Unique Paths

Identifying Problem Isomorphism

“Unique Paths” has an approximate isomorphism “Minimum Path Sum”.

In “Unique Paths”, a robot is located at the top-left corner of a m x n grid. The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid. Your task is to find the number of unique paths the robot can take to reach the destination.

In “Minimum Path Sum”, you’re given a m x n grid filled with non-negative numbers. You need to find a path from top left to bottom right which minimizes the sum of all numbers along its path. You can only move either down or right at any point in time.

Both share the same constraint of only being able to move down or right, and they share the same goal of reaching the bottom-right corner from the top-left. However, while “Unique Paths” is interested in the number of possible paths, “Minimum Path Sum” is interested in the path with the minimum total sum.

“Minimum Path Sum” is more complex as it requires you to not only find a path but to optimize the path such that the total sum of the path is minimized. On the other hand, “Unique Paths” only requires finding the total number of paths, which can be solved using combinatorics or dynamic programming techniques without the need for optimization.

Given an M X N grid, with the starting point in the upper left corner and finish point in the bottom right, notice that if M > N, then at first glance, the number of possible paths is simply M choose N. This is because of the M columns, pick N of them to move downwards (each combination of N columns to move down one space corresponds to exactly 1 path from Start to Finish). There are two wrinkles here though: First, we don’t actually move down N time, but N - 1 times (as we are already occupying the first row). Second, the formula for ’n choose k’ assumes that any of the k elements can only be chosen once. But we can move downward N times all in the same column. What we need is the formula for ’n choose k’ with replacement (allowing for the same value of k to be chosen multiple times). That formula is (m + n - 1) choose n. So our solution is the value of (m + (n-1) - 1) choose (n - 1).

Python Solution

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class Solution:
    def uniquePaths(self, m: int, n: int) -> int:

        def factorial(n):
            if n <= 1:
                return 1
            return n * factorial(n - 1)

        def n_choose_k(n, k):
            return factorial(n) / (factorial(k) * factorial(n - k))

        return int(n_choose_k(m + (n - 1) - 1, n - 1))

Explanation

The solution implements the formula for combinations with replacement, also known as the formula for counting the number of ways to move in a grid.

This is how the code works:

  • The factorial() function is a recursive function that calculates the factorial of a number.
  • The n_choose_k() function uses the factorial() function to calculate the number of combinations (or ways to choose k elements from a set of n elements).
  • In the uniquePaths() function, n_choose_k() is used with parameters (m + n - 2, n - 1) to calculate the number of unique paths in a grid from the top left to the bottom right corner. This is based on the formula for combinations with replacement: (m + n - 1) choose (n - 1). The result is then cast to an integer with int() and returned.

So, when you call uniquePaths(m, n), the function returns the number of unique paths from the top left to the bottom right in an m x n grid.

This is a correct and efficient solution for counting the number of paths in a grid when only moving down or right.

Language Agnostic Coding Drills

  1. Understanding Variables and Data Types: The concept of variables and different data types forms the basis of all programming. Here, you need to understand integers as they are used for counting the grid size and the number of paths.

    • Drill: Create and manipulate integer variables.

  2. Recursive Functions: The factorial function is implemented as a recursive function. Understanding how recursion works is key here.

    • Drill: Write a simple recursive function, like a function to compute factorials.

  3. Function Definitions and Calls: Functions are defined to perform certain tasks in the code. In the given code, factorial and combination calculation tasks are defined as functions.

    • Drill: Define a simple function and call it.

  4. Understanding Mathematical Concepts - Factorials: The calculation of the factorial of a number is an essential part of the code as it is used in the combination formula.

    • Drill: Implement a factorial calculation function.

  5. Understanding Mathematical Concepts - Combinations: Combinations are used to calculate the number of ways to choose k elements from a set of n elements. Here, it is used to calculate the number of unique paths in a grid.

    • Drill: Implement a function to calculate combinations.

  6. Applying Mathematical Concepts to Programming Problems: The main task here is to apply the mathematical concept of combinations to solve a programming problem - calculating the number of unique paths in a grid.

    • Drill: Implement a function that takes the size of a grid as input and returns the number of unique paths from the top left to the bottom right of the grid.

The drills start with understanding basic concepts like variables and simple functions, then move to understanding and implementing more complex mathematical concepts like factorials and combinations, and finally, to applying these mathematical concepts to solve a programming problem.

Solution for Coding Drills in Python

  1. Understanding Variables and Data Types:
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# Create and manipulate integer variables
m = 5
n = 5
print(m, n)
  1. Recursive Functions:
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# Write a simple recursive function to compute factorials
def factorial(n):
    if n <= 1:
        return 1
    else:
        return n * factorial(n - 1)

print(factorial(5))  # Output: 120
  1. Function Definitions and Calls:
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# Define a simple function and call it
def greet():
    print("Hello, world!")

greet()  # Output: Hello, world!
  1. Understanding Mathematical Concepts - Factorials:
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# Implement a factorial calculation function
def factorial(n):
    if n <= 1:
        return 1
    else:
        return n * factorial(n - 1)

print(factorial(5))  # Output: 120
  1. Understanding Mathematical Concepts - Combinations:
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# Implement a function to calculate combinations
def n_choose_k(n, k):
    return factorial(n) / (factorial(k) * factorial(n - k))

print(n_choose_k(5, 2))  # Output: 10.0
  1. Applying Mathematical Concepts to Programming Problems:
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# Implement a function that takes the size of a grid as input and returns the number of unique paths
def uniquePaths(m, n):
    return int(n_choose_k(m + n - 2, n - 1))

print(uniquePaths(5, 5))  # Output: 70

These examples cover the drills related to this problem and they build on each other to form the final solution.

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from math import factorial

class Solution:
    def uniquePaths(self, m: int, n: int) -> int:
        rows, cols = m, n        

        return factorial( m+n-2 ) // ( factorial( m-1 ) * factorial( n-1 ) )

Problem Classification

Language Agnostic Coding Drills

  1. Variables: Understand how to declare and assign values to variables. This is fundamental to any programming task.

  2. Data Types: Learn about the basic data types in a language, including integers and floating-point numbers.

  3. Arithmetic Operations: Learn how to perform basic arithmetic operations like addition, subtraction, multiplication, and division.

  4. Importing Libraries: Understand how to import libraries or modules in a programming language.

  5. Using Functions from Libraries: Learn how to use functions provided by the libraries. This includes understanding the function parameters and the return value.

  6. Writing Functions: Learn to write your own functions. This includes understanding parameters, return values, and the scope of variables.

  7. Mathematical Concepts: Understand the concept of a factorial in mathematics and how it can be used in combination formulas.

  8. Object-Oriented Programming: Learn the basics of object-oriented programming. This includes understanding the concepts of classes, objects, methods, and properties.

  9. Combination Formula: Understand the concept of combination in mathematics and how to calculate it using factorials.

  10. Complex Arithmetic Operations: Understand how to perform complex arithmetic operations involving multiple steps and precedence of operations.

  11. Problem Solving: Learn how to approach a problem, break it down, and use the above concepts to devise a solution. This involves analytical thinking and some creativity.

In the given code, these concepts are applied to solve the problem of finding unique paths in a grid using combination formula in mathematics.

Targeted Drills in Python

  1. Variables
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# declare a variable and assign a value
variable = 10
print(variable)  # output: 10
  1. Data Types
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integer_variable = 5
print(type(integer_variable))  # output: <class 'int'>

float_variable = 5.0
print(type(float_variable))  # output: <class 'float'>
  1. Arithmetic Operations
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a = 10
b = 5

# perform basic arithmetic operations
print(a + b)  # output: 15
print(a - b)  # output: 5
print(a * b)  # output: 50
print(a / b)  # output: 2.0
  1. Importing Libraries
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# import the math library
import math
  1. Using Functions from Libraries
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import math

# use the sqrt function from the math library
print(math.sqrt(25))  # output: 5.0
  1. Writing Functions
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def add_two_numbers(a, b):
    return a + b

# use the function
print(add_two_numbers(5, 10))  # output: 15
  1. Mathematical Concepts
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# calculate the factorial of a number
import math
print(math.factorial(5))  # output: 120
  1. Object-Oriented Programming
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class MyClass:
    def __init__(self, variable):
        self.variable = variable

    def print_variable(self):
        print(self.variable)

# create an object of MyClass
obj = MyClass(10)

# call a method of the object
obj.print_variable()  # output: 10
  1. Combination Formula
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# calculate combination C(n, k)
import math
n = 10
k = 5
combination = math.factorial(n) / (math.factorial(k) * math.factorial(n - k))
print(combination)  # output: 252.0
  1. Complex Arithmetic Operations
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a = 10
b = 5
c = 2

# perform complex arithmetic operation
result = (a + b) / c
print(result)  # output: 7.5
  1. Problem Solving This step is about applying all the above concepts to solve a problem. For example, the function uniquePaths in the provided code applies most of these concepts to solve a problem.

The problem statement is as follows:

A robot is located at the top-left corner of a m x n grid. The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid.

The problem is asking how many unique paths the robot can take to reach the destination from the starting position.

To start, we might first visualize the problem and the grid. A good starting point could be a 2 x 2 grid where the robot starts at position (0,0) and needs to reach the position (1,1).

The robot can either move right to (0,1) or down to (1,0). From (0,1), the robot can only move down to (1,1) and from (1,0), the robot can only move right to reach (1,1). So, there are 2 unique paths that the robot can take in a 2 x 2 grid.

Next, we can try with a larger grid like a 3 x 3 grid to try and identify a pattern. A manual calculation shows that there are 6 unique paths in a 3 x 3 grid.

From this exercise, we can infer that this problem involves combinations. The robot needs to move m-1 steps down and n-1 steps right, regardless of the order.

In total, the robot needs to take m+n-2 steps. We just need to choose m-1 steps to move down (or equivalently, n-1 steps to move right) among these m+n-2 steps.

The number of ways to choose k items from n items (order doesn’t matter) is given by the combination formula, which is n! / [k!(n-k)!]. Here, n = m+n-2, and k = m-1 (or k = n-1, both are equivalent).

So the number of unique paths that the robot can take is given by the formula: (m+n-2)! / [(m-1)!(n-1)!]

This is why the uniquePaths function uses factorials to compute the number of unique paths. The factorial function is imported from the math module in Python.

This solution has a time complexity of O(1) since it involves constant time operations (calculations based on the formula).