Scala algorithm: Check a binary tree is a search tree

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Algorithm goal

A binary tree is a tree that can only have at most 2 children. A search tree is a tree in which all the children to the left of the node are < than the value of this node, and all items of the right are >=, and this property also applies to their children nodes as well.

This property enables very fast lookups (hence the name a search tree) due to the ordering guarantees.

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Algorithm in Scala

29 lines of Scala (compatible versions 2.13 & 3.0), showing how concise Scala can be!

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Explanation

A common mistake in implementing this algorithm is to think that comparing a node's value with the value of its direct child is enough.

In fact, one has to ensure that the property applies to all the children. While it is possible to verify this by traversing the tree top-down, in fact a better approach is to perform an in-order traversal of the tree and check that it is sorted. In this algorithm, we do exactly that using LazyLists and implicit classes provided by Scala. (this is © from www.scala-algorithms.com)

Scala concepts & Hints

  1. 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').

  2. 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)
    
  3. 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))
    
  4. Sliding / Sliding Window

    Get fixed-length sliding sub-sequences (sliding windows) from another sequence

  5. 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")
    

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

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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. Sliding Window Rate Limiter
  54. Tic Tac Toe board check
  55. Find an unpaired number in an array
  56. Check if a String is a palindrome
  57. Count binary gap size of a number using tail recursion
  58. Remove duplicates from a sorted list (Sliding)
  59. Monitor success rate of a process that may fail
  60. Find sub-array with the maximum sum
  61. Find the minimum absolute difference of two partitions
  62. Find maximum potential profit from an array of stock price
  63. Fibonacci in purely functional immutable Scala
  64. Fizz Buzz in purely functional immutable Scala
  65. Find combinations adding up to N (non-unique)
  66. Make a binary search tree (Red-Black tree)
  67. Remove duplicates from an unsorted List
  68. Mars Rover
  69. Find combinations adding up to N (unique)
  70. Find indices of tuples that sum to a target (Two Sum)
  71. Count factors/divisors of an integer
  72. Compute single-digit sum of digits
  73. Traverse a tree Depth-First
  74. Reverse bits of an integer
  75. Find k closest elements to a value in a sorted Array
  76. QuickSelect Selection Algorithm (kth smallest item/order statistic)
  77. 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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