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 \(\sqrt{(2-1)^2+(4-2)^2} = \sqrt{5} \).

Test cases in Scala

assert(findClosestPair(Set.empty).isEmpty)
assert(findClosestPair(Set(Coordinate(x = 2, y = 2))).isEmpty)
assert(
  findClosestPair(Set(Coordinate(x = 2, y = 2), Coordinate(x = 2, y = 3)))
    .contains(Coordinate(x = 2, y = 2) -> Coordinate(x = 2, y = 3))
)
assert(
  findClosestPair(
    Set(
      Coordinate(x = 1, y = 2),
      Coordinate(x = 2, y = 4),
      Coordinate(x = 5, y = 6),
      Coordinate(x = -2, y = -2)
    )
  ).map(distance)
    .contains(
      distance(
        Coordinate(x = 1, y = 2) -> Coordinate(x = 2, y = 4)
      )
    )
)
assert(
  findClosestPair(
    Set(Coordinate(0, 500), Coordinate(261, -1), Coordinate(0, -158))
  ).map(distance)
    .contains(
      distance(
        Coordinate(0, -158) -> Coordinate(261, -1)
      )
    )
)
assert(
  findClosestPair(
    Set(
      Coordinate(500, 0),
      Coordinate(28, 0),
      Coordinate(1, -167),
      Coordinate(500, 1)
    )
  ).map(distance)
    .contains(
      distance(
        Coordinate(500, 0) -> Coordinate(500, 1)
      )
    )
)
assert(
  findClosestPairBruteForce(
    Set(
      Coordinate(500, 0),
      Coordinate(28, 0),
      Coordinate(1, -167),
      Coordinate(500, 1)
    )
  ).map(distance)
    .contains(
      distance(
        Coordinate(500, 0) -> Coordinate(500, 1)
      )
    )
)
assert(
  findClosestPairBruteForce(
    Set(Coordinate(0, -1), Coordinate(0, 0), Coordinate(1, -1))
  ).map(distance)
    .contains(
      distance(
        Coordinate(500, 0) -> Coordinate(500, 1)
      )
    )
)

Algorithm in Scala

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

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.

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")
   

   ▶ Run code in the IDE

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-")
   

   ▶ Run code in the IDE

   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))
   

   ▶ Run code in the IDE

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)
   

   ▶ Run code in the IDE

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))
   

   ▶ Run code in the IDE

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)
   

   ▶ Run code in the IDE

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")
   

   ▶ Run code in the IDE

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)
   

   ▶ Run code in the IDE

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)
    

    ▶ Run code in the IDE

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")
    

    ▶ Run code in the IDE

12. View

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