Given n real numbers x1, x2, …, xn, find the two that are closest together by a) a brute force a

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Given n real numbers x1, x2, …, xn, find the two that are...

Key Concepts

Optimized Pairwise Comparison

Once data is sorted, the problem of finding the closest pair can be optimized by only comparing adjacent elements. This capitalizes on the fact that in an ordered sequence, any two elements that are not next to each other must have an intervening element with an even smaller difference, significantly reducing the computational effort compared to checking all pairs.

Sorting Algorithms

Sorting algorithms arrange data into a defined order (usually ascending or descending), which can simplify subsequent operations. In optimization problems, sorting can be used to bring related elements closer together, thereby reducing the search space and the number of comparisons needed, which is crucial for improving efficiency in finding the closest pair.

Brute Force Algorithm

A brute force algorithm examines every possible pair of elements to find a solution, relying on exhaustive search rather than employing any shortcuts or heuristics. This approach is simple to implement but often inefficient when the size of the dataset increases because the number of comparisons grows rapidly, typically leading to quadratic time complexity.

Time Complexity Analysis

Time complexity analysis helps in understanding how the number of operations or comparisons in an algorithm increases with the input size. The brute force method typically has a time complexity of O(n²), whereas more optimized methods, such as using sorted order, can achieve a time complexity of O(n log n) or better by reducing the number of necessary comparisons.

Transcript

00:01 All right, so here we have the same algorithm under two conditions.

00:05 The first is that it will not be sorted, but we're looking for finding the closest two numbers, right? which is really just finding the distance between them.

00:16 So i'm going to call this algorithm closest, and we need a list of n real valued numbers.

00:29 And in order to have a distance, we need at least two elements.

00:35 So there must be at least two elements in this list.

00:40 So first, i'm going to initialize some parameters.

00:44 We'll set k equal to 1, the first index.

00:51 Set l equal to 2, the second index.

00:58 And then we're going to set m equal to their distance.

01:11 Okay, so that's just going to be a 2 minus a1.

01:21 And then we want the absolute value of this distance, because distance should be positive.

01:31 So they want to start iterating through every possible pair in this list.

01:37 So we'll have 4i as it runs from 1 to n.

01:47 And then we'll have a nested list.

01:55 So in this case we're going to have, we'll say j.

01:58 This one's going to run from i plus 1.

02:12 To n right and so what's happening here is that we're in i is equal to one it'll be at a sub one and then it's gonna compare a sub one to a a sub two a three four a five a six seven etc and then it's gonna jump to a two and then it'll compare a a sub two to a three eight eight four a five a six all the way okay so this is gonna allow us to compare every pair of them and we want to check their distance okay so we'll say if and with the absolute value here, a subj minus a sub i is less than m.

03:03 Then we will reset k equal to i, l equal to j, and m is now going to be equal to the absolute value of a so j minus a sub i.

03:36 And so this has gone through and made this comparison with every pair of numbers in this list, and then we want to return a sub i and a subjax.

04:01 We are finding the two closest numbers...

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