs of rows and columns in the matrix that contain the same integers in the same order.

  3. Output: A single integer representing the count of such row-column pairs.

  4. Constraints: The matrix size and the range of the integers are specified, but there are no additional constraints on the distribution or frequency of the integers in the matrix.

  5. Complexity: Due to the size constraint of 200 x 200, the algorithm should preferably have a time complexity of O(n^2) or better to be efficient.

  6. Matching Criterion: A pair is considered valid only if the elements in both the row and column are in the same sequence.

The problem doesn’t involve any external data sources, side-effects, or additional parameters. It’s a self-contained problem aiming to test array manipulation, comparison, and perhaps hashing techniques in an efficient manner.

Establishing the boundary of the problem involves clearly defining the limits within which the solution must operate. For this problem, the boundaries can be established as follows:

  1. Matrix Dimensions: The grid is a square matrix with dimensions n x n, where ( 1 \leq n \leq 200 ). This sets the physical boundary of the data you’ll be working with.

  2. Element Range: Each element of the grid ( \text{grid}[i][j] ) is an integer with ( 1 \leq \text{grid}[i][j] \leq 10^5 ).

  3. Output Boundaries: The output is a single integer, which can range from 0 to ( n ) (if all rows match with all columns, which is highly unlikely).

  4. Time Complexity: Given the constraint ( n \leq 200 ), the algorithm should have a reasonable time complexity to solve the problem within these boundaries, ideally ( O(n^2) ) or better.

  5. Functional Boundaries: The solution should solely focus on finding and counting row-column pairs that contain the same integers in the same order.

  6. No External Dependencies: The problem is self-contained, meaning there are no external data sources or APIs to interact with.

  7. Comparison Rules: A row and a column are considered a matching pair only if their elements are identical in both value and order.

By defining these boundaries, you’re setting clear ‘rules of the game’, thereby constraining the problem space. This makes it easier to conceptualize a solution that operates efficiently within these limits.

Problem Classification

The problem falls under the domain of “Arrays and Matrices” in computer science. It’s closely related to combinatorial analysis, as it involves finding and counting specific combinations of rows and columns.

‘What’ Components

  1. Input Matrix: A square matrix, grid, of integers is given.
  2. Rows and Columns: The matrix contains rows and columns that can be compared.
  3. Matching Criteria: A row and column pair is considered a match if they contain the same integers in the same order.
  4. Output: The number of such matching pairs needs to be returned.

The problem can be classified as a “Counting and Matching” problem within the broader category of “Array and Matrix Manipulation”. It involves iterating through the matrix to find combinations that meet a certain criteria (matching rows and columns) and then counting those combinations.

Distilling the Problem to Its Core Elements

Fundamental Concept or Principle

The fundamental concept underlying this problem is “Element-wise Comparison” of arrays (rows and columns in this case). The principle of iteration and comparison is at its core.

Simplest Description

Imagine you have a square grid of numbers. You have to count how many rows have the exact same numbers, in the exact same order, as any column in the grid.

Core Problem

The core problem is identifying and counting the number of rows in a square grid that have the same sequence of integers as any column.

Simplified Problem Statement

Count the number of rows in a square grid of numbers that have the same elements, in the same order, as any column in the same grid.

Key Components

  1. Square Grid: A 2D array of integers.
  2. Rows: Horizontal sequences of integers in the grid.
  3. Columns: Vertical sequences of integers in the grid.
  4. Element-wise Comparison: Compare each element of a row with each element of a column.
  5. Count: Keep track of the number of matching rows and columns.

Minimal Set of Operations

  1. Iterate through each row in the grid.
  2. For each row, iterate through each column.
  3. Perform element-wise comparison between the row and column.
  4. If all elements match, increment a counter.
  5. Return the counter value.

Visual Model of the Problem

Visualization Technique: Grid Mapping

Visualizing this problem can help a lot in understanding it. Consider the square grid as a matrix that can be drawn on paper or visualized in your mind. Rows are horizontal sequences and columns are vertical sequences in this grid.

Example 1:

For the grid [[3,2,1],[1,7,6],[2,7,7]], you can visualize it as:

3  2  1
1  7  6
2  7  7
  • Here, Row 2: [2, 7, 7] is the same as Column 1: [2, 7, 7]

Example 2:

For the grid [[3,1,2,2],[1,4,4,5],[2,4,2,2],[2,4,2,2]], visualize it as:

3  1  2  2
1  4  4  5
2  4  2  2
2  4  2  2
  • Row 0: [3, 1, 2, 2] matches with Column 0: [3, 1, 2, 2]
  • Row 2: [2, 4, 2, 2] matches with Column 2: [2, 4, 2, 2]
  • Row 3: [2, 4, 2, 2] also matches with Column 2: [2, 4, 2, 2]

In this visualization, you can physically (or mentally) go through each row and then run your eyes vertically down each column, checking for matches. This helps you make the problem tangible and sets the stage for coding the solution.

Problem Restatement

In a square grid filled with integers, you need to find how many row-column pairs have the same set of integers in the exact same sequence.

Essential Elements:

  • The grid is square, meaning the number of rows is equal to the number of columns.
  • Rows are horizontal sequences, and columns are vertical sequences in the grid.
  • A row and column are considered equal if their elements match exactly, both in value and order.

Requirements:

  • You have to return the total count of such row-column pairs.

Constraints:

  • The grid size is at most 200x200.
  • The integers in the grid are between 1 and 105.

Abstract Representation of the Problem

We have a square 2D array, where each cell contains an integer. The goal is to count the number of unique ordered sequences that appear both as a row and as a column within this array.

Key Elements:

  • Square 2D array
  • Rows and columns as ordered sequences
  • Matching criteria: Ordered sequences must be identical in terms of elements and their order.

Objective:

  • Count the number of such matching ordered sequences.

Constraints:

  • Fixed upper limit on the dimensions of the array.
  • Fixed range for the integer values.

Terminology

  1. 2D Array (Matrix): A data structure used to represent the grid where each cell contains an integer. The concept of rows and columns comes from this structure.

  2. Row: A horizontal sequence of integers in the 2D array. Rows are key elements to compare against columns.

  3. Column: A vertical sequence of integers in the 2D array. Columns are key elements to compare against rows.

  4. Ordered Sequence: A list of integers where the order matters. Both rows and columns represent ordered sequences in this problem.

  5. Pair (ri, cj): A combination of a row indexed by ‘ri’ and a column indexed by ‘cj’. The problem is focused on finding such pairs where the row and column are equal as ordered sequences.

  6. Constraints: Specific limitations defined in the problem, like the maximum size of the grid and the range of integers it can contain. These affect the possible solution space and computational complexity.

Understanding these terms is essential for grasping both the problem statement and any possible solutions. They form the vocabulary with which the problem is described and solved.

Problem Simplification and Explanation

Key Concepts:

  1. Grid: Think of the grid as a chessboard, where each cell has a number instead of a chess piece.

  2. Row and Column: In this chessboard, each horizontal line of cells is a “row,” and each vertical line of cells is a “column.”

  3. Ordered Sequence: Imagine each row and each column as a unique “code” made up of the numbers in that line. The order of numbers in this code matters.

  4. Pair (ri, cj): The problem asks you to find combinations where a row’s code and a column’s code are identical.

Interaction:

You need to compare every row with every column to find these matching “codes.”

Metaphor:

Imagine you have several locks and several keys. Each row is a lock, and each column is a key. You have to find which lock (row) can be opened by which key (column), but the trick is, the pattern of the teeth on the key must exactly match the pattern inside the lock for it to work.

In simpler terms, you’re trying to find out which rows and columns have the exact same numbers in the exact same order.

Constraints

  1. Square Matrix: The grid is a square (n x n), which means the number of rows equals the number of columns. This symmetry could simplify the logic needed for comparison.

  2. Limited Range of Elements: Each grid element has a value between 1 and 105. You could use this to possibly use a hash-based approach for quick comparisons.

  3. Ordered Sequence: Since rows and columns need to be identical in terms of both elements and their sequence, you don’t need to sort or rearrange them. Simple one-to-one comparisons will suffice.

  4. Small Size: The maximum size of the grid is 200x200. While this isn’t tiny, it’s small enough that an algorithm with moderate time complexity won’t run for an excessively long time.

  5. Counting Pairs: The problem only asks for the number of pairs, not the pairs themselves. This means we don’t need to store these pairs, reducing space complexity.

  6. Same Data, Different Context: Rows and columns use the same set of numbers but in different configurations. By exploiting this, we can potentially save computational effort.

These characteristics offer avenues for efficient problem-solving approaches.

  1. Square Matrix: The n x n dimension tells us that the number of rows and columns are equal. This helps in limiting the search space; we only need to compare each row with each column, not every possible pair of rows and columns.

  2. Element Range: Elements in the grid are between 1 and 105. This fixed range might allow for optimizations, like using hash-based or array-based methods for quicker comparisons.

  3. Size Boundaries: The maximum size being 200x200 allows us to gauge the time complexity we can afford. A solution that is O(n^2) or better will be efficient enough given these constraints.

  4. Output Simplicity: The task asks for a count of pairs, not the pairs themselves, which simplifies the problem by allowing us to focus on the numerical aspect of the solution, without worrying about data structures to hold the pairs.

By considering these constraints, we can guide our problem-solving strategy towards a more optimized and efficient solution.

Case Analysis

Additional Examples and Test Cases

1. Smallest Grid Case

  • Input: [[1]]
  • Output: 1
  • Reasoning: The smallest grid is a 1x1 matrix. Both the single row and the single column contain the same element. So, the count is 1.

2. All Different Elements

  • Input: [[1,2],[3,4]]
  • Output: 0
  • Reasoning: All elements are distinct, so no row and column can have the same elements.

3. All Same Elements

  • Input: [[1,1],[1,1]]
  • Output: 4
  • Reasoning: All rows and columns have the same elements (two 1s), so every row can pair with every column.

4. Single Matching Pair

  • Input: [[1,2,3],[4,5,6],[1,2,3]]
  • Output: 1
  • Reasoning: The first row and the last row are the same, and they form one pair.

5. Multiple Matching Pairs

  • Input: [[1,2,1],[2,1,2],[1,2,1]]
  • Output: 3
  • Reasoning: Each row can match with each diagonal column (from top-left to bottom-right or top-right to bottom-left).

6. Largest Grid Case (Boundary Condition)

  • Input: A 200x200 grid filled with 1s.
  • Output: 40,000 (200 rows * 200 columns)
  • Reasoning: With all elements being 1, every row and every column can be paired with every other row and column.

Edge Cases:

  1. Smallest Grid Case: As demonstrated above, this tests the lower boundary of the problem constraints.

  2. All Different Elements: Ensures the solution properly handles the situation where no rows and columns are the same.

  3. All Same Elements: Ensures the solution handles situations where every element is the same.

  4. Largest Grid Case (Boundary Condition): Tests the upper boundary of the problem constraints to ensure efficiency and correctness.

By analyzing these additional test cases and edge cases, you can gain deeper insights into the problem and ensure the solution covers all scenarios.

Visualizing the cases can help understand the problem better. Let’s visualize some key scenarios:

  1. Smallest Grid Case

    [1]

    Visualizing this as a point or a single cell can suffice.

  2. All Different Elements

    [1, 2]
[3, 4]

    Think of this as a 2x2 square where each cell has a unique value. No row or column shares the same set of values.

  3. All Same Elements

    [1, 1]
[1, 1]

    Picture this as a 2x2 square, where each cell has the same value. Every row can pair with every column, indicated by lines connecting them.

  4. Single Matching Pair

    [1, 2, 3]
[4, 5, 6]
[1, 2, 3]

    Imagine three rows and three columns, with the first and last rows having the same values. You can draw a line between these two to signify their match.

  5. Multiple Matching Pairs

    [1, 2, 1]
[2, 1, 2]
[1, 2, 1]

    Here, visualize diagonals that run from the top-left to bottom-right and from the top-right to bottom-left. These diagonals can pair with rows, which can be represented by lines connecting them.

  6. Largest Grid Case

    200x200 grid filled with 1s

    Picture this as a large square, all filled with the same number. You can visualize lines connecting each row to every column, indicating they all match.

Visualization helps clarify the relations between rows and columns, making it easier to devise a solution.

  1. Smallest Grid Case: In a 1x1 grid, there’s only one row and one column. They will always match. This shows that the minimum output is 1 for n=1.

  2. All Different Elements: When all elements in a grid are different, there will be no matching rows and columns. This highlights that diversity in the grid elements leads to zero matching pairs.

  3. All Same Elements: When all elements in the grid are the same, every row matches every column. This extreme case is useful to consider for performance optimization, as the number of pairs would be n^2.

  4. Single Matching Pair: Even if the grid is diverse, there can be instances where rows and columns match. It indicates that partial similarity can exist in the grid.

  5. Multiple Matching Pairs: Multiple rows can match with multiple columns. We must be careful to count these accurately to get the correct total number of pairs.

  6. Largest Grid Case: Understanding the largest grid case (n=200) is useful for considering performance. This constraint informs us that our algorithm must be efficient enough to handle large numbers of rows and columns.

Key Takeaways:

  • A single row can potentially match with multiple columns and vice versa.
  • The grid’s size and the diversity of its elements can vastly affect the number of matching pairs.
  • Extreme cases like smallest and largest grids help in understanding the boundaries and performance considerations.

Identification of Applicable Theoretical Concepts

  1. Hashing: The need to compare rows and columns suggests that hashing could be an efficient way to store and look up sequences for comparison.

  2. Sorting: Sorting each row and each column can reduce the problem to simpler comparison tasks. Note that sorting changes the problem slightly since the order matters in the original problem.

  3. Array Manipulation: As the problem revolves around arrays, understanding their properties and manipulation can be useful.

  4. Counting: This is essentially a counting problem where we have to tally the number of times a row matches a column.

  5. Dynamic Programming: Though not directly applicable, the concept of storing previously computed information to avoid redundant calculations could be considered.

  6. Matrix Properties: Understanding the properties of a square matrix might provide additional insights, though the problem mainly revolves around individual rows and columns.

  7. Set Theory: The uniqueness of row and column elements in this problem can be looked at through the lens of sets, particularly for optimization.

  8. Brute-force Enumeration: For smaller grids, brute-force approaches can easily tackle this problem by making every possible comparison. However, this is not scalable.

By recognizing these mathematical and algorithmic properties, one can break down the problem into smaller, more manageable tasks, leading to a more efficient and optimized solution.

Simple Explanation

Imagine you have a big square board with rows and columns, like a checkerboard. Each square on the board has a number written in it. Your job is to find how many rows have the exact same numbers as the columns, and in the same order.

For example, let’s say you have three rows and three columns. One row has the numbers 2, 7, 7 and there’s also a column with the numbers 2, 7, 7. That’s a match!

Think of it like bingo but instead of just one number matching, an entire row has to match an entire column.

So, the challenge is to count how many of these matching rows and columns you have on your square board.

Problem Breakdown and Solution Methodology

Approaching this problem involves several key steps:

  1. Initial Understanding: First, realize that you’re dealing with a square matrix. This means the number of rows is equal to the number of columns. Think of it as a tic-tac-toe board where you have to find matching rows and columns.

  2. Data Collection: For each row, collect the sequence of numbers it contains. Do the same for each column. Imagine it as collecting horizontal and vertical number sequences on the tic-tac-toe board.

  3. Comparison: Compare each row sequence with each column sequence to see if they match. It’s like checking if a horizontal line on your tic-tac-toe board has the same numbers as a vertical line.

  4. Counting: Keep a counter to note how many such matches you find. It’s like scoring a point each time you find a matching row and column.

  5. Output: At the end, your counter will have the total number of matching row-column pairs, which is your answer.

Effects of Changes in Parameters:

  • If the size of the matrix increases, the number of comparisons will increase exponentially. For a matrix of size ’n’, you’d potentially have to make n^2 comparisons.
  • If the range of numbers in the cells increases, the complexity remains the same, but the actual comparison might take longer depending on how the numbers are stored or represented.

Example Walkthrough: Suppose you have the matrix:

3 1 2 2
1 4 4 5
2 4 2 2
2 4 2 2

Step 1: Recognize it’s a 4x4 matrix. Step 2: Collect rows as [3,1,2,2], [1,4,4,5], [2,4,2,2], [2,4,2,2] and columns as [3,1,2,2], [1,4,4,5], [2,4,2,2], [2,5,2,2]. Step 3: Compare each row with each column:

  • Row 1 ([3,1,2,2]) matches with Column 1 ([3,1,2,2])
  • Row 3 ([2,4,2,2]) matches with Column 3 ([2,4,2,2])
  • Row 4 ([2,4,2,2]) also matches with Column 3 ([2,4,2,2]) Step 4: You found 3 matches, so your counter will be 3. Step 5: Your output is 3.

By breaking down the problem in this manner, you can better conceptualize and solve it.

Inference of Problem-Solving Approach from the Problem Statement

  1. 0-indexed Matrix: The term implies that the matrix’s rows and columns start at index 0. This informs us to loop from 0 to n-1 for both rows and columns.

  2. n x n Integer Matrix: The term implies that the number of rows and columns are the same, simplifying our looping and comparison logic. Being integer means straightforward equality comparisons.

  3. Pairs (ri, cj): The objective is to find pairs, suggesting a double loop might be necessary—one for rows and another for columns.

  4. Row and Column are Equal: This specifies the criteria for what we are counting, guiding us to focus on element-by-element comparison between rows and columns.

  5. Same Elements in the Same Order: This tells us that not just the elements, but also their sequence, must be identical, steering us toward linear comparison methods like list comparison in Python.

  6. Constraints: Knowing that ’n’ will not be greater than 200 and the elements will be integers between 1 and 105 helps in understanding that simple nested loops will not cause performance issues.

  7. Return the Number of Pairs: The final output is a single integer, informing us that a counter variable can suffice to store the output.

Each keyword or phrase leads to certain strategies:

  • Loops starting from 0 for indexing.
  • Nested loops for comparing rows and columns.
  • Direct integer comparison and sequence validation.
  • Using a simple integer counter to store the number of valid pairs.

These concepts collectively shape our approach, emphasizing direct comparisons within nested loops.

  1. 0-indexed Matrix: Draw a grid and label the rows and columns starting from 0 to n-1. This visually confirms that your loop structures should begin from 0.

  2. n x n Integer Matrix: Draw a square grid to emphasize that the number of rows and columns are the same. This sets the expectation that your loop structures will be symmetrical for rows and columns.

  3. Pairs (ri, cj): Use color-coding or markers to highlight a row and a column you’re comparing. This helps you visually match each row with each column, reinforcing the idea of nested loops.

  4. Row and Column are Equal: Create two separate arrays on paper, one representing a row and the other a column. Put ticks or crosses next to each element to visually compare them. This illustrates the concept of element-wise comparison.

  5. Same Elements in the Same Order: Number the elements in your arrays from the previous step to emphasize the importance of order. This will remind you to keep track of the sequence during the comparison.

  6. Constraints: On the side, jot down the constraints like “n <= 200” or “1 <= grid[i][j] <= 105” to keep them visible while solving. You can use this as a quick reference to ensure your solution will meet the performance requirements.

  7. Return the Number of Pairs: Keep a separate box or circle where you update a count every time a matching pair is found in your diagram or table. This visually represents the counter variable in your code.

Using tables and diagrams to represent these concepts will serve as a visual aid while you go through the process of solving the problem. They can help ensure that you’re taking all the important aspects into account.

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?

Stepwise Refinement

Stepwise Refinement of the Approach

  1. Initialize Counter: Start with a variable to count the number of matching row-column pairs.

  2. Outer Loop for Rows: Loop through each row in the matrix.

    2.1. Capture Current Row: Take the current row and store it in a temporary array.

  3. Inner Loop for Columns: Inside the row loop, loop through each column.

    3.1. Capture Current Column: Take the current column and store it in another temporary array.

    3.2. Compare Row and Column: Perform element-wise comparison between the current row and column arrays.

    3.2.1. Element Matching: If all elements match and are in the same sequence, increment the counter.

  4. Return the Counter: After the loops are finished, return the counter as the answer.

Granular, Actionable Steps

  1. Initialization: Declare an integer counter and set it to zero.

  2. Iterate through Rows: Use a for loop to iterate through each row i from 0 to n-1.

    • Capture Row: Store row i in a list called current_row.

  3. Iterate through Columns: Inside the row loop, use another for loop to iterate through each column j from 0 to n-1.

    • Capture Column: Traverse down column j and store the elements in a list called current_column.

  4. Element-wise Comparison: Use a function to compare current_row and current_column.

    • Increment Counter: If the function returns true, increment the counter by 1.

  5. Return Counter: After both loops have terminated, return the counter value.

Independent Parts

  • Capture Row and Column: The tasks of capturing a row or column are independent of each other.
  • Element-wise Comparison: This can also be considered as an independent task, as it only depends on the current row and column arrays.

Repeatable Patterns

  • Row and Column Iteration: The logic to iterate through rows and columns is repetitive, which suggests the use of nested loops.

  • Element-wise Comparison: The process of comparing a row with a column is the same every time; this can be encapsulated in a reusable function.

By breaking down the problem in this way, you can tackle each piece independently, ensure that each step is actionable, and that the entire solution is structured logically.

Solution Approach and Analysis

Detailed Approach to Solving the Problem

Step 1: Initialization

Start by initializing a counter variable to zero. This counter will keep track of the number of matching row and column pairs.

Step 2: Loop Through Rows

For each row in the matrix, extract the row elements into a list. Consider this list as your “reference row.”

Step 3: Loop Through Columns for Each Row

For each row, also loop through each column of the matrix. Extract the column elements into a list, which we can call the “current column.”

Step 4: Compare Row and Column

Compare the “reference row” with the “current column.” If they are equal, increment the counter by one.

Step 5: Return Counter

After all loops are completed, return the counter value as the answer.

Intuitive Explanation with Metaphor

Imagine each row and column as a train with carriages representing numbers. You need to find trains (rows and columns) that have the same sequence of carriages. For each train (row), you’re checking every other vertical train track (column) to see if they match.

Effects of Changes in Problem Parameters

  1. Size of Matrix (n): As the size increases, the time complexity would increase, as you’d have to perform more comparisons.
  2. Range of Numbers: A larger range doesn’t directly impact the solution’s complexity but might affect memory if a hashing-based optimization is implemented.

Example Case:

Input

grid = [[3, 2, 1], 
        [1, 7, 6], 
        [2, 7, 7]]

Step-by-Step Walkthrough

  1. Initialization: Counter = 0
  2. First Row: Reference Row = [3, 2, 1]
    • Compare with all columns: No match
  3. Second Row: Reference Row = [1, 7, 6]
    • Compare with all columns: No match
  4. Third Row: Reference Row = [2, 7, 7]
    • Compare with second column: Match found
    • Counter incremented to 1
  5. Return Counter: Output is 1

Following this structured approach ensures that all row-column pairs are checked for matching sequences, thereby solving the problem comprehensively.

Identify Invariant

The invariant in this problem is the sequence of elements in each row or column. Specifically, the requirement that a row and column are considered equal if they contain the same elements in the same order establishes this invariant.

In simpler terms, an invariant is a property that remains unchanged during the execution of an algorithm. In the case of this problem, regardless of which row or column you are examining, the rule for comparison (same elements, same order) stays constant throughout the problem-solving process. This invariant guides the algorithm by setting a clear, unchanging criterion for comparing rows and columns.

Identify Loop Invariant

A loop invariant for this problem could be a counter that keeps track of the number of row-column pairs that are equal. This counter would start at zero and could only increase (or stay the same) during each iteration of the loop that checks row-column equality.

In essence, the loop invariant ensures that at any given point in the iteration, the counter accurately reflects the number of row-column pairs that have been found to be equal up to that point in the loop. This loop invariant holds true before the loop starts (base case), during each iteration (maintenance), and after the loop terminates (termination).

By maintaining this loop invariant, you can confidently assert that the final value of the counter will be the total number of equal row-column pairs, thus fulfilling the problem’s requirements.

Thought Process

Thought Process and Steps

  1. Understand the Problem: The first step is to comprehend the problem statement. We need to find pairs of rows and columns that contain the same elements in the same order.

  2. Identify Cues: The problem’s constraints, such as 0-based index and n x n matrix, give cues. The 0-based index implies that we should be careful with the starting and ending indices. The square matrix suggests that the number of rows equals the number of columns, which simplifies traversal.

  3. Insights:

    • Each row has ’n’ elements, and each column has ’n’ elements. This uniformity can help us in making comparisons.
    • The grid’s size constraint up to 200x200 indicates that an O(n^2) solution would likely be sufficient.

  4. Algorithm Approach:

    • We could directly compare each row with each column, but this would be an O(n^3) operation, which might be too slow.
    • A more efficient approach is to create a hash table to store string representations of rows and columns for quick look-up and comparison.

  5. Start Coding: After deciding on the algorithm, the next step is to translate this logic into code.

Python3 Code - PENDING TEST

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def count_equal_row_col_pairs(grid):
    n = len(grid)
    counter = 0
    row_dict = {}
    col_dict = {}

    # Convert each row to a string and store its occurrences in row_dict
    for row in grid:
        row_str = ','.join(map(str, row))
        row_dict[row_str] = row_dict.get(row_str, 0) + 1

    # Convert each column to a string and store its occurrences in col_dict
    for j in range(n):
        col = [grid[i][j] for i in range(n)]
        col_str = ','.join(map(str, col))
        col_dict[col_str] = col_dict.get(col_str, 0) + 1

    # Compare rows and columns
    for row_str in row_dict:
        if row_str in col_dict:
            counter += row_dict[row_str] * col_dict[row_str]

    return counter

# Test the function
grid1 = [[3,2,1],[1,7,6],[2,7,7]]
print(count_equal_row_col_pairs(grid1))  # Output should be 1

grid2 = [[3,1,2,2],[1,4,4,5],[2,4,2,2],[2,4,2,2]]
print(count_equal_row_col_pairs(grid2))  # Output should be 3

Effects of Changing Parameters

  • If the grid size increases, the time complexity remains O(n^2), but the computation will take longer.
  • If the element values change but the order remains the same, the solution will still work, as we care about the order for row-column pairs.

By following this step-by-step guide, we transform the problem statement into a working code solution efficiently.

Establishing Preconditions and Postconditions

Parameters

  1. Inputs: The input to the method is a 2D list called grid.
  2. Types: This 2D list consists of integers.
  3. Representation: In the context of the problem, the 2D list represents an n x n grid.

Preconditions

  1. State: Before calling this method, no specific state of the program needs to be maintained.
  2. Constraints:
    • The 2D list, grid, must be square (n x n).
    • 1 <= n <= 200
    • 1 <= grid[i][j] <= 10^5
  3. Program State: No specific state is required.

Method Functionality

  1. Expected Action: The method is expected to return the number of pairs (row, column) that contain the same elements in the same order.
  2. Interaction: It scans through the rows and columns of the grid to compare them.

Postconditions

  1. State: No change in the state of the program.
  2. Return Value: The return value is an integer that represents the number of row-column pairs that are equal.
  3. Side Effects: There are no side effects; the method is pure.

Error Handling

  1. Preconditions Not Met: If the constraints on the input parameters are violated, the behavior is undefined. The method assumes that the input meets the constraints.
  2. Response: The method doesn’t explicitly handle errors or throw exceptions. Error handling should be done by the caller if necessary.

Problem Decomposition

Problem Understanding

  1. In Own Words: The problem asks for the number of rows and columns in a given 2D grid that have the same sequence of integers.
  2. Key Components:
    • 2D grid of integers
    • Rows and columns to compare
    • Number of matching row-column pairs

Initial Breakdown

  1. Major Parts:
    • Reading the grid
    • Comparing rows with columns
    • Counting matching pairs

Subproblem Refinement

  1. Reading the Grid:
    • No further refinement needed.
  2. Comparing Rows with Columns:
    • Extract a row.
    • Extract a column.
    • Compare the two.
  3. Counting Matching Pairs:
    • Initialize a counter.
    • Increment the counter if a match is found.

Task Identification

  1. Row Extraction: Repeated for each row.
  2. Column Extraction: Repeated for each column.
  3. Comparison: Repeated for each row-column pair.
  4. Counter Initialization: Done once.
  5. Counter Increment: Repeated each time a match is found.

Task Abstraction

  1. Row Extraction: Specific enough to be clear but abstract enough for reuse.
  2. Column Extraction: Same as above.
  3. Comparison: Specific in the context but can be abstracted for reuse.
  4. Counter Initialization: Clear and straightforward.
  5. Counter Increment: Clear and straightforward.

Method Naming

  1. Row Extraction: extract_row()
  2. Column Extraction: extract_column()
  3. Comparison: compare_sequences()
  4. Counter Initialization: initialize_counter()
  5. Counter Increment: increment_counter()

Subproblem Interactions

  1. Order:
    • Initialize counter
    • Extract row
    • Extract column
    • Compare sequences
    • If match found, increment counter
  2. Dependencies:
    • Counter must be initialized before incrementing.
    • Row and column must be extracted before comparison.

From Brute Force to Optimal Solution

Brute Force Solution

First, let’s outline a brute-force solution.

  1. Initialize a counter: Let’s start by setting a counter to zero that will keep track of the number of row-column pairs that are equal.

  2. Nested Loops for Rows and Columns: Run a nested loop, where the outer loop iterates through each row and the inner loop iterates through each column.

  3. Compare Row and Column: For each pair of row and column, compare them. If they are equal, increment the counter.

  4. Return Counter: Finally, return the counter as the output.

Here is a Python3 code snippet illustrating the brute-force approach:

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def find_equal_pairs(grid):
    counter = 0
    n = len(grid)
    
    for i in range(n):
        for j in range(n):
            row = grid[i]
            col = [grid[x][j] for x in range(n)]
            
            if row == col:
                counter += 1
                
    return counter

Inefficiencies in the Brute Force Approach

  1. Time Complexity: The nested loops give us a time complexity of (O(n^2)). Inside these loops, we are also comparing two arrays of size (n), which makes the time complexity (O(n^3)).

  2. Space Complexity: We are also using extra space to store the column in each iteration, giving us a space complexity of (O(n)).

Optimization Steps

Step 1: Use Hashing to Store Rows and Columns

  1. We can use sets to keep track of unique rows and columns as strings.
  2. Convert each row and each column to a string and store it in the set.

Step 2: Count Matching Pairs

  1. For each unique row string, see if a similar column string exists in the set.
  2. If it does, increment the counter.

Here’s a Python3 code snippet with these optimizations:

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def find_equal_pairs(grid):
    counter = 0
    n = len(grid)

    row_set = set()
    col_set = set()

    # Convert rows and columns to strings and store them in sets
    for i in range(n):
        row_str = ','.join(map(str, grid[i]))
        row_set.add(row_str)
        
        col = [grid[x][i] for x in range(n)]
        col_str = ','.join(map(str, col))
        col_set.add(col_str)

    # Count matching pairs
    for row_str in row_set:
        if row_str in col_set:
            counter += 1

    return counter

Complexity Analysis of Optimized Solution

  1. Time Complexity: We’ve reduced it to (O(n^2)) because we go through the grid only once to initialize our sets and another time to count matching pairs.

  2. Space Complexity: We use two sets that could potentially hold all rows and all columns, so (O(n^2)).

This optimized version is both faster and a more efficient use of space compared to the brute-force solution.

Code Explanation and Design Decisions

Initial Parameters

  1. grid: A 2D list containing integers. It represents the square grid that we have to analyze to find equal rows and columns.

Primary Loop

  1. The outer loop iterates over the rows of the grid, and the inner loop iterates over the columns. Each iteration represents a row or a column in the grid.
  2. These iterations populate two sets (row_set and col_set) which are used to efficiently find matching rows and columns.

Conditions and Branches

  1. The condition if row_str in col_set: checks whether a row string is present in the set of column strings.
  2. This is vital for counting the number of equal row-column pairs, fulfilling the problem’s primary constraint.

Updates to Parameters

  1. We update the counter whenever we find a row that matches a column.
  2. The update reflects a successful match and contributes to solving the main problem.

Invariant

  1. The invariant here is the uniqueness of rows and columns stored in row_set and col_set. This helps us by reducing the space we have to search to find matching rows and columns, thereby improving efficiency.

Significance of Final Output

  1. The final output is the variable counter, which is returned.
  2. It represents the total number of matching row-column pairs, satisfying the primary requirement of the problem statement.

Coding Constructs

High-Level Strategies

  1. The code uses a hashing technique by converting rows and columns to strings and storing them in sets for quick lookups.

Purpose for Non-Programmer

  1. Imagine you have a grid made up of numbers. The code checks how many rows in this grid look exactly the same as any column. It counts these special cases for you.

Logical Elements

  1. Iteration: Looping through each row and column.
  2. Hashing: Transforming each row and column into a unique string.
  3. Condition Checking: Checking for matching rows and columns.
  4. Counting: Counting the number of matches.

Algorithmic Approach in Plain English

  1. Take each row, make a string out of it, and remember it.
  2. Do the same for each column.
  3. Now check for each row if it looks like any column. Keep a count of these.
  4. Report back this count.

Key Steps on Input Data

  1. Convert each row and column into a unique string.
  2. Store these strings in separate sets.
  3. Loop through each row string and check if it is present in the set of column strings.
  4. If yes, increase a counter.

Algorithmic Patterns

  1. Iteration: Looping through rows and columns.
  2. Hashing: Creating unique identifiers for rows and columns.
  3. Lookup: Using sets for constant-time search.
  4. Counting: A simple counter variable to keep track of matches.

Language Agnostic Coding Drills

Dissected Concepts

  1. Variable Initialization:

    • Description: Declaring variables and initializing them.
    • Difficulty: Easy. This is fundamental and almost every program will need variables.

  2. Loops and Iteration:

    • Description: Using loops to go through rows and columns.
    • Difficulty: Easy. Loops are a basic construct used to iterate through data.

  3. Data Conversion:

    • Description: Converting numerical arrays or lists to string form.
    • Difficulty: Easy to Medium. While the operation is straightforward, understanding why it’s necessary could require additional knowledge.

  4. Data Storage (HashSet or Set Operations):

    • Description: Storing the converted strings in a set.
    • Difficulty: Medium. Requires understanding of data structures and why a set is optimal here.

  5. Conditional Checks:

    • Description: Using if-statements to check conditions.
    • Difficulty: Easy. Basic but vital to any program.

  6. Counter Mechanism:

    • Description: Implementing a counter to keep track of matches.
    • Difficulty: Easy. Simple but crucial to the problem’s solution.

  7. Lookup Operations:

    • Description: Searching a set to check for matching strings.
    • Difficulty: Medium. Requires understanding how hashsets work and why they provide constant time look-up.

Problem-Solving Approach

  1. Variable Initialization:

    • Start by initializing the counter and sets for rows and columns.

  2. Loops and Iteration:

    • Iterate through each row to process it. This will be your first major loop.

  3. Data Conversion:

    • Inside the loop, convert each row into a unique string representation.

  4. Data Storage (HashSet or Set Operations):

    • Store these unique strings in a set dedicated for rows.

  5. Loops and Iteration (again):

    • After rows, iterate through each column in a similar fashion. This will be your second major loop.

  6. Data Conversion (again):

    • Convert each column into a string just like you did with rows.

  7. Data Storage (again):

    • Store these column strings in a separate set dedicated for columns.

  8. Loops and Iteration (once more):

    • Iterate through the set of row strings.

  9. Conditional Checks and Lookup Operations:

    • For each row string, check if a matching string exists in the set of column strings.

  10. Counter Mechanism:

    • If a match is found, increment the counter.

By performing these “drills” in the prescribed order, you can build up to the final solution to the problem. Each drill contributes a piece to the puzzle, and when they are combined, they solve the problem in an efficient manner.

Targeted Drills in Python

General Coding Drills in Python

1. Variable Initialization

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# Initialize a variable
count = 0

2. Loops and Iteration

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# Loop through a list
for i in range(5):
    print(i)

3. Data Conversion

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# Convert an integer list to string
integer_list = [1, 2, 3]
string_list = ''.join(map(str, integer_list))

4. Data Storage (HashSet or Set Operations)

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# Store data in a set
my_set = set()
my_set.add("hello")

5. Conditional Checks

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# Use of if-statement
x = 5
if x == 5:
    print("x is 5")

6. Counter Mechanism

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# Implement a counter
counter = 0
for i in range(5):
    counter += 1

7. Lookup Operations

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# Use a set to look up an element
my_set = {"hello", "world"}
if "hello" in my_set:
    print("Found")

Problem-Specific Drills in Python

1. Data Conversion for Matrix

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# Convert a matrix row to a string
row = [1, 0, 1]
row_string = ''.join(map(str, row))
  • This drill is essential for creating a unique representation of matrix rows or columns.

2. Row and Column Separation

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# Extract a column from a matrix
matrix = [[1, 0], [0, 1]]
column = [row[0] for row in matrix]
  • This drill helps in handling rows and columns separately, a requirement for this problem.

Integration Steps

  1. Initialize Variables: Use the Variable Initialization drill to set up counter and row_set and column_set.

  2. First Major Loop: Apply the Loops and Iteration drill to iterate through each row of the matrix.

  3. Convert Row to String: Utilize the Data Conversion for Matrix drill to convert the current row to a string.

  4. Store Row Strings: Utilize the Data Storage drill to store the row strings in row_set.

  5. Second Major Loop: Similar to step 2, use another loop to go through each column.

  6. Convert Column to String: Use the Data Conversion for Matrix and Row and Column Separation drills to convert the column to string.

  7. Store Column Strings: Use Data Storage to store these in column_set.

  8. Third Major Loop: Use another loop to iterate through row_set.

  9. Look for Matches: Apply the Lookup Operations drill to look for matching column strings in column_set.

  10. Count Matches: Finally, use the Counter Mechanism to count the matches.

By carefully assembling these drills in the specified sequence, you’ll create a cohesive program that effectively solves the problem.

Q&A

Similar Problems

Here are 10 problems that involve similar problem-solving strategies or underlying concepts:

  1. Two Sum: Similar to using HashSets or dictionaries for quick lookup, the Two Sum problem also involves storing numbers in a data structure for quick query.

  2. Longest Substring Without Repeating Characters: This problem requires you to iterate through a string while keeping track of unique characters. The usage of a HashSet to store unique characters and the iteration process is similar.

  3. Find All Duplicates in an Array: This involves iterating through an array and using a data structure to keep track of elements that appear more than once. It uses similar counter mechanisms and data storage techniques.

  4. Valid Sudoku: You need to validate rows, columns, and sub-grids in a Sudoku puzzle, similar to our row and column checks. Data storage and quick lookup are essential here as well.

  5. Isomorphic Strings: This problem requires you to map characters from one string to another, much like mapping rows to columns in our problem. It involves data storage and conditional checks.

  6. Group Anagrams: This problem involves converting strings to a unique representation (like sorting each string) and storing them in a hash table. This is similar to the way we stored unique rows and columns in our problem.

  7. Palindrome Pairs: Like our problem, this also involves string manipulation and data storage. You need to find pairs of strings that form palindromes.

  8. First Missing Positive: This involves iterating through an array while swapping elements to their correct positions. The concept of placing elements correctly is similar to rearranging rows and columns in our problem.

  9. 3Sum: This problem requires you to find all unique triplets in an array that sum up to a specific target. The use of HashSets or similar data structures for quick lookup is a common technique.

  10. Number of Islands: This problem involves scanning a 2D grid to identify clusters of connected components, similar to identifying unique rows and columns in a matrix.

Each of these problems involves at least one of the following: iteration over a data structure, use of HashSets or dictionaries for quick lookup, conditional checks based on constraints, or transformation of data for easier processing. These are the concepts and techniques that were also essential in solving our original problem.