Range Addition II
The idea is to find the minimum operation size for both row and column because that’s where the maximum number will be after performing all operations:


The above solution is based on the fact that all operations increment an equalsized rectangle from the topleft corner of the matrix. The maximum value will be in the area where every operation overlaps. This area is represented by a rectangle from the topleft with dimensions equal to the smallest rectangle incremented by any operation. So we just need to find this smallest rectangle.


Problem Classification
The problem is based on data manipulation and involves the concept of matrix operations. It falls under the category of numerical computing.
‘What’ components:
 Matrix M: An m x n matrix initialized with all 0’s.
 Operations ops: An array of operations where each operation is an array of two elements [ai, bi]. Each operation represents a modification of the matrix M  all elements M [y] for all 0 <= x < ai and 0 <= y < bi should be incremented by one.
 Maximum integers in the matrix: After all operations have been performed on the matrix, the problem asks for the count of maximum integers in the matrix.
Problem Classification:
 Numerical computing: This problem involves computations on a numerical matrix, with each operation modifying the matrix elements.
 Data manipulation: The problem involves modifying and interpreting data (a matrix and a list of operations in this case).
 Array Processing: The problem involves processing the operations array and performing them on the matrix M.
The problem is a combination of understanding matrix manipulation, data interpretation, and array processing. Its solution requires an understanding of how to perform operations on matrices and how to extract information from the modified matrix.
Language Agnostic Coding Drills
Distinct Concepts:
 Variable declaration: The code begins by declaring variables
min_row
andmin_col
.  Looping through a list: The code then loops through each operation in
ops
.  Accessing elements in a list: Inside the loop, it accesses elements in the operations list using an index.
 Conditional check and assignment: In the loop, a condition checks if the current operation’s row or column is less than
min_row
ormin_col
. If it is,min_row
ormin_col
is updated to that value.  Multiplication and return statement: After the loop, the code multiplies
min_row
andmin_col
and returns the result.
 Variable declaration: The code begins by declaring variables
Coding Concepts or Drills:
 Basic Python syntax: (Difficulty Level  1) Understanding the basic syntax is the starting point. This includes knowing how to declare variables, loop through a list, and access elements in a list.
 Conditional statements: (Difficulty Level  2) Writing conditions to compare values and using them to update variables is a common pattern in programming. This is a bit more challenging as it requires logical thinking to determine the appropriate condition.
 Working with nested lists: (Difficulty Level  3) The operations are given as a list of lists, which requires understanding how to access elements in a nested list.
 Returning the result: (Difficulty Level  4) Knowing what value to return and when to return it is crucial to solve the problem.
Problemsolving Approach:
 Understand the problem: Recognize that the operations always start at the top left corner of the matrix (0,0) and that every operation increments the values in a submatrix. The submatrix is defined by the rows and columns in the operation, so the overlapping area of all submatrices will have the maximum value.
 Initialize variables: Start with
min_row
andmin_col
as the full size of the matrix. The reasoning behind this is to find the smallest submatrix affected by the operations.  Loop through the operations: For each operation, check if the operation’s row or column is smaller than the current
min_row
ormin_col
. If it is, updatemin_row
andmin_col
. This is because we are looking for the smallest common submatrix affected by all operations.  Return the result: At the end, return the area of the smallest common submatrix, which is
min_row * min_col
. This represents the count of the maximum integers in the matrix after performing all the operations.
Targeted Drills in Python
Python Coding Drills:
Basic Python syntax:
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# Declaring a variable my_var = 10 # Looping through a list my_list = [1, 2, 3, 4, 5] for i in my_list: print(i) # Accessing elements in a list print(my_list[1]) # Outputs: 2
Conditional statements:
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a = 10 b = 20 # Conditional check and assignment if b > a: a = b print(a) # Outputs: 20
Working with nested lists:
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# Nested list my_list = [[1, 2], [3, 4], [5, 6]] for sublist in my_list: for i in sublist: print(i) # Outputs each number in the sublists
Returning the result:
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def calculate_area(length, width): return length * width print(calculate_area(4, 5)) # Outputs: 20
ProblemSpecific Concepts:
This problem doesn’t have any unique problemspecific concepts outside of the general programming concepts identified. The key to this problem is understanding how operations on the matrix are performed and how they affect the matrix. This is an important concept because it allows us to determine the smallest submatrix affected by all operations, and to calculate the number of maximum integers in the matrix.
Integration of Concepts:
All the drills above can be integrated together in a stepbystep manner to build the final solution to the problem.
 Begin with variable declaration and initialization. Initialize
min_row
andmin_col
as the full size of the matrix.  Next, loop through the list of operations. This will involve the looping through a list drill.
 Within this loop, for each operation, check if the operation’s row or column is smaller than the current
min_row
ormin_col
. If it is, updatemin_row
andmin_col
. This involves using conditional checks and assignments.  After the loop, return the result of
min_row * min_col
, which involves returning the result.
Following these steps will lead to the final solution, with each step contributing to solving the problem.
 Begin with variable declaration and initialization. Initialize