Design & Analysis of Algorithms - Vol 2 | 13. Divide and Conquer: Closest Pair of Points by Abraham | Learn Smarter
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.

Typing
Memory
Math
English Adventures
Knowledge
13. Divide and Conquer: Closest Pair of Points

The chapter explores the divide and conquer approach for solving the geometric problem of finding the closest pair of points among a given set of points in two dimensions. It compares a brute force O(n²) solution with an optimized O(n log n) algorithm that utilizes sorting and recursive calls to efficiently identify the closest pairs. Key insights include the importance of spatial partitioning and leveraging sorted lists to minimize comparisons across the dividing line.

Enroll to start learning

You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.

Sections

  • 13.1

    Design And Analysis Of Algorithms

    This section discusses the divide and conquer algorithm for finding the closest pair of points among a set of points in two-dimensional space.

    Learning Practice

  • 13.1.1

    Divide And Conquer: Closest Pair Of Points

    This section introduces the divide and conquer algorithm to find the closest pair of points from a set, optimizing the naive O(n²) approach to O(n log n).

    Learning Practice

  • 13.2

    Introduction To The Problem

    This section introduces the geometric problem of finding the closest pair of points in a two-dimensional space using a divide and conquer algorithm.

    Learning Practice

  • 13.2.1

    Naive Algorithm

    The naive algorithm for finding the closest pair of points in a set computes the distance between every pair of points, resulting in a time complexity of O(n²).

    Learning Practice

  • 13.2.2

    Distance Formula

    The Distance Formula provides a method to calculate the distance between two points in a two-dimensional space using their coordinates.

    Learning Practice

  • 13.2.3

    Assumption For Analysis

    This section discusses the divide and conquer technique to find the closest pair of points among a set of points in two dimensions, improving efficiency over a naive quadratic approach.

    Learning Practice

  • 13.2.4

    Brute Force Solution

    The brute force solution for finding the closest pair of points in a set is an O(n²) algorithm that calculates distances between all pairs of points.

    Learning Practice

  • 13.3

    One Dimensional Case

    This section covers the closest pair of points problem using divide and conquer strategies in both one and two-dimensional contexts.

    Learning Practice

  • 13.3.1

    Sorting And Finding Minimum Distance

    The section discusses the Divide and Conquer algorithm for finding the closest pair of points among a set of points in a two-dimensional space, improving efficiency from O(n^2) to O(n log n).

    Learning Practice

  • 13.4

    Two Dimensional Case

    This section discusses the divide-and-conquer algorithm for finding the closest pair of points in a two-dimensional space, significantly improving efficiency over brute-force methods.

    Learning Practice

  • 13.4.1

    Dividing Points

    This section discusses the divide and conquer algorithm for finding the closest pair of points among a set of two-dimensional points.

    Learning Practice

  • 13.4.2

    Computing Closest Pairs

    This section discusses a divide and conquer algorithm to efficiently find the closest pair of points in a two-dimensional space.

    Learning Practice

  • 13.4.3

    Recursive Call For Q And R

    This section explores the divide-and-conquer algorithm for finding the closest pair of points among a given set of points in two-dimensional space.

    Learning Practice

  • 13.5

    Combining Results

    This section covers the divide and conquer algorithm for finding the closest pair of points in a two-dimensional space, improving upon a naive O(n²) approach to achieve O(n log n) efficiency.

    Learning Practice

  • 13.5.1

    Finding Minimum Distance

    This section explores the divide-and-conquer algorithm to efficiently find the closest pair of points in a set of two-dimensional points.

    Learning Practice

  • 13.5.2

    Candidates For Overall Minimum Distance

    This section explores the divide and conquer algorithm for finding the closest pair of points among a set of points in two dimensions.

    Learning Practice

  • 13.5.3

    Points In The Zone

    This section introduces the divide and conquer algorithm to find the closest pair of points in a set, transitioning from a naive O(n²) method to a more efficient O(n log n) approach.

    Learning Practice

  • 13.6

    Finalizing The Algorithm

    This section introduces a divide and conquer algorithm to efficiently find the closest pair of points in a set of two-dimensional coordinates.

    Learning Practice

  • 13.6.1

    Base Case For Recursion

    The section discusses the base case for recursion within the context of algorithms, particularly focusing on the closest pair of points problem using a divide and conquer approach.

    Learning Practice

  • 13.6.2

    Setting Up S_y

    This section introduces the divide and conquer algorithm for finding the closest pair of points in a two-dimensional space.

    Learning Practice

  • 13.6.3

    Scanning Points

    This section discusses the divide and conquer algorithm for finding the closest pair of points among a set of points in two dimensions, optimizing it from an O(n²) to an O(n log n) complexity.

    Learning Practice

  • 13.6.4

    Return Statement

    This section discusses the divide and conquer algorithm for finding the closest pair of points among a given set of points in two dimensions, improving efficiency from a naive O(n^2) approach to O(n log n).

    Learning Practice

  • 13.7

    Complexity Analysis

    The section introduces the divide and conquer algorithm to determine the closest pair of points in a set, improving efficiency from O(n²) to O(n log n).

    Learning Practice

  • 13.7.1

    Initial Sorting Phase

    This section discusses the divide and conquer algorithm aimed at finding the closest pair of points among a set of two-dimensional points.

    Learning Practice

  • 13.7.2

    Overall Complexity

    This section presents the divide and conquer algorithm used to find the closest pair of points among a given set of points in a two-dimensional space, illustrating the efficiency of the algorithm in contrast to brute-force methods.

    Learning Practice

References

ch38.pdf

Class Notes

Memorization

What we have learnt

  • The closest pair of points ...
  • Sorting the points by their...
  • It is sufficient to conside...

Final Test

Revision Tests