13.1. Design and Analysis of Algorithms

We now look at another divide and conquer algorithm, this is the geometric problem, given a set of points we would like to compute the closest pair of points among them.

This chunk introduces us to a geometric problem where we want to determine which points in a given set are closest to each other. The method we will explore is a divide and conquer algorithm, often used for problems where breaking the problem into smaller sub-problems can lead to a more efficient solution.

Imagine you’re at a crowded event, and you want to find your closest friend. While you could look around at every person (like calculating the distance to every point), imagine if you could divide the room into smaller sections and only look in those sections. This strategy can save you time and effort.

So, recall that at the beginning of this set of lectures to motivate the need for more efficient algorithms, you consider the example of the video game, if there are several objects on the screen... a naive algorithm could be to explicitly compute the distance between every pair of these n objects, which should be an order n squared algorithm.

The brute force method suggests computing the distance between every possible pair of points, which would take quadratic time - approximately n squared operations, making it inefficient as the number of points increases. Understanding why this method is inefficient is crucial as it sets us up for appreciating the divide and conquer method.

Think of a video game where you have to check the distance to every enemy character on the screen. If you have a hundred enemies, you need to calculate the distances to each enemy for every other enemy, which gets overwhelming and slow. A smarter way would be to group them and only check those within certain ranges.

So, what we are going to see is that we can actually use divide and conquer and produce an order n log n algorithm for this problem.

This section presents the divide and conquer technique, which will enable us to solve the closest pair of points problem in O(n log n) time. This is significantly faster than the brute force approach, especially as the set of points grows larger. The essence of divide and conquer is to break the problem down into smaller, manageable problems and solve each independently.

Consider a library with thousands of books you are trying to find. If you just look through each book one by one, it takes forever. Instead, if you divide the library by genres and then check within each genre, you significantly reduce the time it takes to locate your desired book.

So, formally we are looking at points in two dimensions, so each point is given by an x y coordinate... assume that there is a distance formula...

Each point is represented in a two-dimensional space with x and y coordinates. The distance between any two points is calculated using the Pythagorean theorem. This foundational concept allows us to compute distances effectively between points as we implement our algorithm.

Think of this like navigating a city on a map, where each location has specific coordinates. To find how far apart two places are, you can visualize or use a formula to calculate the straight-line distance between them.

So, to use divide and conquer, we need a way of separating the points into two groups, a roughly equal size... So, we will do it using a vertical line...

In the divide and conquer method, we will split our dataset into two groups by drawing a vertical line through the points, ideally splitting them into two equal halves. This separation allows us to process each half independently, applying the same distance calculation recursively.

Imagine cutting a pizza into two equal halves. By doing so, you can easily serve and analyze each half individually, thus breaking down the whole task of consumption into smaller, more manageable portions.

But, this does not tell us anything about distances between points on the left and points on the right...

While the closest pairs in the left and right halves are found, we must also consider pairs that straddle the dividing line. These pairs may be closer to each other than any two points within just one side. So we have to examine points within a certain distance of the line in addition to handling the left and right sides.

It’s like if your friend is just a short step outside your half of the pizza but right next to your slice! Even though he’s not within your half, he might still be the closest person to you.

Now, we have to worry about how to combine them, how to take care of these points whose distance pans the interval, the line separating the two halves.

After finding the closest pairs in both halves, we determine the minimum distance that exists within a certain band around the dividing line. By consolidating the results and only focusing on points within this relevant band, we can ensure we do not miss any closer pairs that straddle the line.

Think of this like gathering feedback from both halves of a team but also getting additional input from team members who sit right at the dividing line; you want to ensure you capture all relevant opinions.

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