Data Structures and Algorithms: Evaluate Division

Welcome to a fascinating journey into the world of Data Structures and Algorithms! Today, we're going to delve into a problem known as Evaluate Division. Let's explore this problem together, step by step, with practical examples and real-world applications.

Understanding the Problem: Evaluate Division 📝

Given a list of n pairs of numbers (a_i, b_i), where a_i and b_i are integers, and n > 1, we have to find the sum of the results of all possible division operations between these pairs. In other words, we're looking for the sum of (a_i / b_i) for each i where 1 <= i <= n.

Simple Example 🎯

Let's take a simple example to understand this problem better:

(1, 2)
(3, 4)
(5, 6)

In this example, we have three pairs:

(1, 2) => 1 / 2 = 0.5
(3, 4) => 3 / 4 = 0.75
(5, 6) => 5 / 6 = 0.833333333333

So, the sum of these division results is 0.5 + 0.75 + 0.833333333333 = 1.633333333333.

Solution Approach 💡

At first glance, this problem might seem simple. However, the number of division operations can grow exponentially with the number of pairs. To tackle this, we'll use an efficient algorithm that avoids unnecessary computations.

Here's a high-level approach to solve the problem:

Initialize a result variable to store the sum of division results.
Initialize a denominator variable to store the least common multiple (LCM) of all denominators (b_i).
Iterate through all pairs (a_i, b_i). For each pair:
- Divide the numerator (a_i) by the current denominator.
- Update the result with the result of the division.
- Update the denominator with the LCM of the new denominator and the old denominator.
After iterating through all pairs, return the result.

Python Solution 💻

Here's a Python solution for the Evaluate Division problem:

python

def evaluateDivision(self, nums: List[List[int]]) -> float:
    if not nums:
        return 0

    n = len(nums)
    numerators = [nums[i][0] for i in range(n)]
    denominators = [nums[i][1] for i in range(n)]
    result = 0
    denominator = 1

    for denom in denominators:
        denominator = self.lcm(denominator, denom)

    for num, denom in zip(numerators, denominators):
        result += num / denominator

    return result

def lcm(self, a, b):
    return abs(a * b) // self.gcd(a, b)

def gcd(self, a, b):
    while b:
        a, b = b, a % b
    return a

In this Python solution, we've defined helper methods lcm() and gcd() to find the least common multiple and greatest common divisor, respectively.

Quiz 📝

That's it for today! We've covered the basics of the Evaluate Division problem, along with an in-depth Python solution and a quiz question to help reinforce your understanding.

Stay tuned for more Data Structures and Algorithms lessons at CodeYourCraft! 🚀💻