Scala algorithm: Closest pair of coordinates in a 2D plane

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Algorithm goal

From a set of coordinates, find the pair that are the closest to each other. (Looking forMergeSortStackSafe?)

For example, in a set of \([(1,2), (2,4), (5,6), (-2, -2)]\), the closest pair is \([(1,2), (2,4)]\), distance \(\sqrt{(2-1)^2+(4-2)^2} = \sqrt{5} \).

an illustration demonstrating 2D closest pairs

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Algorithm in Scala

85 lines of Scala (compatible versions 2.13 & 3.0).

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Explanation

The most straightforward implementation of this algorithm would be to compare every point to another, but that is not efficient, especially for many data points.

For many algorithmic goals, it is worth to have the two questions in your toolbox: (this is © from www.scala-algorithms.com)

  • If I sort this data, does it change how I can read it? (especially useful if you know other parts of the algorithm are going to be at least as slow as \(O(n\log{n})\).
  • If I divide the problem into sub-problems, can I get something useful out of that? (divide-and-conquer; for many problems this is not really possible)

The curious thing here is that we can indeed get something useful out of these two - and follow a similar approach to the related MergeSort and CountInversions problems: When we have sorted the coordinates, say, by their x-axis, we can naturally ask the question: Is the closest pair of coordinates on the left side, the right side, or does it span the two sides?, which has an interesting implication: when we know the closest pair of coordinates on either left or the right side, the cross-side pair cannot be farther apart than either of the sides - this gives us a restriction.

Full explanation is available for subscribers Scala algorithms logo, maze part, which looks quirky

Scala concepts & Hints

  1. Collect

    'collect' allows you to use Pattern Matching, to filter and map items.

    assert("Hello World".collect {
      case character if Character.isUpperCase(character) => character.toLower
    } == "hw")
    
  2. Def Inside Def

    A great aspect of Scala is being able to declare functions inside functions, making it possible to reduce repetition.

    def exampleDef(input: String): String = {
      def surroundInputWith(char: Char): String = s"$char$input$char"
      surroundInputWith('-')
    }
    
    assert(exampleDef("test") == "-test-")
    

    It is also frequently used in combination with Tail Recursion.

  3. Drop, Take, dropRight, takeRight

    Scala's `drop` and `take` methods typically remove or select `n` items from a collection.

    assert(List(1, 2, 3).drop(2) == List(3))
    
    assert(List(1, 2, 3).take(2) == List(1, 2))
    
    assert(List(1, 2, 3).dropRight(2) == List(1))
    
    assert(List(1, 2, 3).takeRight(2) == List(2, 3))
    
    assert((1 to 5).take(2) == (1 to 2))
    
  4. Lazy List

    The 'LazyList' type (previously known as 'Stream' in Scala) is used to describe a potentially infinite list that evaluates only when necessary ('lazily').

  5. Option Type

    The 'Option' type is used to describe a computation that either has a result or does not. In Scala, you can 'chain' Option processing, combine with lists and other data structures. For example, you can also turn a pattern-match into a function that return an Option, and vice-versa!

    assert(Option(1).flatMap(x => Option(x + 2)) == Option(3))
    
    assert(Option(1).flatMap(x => None) == None)
    
  6. Ordering

    In Scala, the 'Ordering' type is a 'type class' that contains methods to determine an ordering of specific types.

    assert(List(3, 2, 1).sorted == List(1, 2, 3))
    
    assert(List(3, 2, 1).sorted(Ordering[Int].reverse) == List(3, 2, 1))
    
    assert(Ordering[Int].lt(1, 2))
    
    assert(!Ordering[Int].lt(2, 1))
    
  7. Partial Function

    A Partial Function in Scala is similar to function type `A => Option[B]` (Option Type).

    def getNaming(num: Int): Option[String] =
      PartialFunction.condOpt(num) { case 1 => "One" }
    
    assert(getNaming(1) == Some("One"))
    
    assert(getNaming(2) == None)
    
  8. Pattern Matching

    Pattern matching in Scala lets you quickly identify what you are looking for in a data, and also extract it.

    assert("Hello World".collect {
      case character if Character.isUpperCase(character) => character.toLower
    } == "hw")
    
  9. Stack Safety

    Stack safety is present where a function cannot crash due to overflowing the limit of number of recursive calls.

    This function will work for n = 5, but will not work for n = 2000 (crash with java.lang.StackOverflowError) - however there is a way to fix it :-)

    In Scala Algorithms, we try to write the algorithms in a stack-safe way, where possible, so that when you use the algorithms, they will not crash on large inputs. However, stack-safe implementations are often more complex, and in some cases, overly complex, for the task at hand.

    def sum(from: Int, until: Int): Int =
      if (from == until) until else from + sum(from + 1, until)
    
    def thisWillSucceed: Int = sum(1, 5)
    
    def thisWillFail: Int = sum(1, 300)
    
  10. Tail Recursion

    In Scala, tail recursion enables you to rewrite a mutable structure such as a while-loop, into an immutable algorithm.

    def fibonacci(n: Int): Int = {
      @scala.annotation.tailrec
      def go(i: Int, previous: Int, beforePrevious: Int): Int =
        if (i >= n) previous else go(i + 1, previous + beforePrevious, previous)
    
      go(i = 1, previous = 1, beforePrevious = 0)
    }
    
    assert(fibonacci(8) == 21)
    
  11. Type Class

    Type classes are one of Scala's most important super-powers: they enable you to add new behaviour to existing classes, without modifying those classes. In many languages, to add a behaviour to a class, you would typically extend it with an interface, and then implement methods against this interface.This, however, does not scale: especially when you have older libraries, you would be forced to make them depend on a new interface, and have to re-build everything.

    Type classes are used heavily in Apple's SwiftUI as "extensions" to enable powerful abstraction capabilities.

    Type classes enable you to do things like this:

    import Ordering.Implicits._
    
    type CommonType = (Int, String, Option[String])
    
    val a: CommonType = (1, "X", None)
    
    val b: CommonType = (2, "A", Some("B"))
    
    assert(a < b, "We can order tuples using Scala-provided type classes")
    
  12. View

    The .view syntax creates a structure that mirrors another structure, until "forced" by an eager operation like .toList, .foreach, .forall, .count.


Scala Algorithms: The most comprehensive library of algorithms in standard pure-functional Scala

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Scala is used at many places, such as AirBnB, Apple, Bank of America, BBC, Barclays, Capital One, Citibank, Coursera, eBay, JP Morgan, LinkedIn, Morgan Stanley, Netflix, Singapore Exchange, Twitter.

Study our 92 Scala Algorithms: 6 fully free, 74 published & 18 upcoming

Fully unit-tested, with explanations and relevant concepts; new algorithms published about once a week.

  1. Compute the length of longest valid parentheses
  2. Check a binary tree is balanced
  3. Make a queue using stacks (Lists in Scala)
  4. Find height of binary tree
  5. Single-elimination tournament tree
  6. Quick Sort sorting algorithm in pure immutable Scala
  7. Find minimum missing positive number in a sequence
  8. Least-recently used cache (LRU)
  9. Count pairs of a given expected sum
  10. Compute a Roman numeral for an Integer, and vice-versa
  11. Compute keypad possibilities
  12. Matching parentheses algorithm with foldLeft and a state machine
  13. Traverse a tree Breadth-First, immutably
  14. Read a matrix as a spiral
  15. Remove duplicates from a sorted list (state machine)
  16. Merge Sort: stack-safe, tail-recursive, in pure immutable Scala, N-way
  17. Longest increasing sub-sequence length
  18. Reverse first n elements of a queue
  19. Binary search a generic Array
  20. Merge Sort: in pure immutable Scala
  21. Make a queue using Maps
  22. Is an Array a permutation?
  23. Count number of contiguous countries by colors
  24. Add numbers without using addition (plus sign)
  25. Tic Tac Toe MinMax solve
  26. Run-length encoding (RLE) Encoder
  27. Print Alphabet Diamond
  28. Balanced parentheses algorithm with tail-call recursion optimisation
  29. Reverse a String's words efficiently
  30. Count number of changes (manipulations) needed to make an anagram with foldLeft and a MultiSet
  31. Count passing cars
  32. Establish execution order from dependencies
  33. Counting inversions of a sequence (array) using a Merge Sort
  34. Longest common prefix of strings
  35. Check if an array is a palindrome
  36. Check a directed graph has a routing between two nodes (depth-first search)
  37. Compute nth row of Pascal's triangle
  38. Run-length encoding (RLE) Decoder
  39. Check if a number is a palindrome
  40. In a range of numbers, count the numbers divisible by a specific integer
  41. Find the index of a substring ('indexOf')
  42. Reshape a matrix
  43. Compute modulo of an exponent without exponentiation
  44. Closest pair of coordinates in a 2D plane
  45. Find the contiguous slice with the minimum average
  46. Compute maximum sum of subarray (Kadane's algorithm)
  47. Pure-functional double linked list
  48. Binary search in a rotated sorted array
  49. Check if a directed graph has cycles
  50. Rotate Array right in pure-functional Scala - using an unusual immutable efficient approach
  51. Check a binary tree is a search tree
  52. Length of the longest common substring
  53. Tic Tac Toe board check
  54. Find an unpaired number in an array
  55. Check if a String is a palindrome
  56. Count binary gap size of a number using tail recursion
  57. Remove duplicates from a sorted list (Sliding)
  58. Monitor success rate of a process that may fail
  59. Find sub-array with the maximum sum
  60. Find the minimum absolute difference of two partitions
  61. Find maximum potential profit from an array of stock price
  62. Fibonacci in purely functional immutable Scala
  63. Fizz Buzz in purely functional immutable Scala
  64. Find combinations adding up to N (non-unique)
  65. Make a binary search tree (Red-Black tree)
  66. Remove duplicates from an unsorted List
  67. Find combinations adding up to N (unique)
  68. Count factors/divisors of an integer
  69. Compute single-digit sum of digits
  70. Traverse a tree Depth-First
  71. Reverse bits of an integer
  72. Find k closest elements to a value in a sorted Array
  73. QuickSelect Selection Algorithm (kth smallest item/order statistic)
  74. Rotate a matrix by 90 degrees clockwise

Explore the 21 most useful Scala concepts

To save you going through various tutorials, we cherry-picked the most useful Scala concepts in a consistent form.

  1. Class Inside Class
  2. Class Inside Def
  3. Collect
  4. Def Inside Def
  5. Drop, Take, dropRight, takeRight
  6. foldLeft and foldRight
  7. For-comprehension
  8. Lazy List
  9. Option Type
  10. Ordering
  11. Partial Function
  12. Pattern Matching
  13. Range
  14. scanLeft and scanRight
  15. Sliding / Sliding Window
  16. Stack Safety
  17. State machine
  18. Tail Recursion
  19. Type Class
  20. View
  21. Zip

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