Closest pair of coordinates in a 2D plane

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

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.

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Scala Concepts & Hints

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")

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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('-')
}

It is also frequently used in combination with Tail Recursion.

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))

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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').

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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)

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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))

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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)

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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")

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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)

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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)

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View

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

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Algorithm in Scala

90 lines of Scala (version 2.13).

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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)
      )
    )
)
def findClosestPair(items: Set[Coordinate]): Option[(Coordinate, Coordinate)] =
  ???

def distance(vs: (Coordinate, Coordinate)): Double = ???

def findClosestPairBruteForce(
    set: Set[Coordinate]
): Option[(Coordinate, Coordinate)] = ???