Scala note 4: Huffman Coding

This post is based on Coursera's Scala course homework for week 4 and week 5. The whole problem can be found here.

Some notes:

Case classes: they are regular classes which export their constructor parameters and which provide a recursive decomposition mechanism via pattern matching.
Pattern matching: allows to match on any sort of data with a first-match policy.

List.groupBy: Partitions this traversable collection into a map of traversable collections according to some function.

/: or folderLeft: applying a binary operator to a start value and all elements of this sequence, going left to right. z /: xs is the same as xs foldLeft z

Map.getOrElse(default value): Returns the value associated with a key, or a default value if the key is not contained in the map.

List.sortBy: sorts the sequence according to the Ordering which results from transforming an implicitly given Ordering with a transform function.

List.flatMap: Builds a new collection by applying a function to all elements of this list and flatten the result elements to a list.

List.Map: Builds a new collection by applying a function to all elements of this list.

See here for the difference between flatMap and Map.

Documentation on List can be found here.

object Huffman {

  /**
   * A huffman code is represented by a binary tree.
   *
   * Every `Leaf` node of the tree represents one character of the alphabet that the tree can encode.
   * The weight of a `Leaf` is the frequency of appearance of the character.
   *
   * The branches of the huffman tree, the `Fork` nodes, represent a set containing all the characters
   * present in the leaves below it. The weight of a `Fork` node is the sum of the weights of these
   * leaves.
   */
  abstract class CodeTree
  case class Fork(left: CodeTree, right: CodeTree, chars: List[Char], weight: Int) extends CodeTree
  case class Leaf(char: Char, weight: Int) extends CodeTree



  // Part 1: Basics

  def weight(tree: CodeTree): Int = tree match {
    case Fork(left, right, chars, w) =>  w
    case Leaf(char, weight) => weight
  }
  def chars(tree: CodeTree): List[Char] = tree match {
    case Fork(left, right, chars, weight) => chars
    case Leaf(char, weight) => List(char)
  }

  def makeCodeTree(left: CodeTree, right: CodeTree) =
    Fork(left, right, chars(left) ::: chars(right), weight(left) + weight(right))



  // Part 2: Generating Huffman trees

  /**
   * In this assignment, we are working with lists of characters. This function allows
   * you to easily create a character list from a given string.
   */
  def string2Chars(str: String): List[Char] = str.toList

  /**
   * This function computes for each unique character in the list `chars` the number of
   * times it occurs. For example, the invocation
   *
   *   times(List('a', 'b', 'a'))
   *
   * should return the following (the order of the resulting list is not important):
   *
   *   List(('a', 2), ('b', 1))
   *
   * The type `List[(Char, Int)]` denotes a list of pairs, where each pair consists of a
   * character and an integer. Pairs can be constructed easily using parentheses:
   *
   *   val pair: (Char, Int) = ('c', 1)
   *
   * In order to access the two elements of a pair, you can use the accessors `_1` and `_2`:
   *
   *   val theChar = pair._1
   *   val theInt  = pair._2
   *
   * Another way to deconstruct a pair is using pattern matching:
   *
   *   pair match {
   *     case (theChar, theInt) =>
   *       println("character is: "+ theChar)
   *       println("integer is  : "+ theInt)
   *   }
   */
    def times(chars: List[Char]): List[(Char, Int)] = {
    /*groupBy: Partitions the traversable collection into a map of traversable collections
     * according to some function
     * chars.groupBy(x => x) : return a map of (char, List(char, char....))
     */ 
    chars.groupBy(x => x).map(t => (t._1, t._2.length)).iterator.toList
    }
  /**
   * using map
   */
  def times2(chars: List[Char]): List[(Char, Int)] = {
    def iterate(map: Map[Char, Int], c: Char) = {
      val count = (map get c).getOrElse(0) + 1
      map + ((c, count))
    }
    // /: alternative of chars foldLeft Map[Char, Int]()
    (Map[Char, Int]() /: chars)(iterate).iterator.toList
  }
  


  /**
   * Returns a list of `Leaf` nodes for a given frequency table `freqs`.
   *
   * The returned list should be ordered by ascending weights (i.e. the
   * head of the list should have the smallest weight), where the weight
   * of a leaf is the frequency of the character.
   */
  def makeOrderedLeafList(freqs: List[(Char, Int)]): List[Leaf] = 
    freqs.sortBy(t => (t._2, t._1)).map(leaf => Leaf(leaf._1,leaf._2 ))
    

  /**
   * Checks whether the list `trees` contains only one single code tree.
   */
  def singleton(trees: List[CodeTree]): Boolean = 
    if(trees.length == 1) true else false

  /**
   * The parameter `trees` of this function is a list of code trees ordered
   * by ascending weights.
   *
   * This function takes the first two elements of the list `trees` and combines
   * them into a single `Fork` node. This node is then added back into the
   * remaining elements of `trees` at a position such that the ordering by weights
   * is preserved.
   *
   * If `trees` is a list of less than two elements, that list should be returned
   * unchanged.
   */
  def combine(trees: List[CodeTree]): List[CodeTree] = trees match {
    case left :: right :: rest => (makeCodeTree(left, right) :: rest)
      .sortBy(t => weight(t))
    case _ => trees
    }

  /**
   * This function will be called in the following way:
   *
   *   until(singleton, combine)(trees)
   *
   * where `trees` is of type `List[CodeTree]`, `singleton` and `combine` refer to
   * the two functions defined above.
   *
   * In such an invocation, `until` should call the two functions until the list of
   * code trees contains only one single tree, and then return that singleton list.
   *
   * Hint: before writing the implementation,
   *  - start by defining the parameter types such that the above example invocation
   *    is valid. The parameter types of `until` should match the argument types of
   *    the example invocation. Also define the return type of the `until` function.
   *  - try to find sensible parameter names for `xxx`, `yyy` and `zzz`.
   */
  def until(singleton: List[CodeTree] => Boolean, combine: List[CodeTree] => List[CodeTree])
  (trees: List[CodeTree]): List[CodeTree] = {
    if(singleton(trees)) trees
    else until(singleton, combine)(combine(trees))
  }

  /**
   * This function creates a code tree which is optimal to encode the text `chars`.
   *
   * The parameter `chars` is an arbitrary text. This function extracts the character
   * frequencies from that text and creates a code tree based on them.
   */
  def createCodeTree(chars: List[Char]): CodeTree = 
    until(singleton, combine)(makeOrderedLeafList(times(chars))).head
 



  // Part 3: Decoding

  type Bit = Int

  /**
   * This function decodes the bit sequence `bits` using the code tree `tree` and returns
   * the resulting list of characters.
   */
  def decode(tree: CodeTree, bits: List[Bit]): List[Char] = tree match {
    case Leaf (c, _) => if (bits.isEmpty) List(c) else  c :: decode(tree, bits)
    case Fork(left, right, _, _) => if (bits.head == 0) decode(left, bits.tail)
                                                     else decode(right, bits.tail)
  }

  /**
   * A Huffman coding tree for the French language.
   * Generated from the data given at
   *   http://fr.wikipedia.org/wiki/Fr%C3%A9quence_d%27apparition_des_lettres_en_fran%C3%A7ais
   */
  val frenchCode: CodeTree = Fork(Fork(Fork(Leaf('s',121895),Fork(Leaf('d',56269),Fork(Fork(Fork(Leaf('x',5928),Leaf('j',8351),List('x','j'),14279),Leaf('f',16351),List('x','j','f'),30630),Fork(Fork(Fork(Fork(Leaf('z',2093),Fork(Leaf('k',745),Leaf('w',1747),List('k','w'),2492),List('z','k','w'),4585),Leaf('y',4725),List('z','k','w','y'),9310),Leaf('h',11298),List('z','k','w','y','h'),20608),Leaf('q',20889),List('z','k','w','y','h','q'),41497),List('x','j','f','z','k','w','y','h','q'),72127),List('d','x','j','f','z','k','w','y','h','q'),128396),List('s','d','x','j','f','z','k','w','y','h','q'),250291),Fork(Fork(Leaf('o',82762),Leaf('l',83668),List('o','l'),166430),Fork(Fork(Leaf('m',45521),Leaf('p',46335),List('m','p'),91856),Leaf('u',96785),List('m','p','u'),188641),List('o','l','m','p','u'),355071),List('s','d','x','j','f','z','k','w','y','h','q','o','l','m','p','u'),605362),Fork(Fork(Fork(Leaf('r',100500),Fork(Leaf('c',50003),Fork(Leaf('v',24975),Fork(Leaf('g',13288),Leaf('b',13822),List('g','b'),27110),List('v','g','b'),52085),List('c','v','g','b'),102088),List('r','c','v','g','b'),202588),Fork(Leaf('n',108812),Leaf('t',111103),List('n','t'),219915),List('r','c','v','g','b','n','t'),422503),Fork(Leaf('e',225947),Fork(Leaf('i',115465),Leaf('a',117110),List('i','a'),232575),List('e','i','a'),458522),List('r','c','v','g','b','n','t','e','i','a'),881025),List('s','d','x','j','f','z','k','w','y','h','q','o','l','m','p','u','r','c','v','g','b','n','t','e','i','a'),1486387)

  /**
   * What does the secret message say? Can you decode it?
   * For the decoding use the `frenchCode' Huffman tree defined above.
   */
  val secret: List[Bit] = List(0,0,1,1,1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,1,1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,1)

  /**
   * Write a function that returns the decoded secret
   */
  def decodedSecret: List[Char] = 
    decode(frenchCode, secret)



  // Part 4a: Encoding using Huffman tree

  /**
   * This function encodes `text` using the code tree `tree`
   * into a sequence of bits.
   */
  def encode(tree: CodeTree)(text: List[Char]): List[Bit] = {
    def encodeChar(tree: CodeTree)(char: Char): List[Bit] = tree match {
      case Leaf(_, _) => List()
      case Fork(left, right, _, _) => if (chars(left).contains(text.head)) 0 :: encodeChar(left)(char)
                                                       else 1 :: encodeChar(right)(char)
    }
    text flatMap(encodeChar(tree))
  }


  // Part 4b: Encoding using code table

  type CodeTable = List[(Char, List[Bit])]

  /**
   * This function returns the bit sequence that represents the character `char` in
   * the code table `table`.
   */
  def codeBits(table: CodeTable)(char: Char): List[Bit] = 
    table(table.indexWhere(x => x._1 == char))._2

  /**
   * Given a code tree, create a code table which contains, for every character in the
   * code tree, the sequence of bits representing that character.
   *
   * Hint: think of a recursive solution: every sub-tree of the code tree `tree` is itself
   * a valid code tree that can be represented as a code table. Using the code tables of the
   * sub-trees, think of how to build the code table for the entire tree.
   */
  def convert(tree: CodeTree): CodeTable = tree match {
    case Leaf(char, _) => List((char, List()))
    case Fork(left, right, chars, _) => mergeCodeTables(convert(left), convert(right))
  }

  /**
   * This function takes two code tables and merges them into one. Depending on how you
   * use it in the `convert` method above, this merge method might also do some transformations
   * on the two parameter code tables.
   */
  def mergeCodeTables(a: CodeTable, b: CodeTable): CodeTable = 
     a.map(code => (code._1, 0:: code._2)) ::: b.map(code => (code._1, 1 :: code._2))

  /**
   * This function encodes `text` according to the code tree `tree`.
   *
   * To speed up the encoding process, it first converts the code tree to a code table
   * and then uses it to perform the actual encoding.
   */
  def quickEncode(tree: CodeTree)(text: List[Char]): List[Bit] = 
    text flatMap (codeBits(convert(tree)))
}

Check code on git: Hoffman.scala.

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