3.2.1 Algebraic data types

An algebraic data type is a type formed by combining other types, that is, it is a composite data type. The data type is created by an algebra of operations of two primary kinds:

• a sum operation that constructs values to have one variant among several possible variants. These sum types are also called tagged, disjoint union, or variant types.

  The combining operation is the alternation operator, which denotes the choice of one but not both between two alternatives.

• a product operation that combines several values (i.e., fields) together to construct a single value. These are tuple and record types.

  The combining operation is the Cartesian product from set theory.

We can combine sums and products recursively into arbitrarily large structures.

An enumerated type is a sum type in which the constructors take no arguments. Each constructor corresponds to a single value.

3.2.2 ADT confusion

Although sometimes the acronym ADT is used for both, an algebraic data type is a different concept from an abstract data type.

• We specify an algebraic data type with its syntax (i.e., structure)—with rules on how to compose and decompose them.

• We specify an abstract data type with its semantics (i.e., meaning)—with rules about how the operations behave in relation to one another.

  The modules we build with abstract interfaces, contracts, and data abstraction, such as the Rational Arithmetic modules from the book Exploring Languages with Interpreters and Functional Programming [10, Ch. 7], are abstract data types.

Perhaps to add to the confusion, in functional programming we sometimes use an algebraic data type to help define an abstract data type.

3.2.3 Using a Scala trait

A list consists of a sequence of values, all of which have the same type. It is a hierarchical data structure. It is either empty or it is a pair consisting of a head element and a tail that is itself a list of elements.

We define List as an abstract type using a Scala trait. (We could also use an abstract class instead of a trait.) We define the constructors for the algebraic data type using the Scala case class and case object features.

    sealed trait List[+A]
    case object Nil extends List[Nothing] 
    case class Cons[+A](head: A, tail: List[A]) extends List[A]

Thus List is a sum type with two alternatives:

• Nil constructs the singleton case object that represents the empty list.

• Cons(h,t) constructs a new list from an element h, called the head, and a list t, called the tail.

Cons itself is a product (tuple) type with two fields, one of which is itself a List.

The sealed keyword tells the Scala compiler that all alternative cases (i.e., subtypes) are declared in the current source file. No new cases can be added elsewhere. This enables the compiler to generate safe and efficient code for pattern matching.

As we have seen previously, for each case class and case object, the Scala compiler generates:

• a constructor function (e.g., Cons)

• accessor functions (methods) for each field (e.g., head and tail on Cons)

• new definitions for equals, hashcode, and toString

In addition, the case object construct generates a singleton object—a new type with exactly one instance.

Programs can use the constructors to build instances and use the pattern matching to recognize the structure of instances and decompose them for processing.

List is a polymorphic type. What does polymorphic mean? We examine that in Section 3.2.4.

    data List a = Nil | Cons a (List a)
                  deriving (Show, Eq)

3.2.4 Polymorphism

Polymorphism refers to the property of having “many shapes”. In programming languages, we are primarily interested in how polymorphic function names (or operator symbols) are associated with implementations of the functions (or operations).

Scala is a hybrid, object-functional language. Its type system supports all three types of polymorphism discussed in the notes on type systems concepts [9,10:5.2].

• subtyping by extending classes and traits

• parametric polymorphism by using generic type parameters,

• overloading through both the Java-like mechanisms and Haskell-like “type classes”

Scala’s type class pattern builds on the languages’s implicit classes and conversions. A type class enables a programmer to enrich an existing class with an extended interface and new methods without redefining the class or subclassing it.

For example, Scala extends the Java String class (which is final and thus cannot be subclassed) with new features from the RichString wrapper class. The Scala implicit mechanisms associate the two classes “behind the scene”. We defer further discussion of implicits until later in the semester.

Note: The type class feature arose from the language Haskell. Similar capabilities are called extension methods in C# and protocols in Clojure and Elixir.

The List data type defined above is polymorphic; it exhibits both subtyping and parametric polymorphism. Nil and Cons are subtypes of List. The generic type parameter A denotes the type of the elements that occur in the list. For example, List[Double] denotes a list of double-precision floating point numbers.

What does the + annotation mean in the definition List[+A]?

3.2.5 Variance

The presence of both subtyping and parametric polymorphism leads to the question of how these features interact—that is, the concept of variance.

3.2.6 Defining functions in companion object

The companion object for a trait or class is a singleton object with the same name as the trait or class. The companion object for the List trait is a convenient place to define functions for manipulating the lists.

Because List is a Scala algebraic data type (implemented with case classes), we can use pattern matching in our function definitions. Pattern matching helps enable the form of the algorithm to match the form of the data structure. Or, in terms that Chiusano and Bjarnason use, it helps in following types to implementations [2].

Note: Other writers call this design approach type-driven development [1] or type-first development [13].

This is considered elegant. It is also pragmatic. The structure of the data often suggests the algorithm needed for a task.

In general, lists have two cases that must be handled: the empty list (represented by Nil) and the nonempty list (represented by Cons). The first yields a base leg of a recursive algorithm; the second yields a recursive leg.

Breaking a definition for a list-processing function into these two cases is usually a good place to begin. We must ensure the recursion terminates—that each successive recursive call gets closer to the base case.

3.2.7 Function to sum a list

Consider a function sum to add together all the elements in a list of integers. That is, if the list is $v_{1}, v_{2}, v_{3}, \cdots, v_{n}$, then the sum of the list is the value resulting from inserting the addition operator between consecutive elements of the list:

$v_{1} + v_{2} + v_{3} + \cdots + v_{n}$

Because addition is an associative operation, the additions can be computed in any order. That is, for any integers $x$, $y$, and $z$:

$(x + y) + z = x + (y + z)$

We can use the form of the data to guide the form of the algorithm—or follow the type to the implementation of the function.

What is the sum of an empty list?

Because there are no numbers to add, then, intuitively, zero seems to be the proper value for the sum.

In general, if some binary operation is inserted between the elements of a list, then the result for an empty list is the identity element for the operation. Zero is the identity element for addition because, for all integers $x$:

$0 + x = x = x + 0$

Now, how can we compute the sum of a nonempty list?

Because a nonempty list has at least one element, we can remove one element and add it to the sum of the rest of the list. Note that the “rest of the list” is a simpler (i.e., shorter) list than the original list. This suggests a recursive definition.

The fact that we define lists recursively as a Cons of a head element with a tail list suggests that we structure the algorithm around the structure of the beginning of the list.

Bringing together the two cases above, we can define the function sum in Scala using pattern matching as follows:

    def sum(ints: List[Int]): Int = ints match {
        case Nil        => 0 
        case Cons(x,xs) => x + sum(xs)
    }

The length of a non-nil argument decreases by one for each successive recursive application. Thus sum will eventually be applied to a Nil argument and terminate.

For a list consisting of elements 2, 4, 6, and 8, that is,

    Cons(2,Cons(4,Cons(6,Cons(8,Nil))))

function sum computes:

    2 + (4 + (6 + (8 + 0)))

Function sum is backward linear recursive; its time and space complexity are both O(n), where n is the length of the input list.

We could, of course, redefine this to use a tail-recursive auxiliary function. With tail call optimization, the recursion could be converted into a loop. It would still be order O(n) in time complexity (but with a smaller constant factor) and O(1) space.

3.2.8 Function to multiply a list

Now consider a function product to multiply together a list of floating point numbers. The product of an empty list is 1 (which is the identity element for multiplication). The product of a nonempty list is the head of the list multiplied by the product of the tail of the list, except that, if a 0 occurs anywhere in the list, the product of the list is 0. We can thus define product with two bases cases and one recursive case, as follows:

    def product(ds: List[Double]): Double = ds match {
        case Nil          => 1.0
        case Cons(0.0, _) => 0.0
        case Cons(x,xs)   => x * product(xs)
    }

Note: 0 is the zero element for the multiplication operation on real numbers. That is, for all real numbers $x$:

$0 * x = x * 0 = 0$

For a list consisting of elements 2.0, 4.0, 6.0, and 8.0, that is,

    Cons(2.0,Cons(4.0,Cons(6.0,Cons(8.0,Nil))))

function product computes:

    2.0 * (4.0 * (6.0 * (8.0 * 1.0))) 

For a list consisting of elements 2.0, 0.0, 6.0, and 8.0, function product “short circuits” the computation as:

    2.0 * 0.0

Like sum, function product is backward linear recursive; it has a worst-case time complexity of O(n), where n is the length of the input list. It terminates because the argument of each successive recursive call is one element shorter than the previous call, approaching one of the base cases.

3.2.9 Function to remove adjacent duplicates

Consider the problem of removing adjacent duplicate elements from a list. That is, we want to replace a group of adjacent elements having the same value by a single occurrence of that value.

As with the above functions, we let the form of the data guide the form of the algorithm, following the type to the implementation.

The notion of adjacency is only meaningful when there are two or more of something. Thus, in approaching this problem, there seem to be three cases to consider:

• The argument is a list whose first two elements are duplicates; in which case one of them should be removed from the result.

• The argument is a list whose first two elements are not duplicates; in which case both elements are needed in the result.

• The argument is a list with fewer than two elements; in which case the remaining element, if any, is needed in the result.

Of course, we must be careful that sequences of more than two duplicates are handled properly.

Our algorithm thus can examine the first two elements of the list. If they are equal, then the first is discarded and the process is repeated recursively on the list remaining. If they are not equal, then the first element is retained in the result and the process is repeated on the list remaining. In either case the remaining list is one element shorter than the original list. When the list has fewer than two elements, it is simply returned as the result.

In Scala, we can define function remdups as follows:

    def remdups[A](ls: List[A]): List[A] = ls match {
        case Cons(x, Cons(y,ys)) =>
            if (x == y)
                remdups(Cons(y,ys))         // duplicate
            else
                Cons(x,remdups(Cons(y,ys))) // non-duplicate
        case _                   => ls
    }

Function remdups puts the base case last in the pattern match to take advantage of the wildcard match using _. This needs to match either Nil and Cons(_,Nil).

The function also depends upon the ability to compare any two elements of the list for equality. Because equals is builtin operation on all types in Scala, we can define this function polymorphically without constraints on the type variable A.

Like the previous functions, remdups is backward linear recursive; it takes a number of steps that is proportional to the length of the list. This function has a recursive call on both the duplicate and non-duplicate legs. Each of these recursive calls uses a list that is shorter than the previous call, thus moving closer to the base case.

3.2.10 Variadic function apply

We can also add a function apply to the companion object List.

    def apply[A](as: A*): List[A] = 
        if (as.isEmpty) 
            Nil 
        else 
            Cons(as.head, apply(as.tail: _*)) 

Scala treats an apply method in an object specially. We can invoke the apply method using a postfix () operator. Given a singleton object X with an apply method, the Scala complier translates the notation X(p) into the method call X.apply(p).

In the List data type, function apply is a variadic function. It accepts zero or more arguments of type A as denoted by the type annotation A* in the parameter list. Scala collects these arguments into a Seq (sequence) data type for processing within the function. The special syntax _* reverses this and passes a sequence to another function as variadic parameters. Builtin Scala data structures such as lists, queues, and vectors implement Seq. It provides methods such as the isEmpty, head, and tail methods used in apply.

It is common to define a variadic apply methods for algebraic data types. This method enables us to create instances of the data type conveniently. For example, List(1,2,3) creates a three-element list of integers with 1 at the head.

3.3.1 Function to take tail of list

Consider a function that takes a List and returns its tail List. (This is different from the tail accessor method on Cons.)

If the List is a Cons, then the function can return the tail element of the cell. What should it do if the list is a Nil?

There are several possibilities:

• return Nil

• throw an exception (with perhaps a custom error string)

• leave the function undefined in this case (which would result with a standard pattern match exception)

Generally speaking, the first choice seems misleading. It seems illogical for an empty list to have a tail. And consider a typical usage of the function. It is normally an error for a program to attempt to get the tail of an empty list. A program can efficiently check whether a list is empty or not. So, in this case, it is probably better to take the second or third approach.

We choose to implement tailList so that it explicitly throws an exception. It can be defined in the companion object for List as follows:

    def tailList[A](ls: List[A]): List[A] = ls match {
        case Nil        => sys.error("tail of empty list")
        case Cons(_,xs) => xs
    }

Above, the value of the head field of the Cons pattern is irrelevant in the computation on the right-hand side. There is no need to introduce a new variable for that value, so we use the wildcard variable _ to indicate that the value is not needed.

Function tailList is O(1) in time complexity. It does not need to copy the list. It is sufficient for it to just return a reference to the tail of the original immutable list. This return value shares the data with its input argument.

3.3.2 Function to drop from beginning of list

We can generalize tailList to a function drop that removes the first n elements of a list, as follows:

    def drop[A](ls: List[A], n: Int): List[A] =
        if (n <= 0) ls
        else ls match {
            case Nil        => Nil
            case Cons(_,xs) => drop(xs, n-1)
    }

The drop function terminates when either the list argument is Nil or the integer argument 0 or negative. The function eventually terminates because each recursive call both shortens the list and decrements the integer.

This function takes a different approach to the empty list issue than tailList does. Although it seems illogical to take the tailList of an empty list, dropping the first element from an empty list seems subtly different. Given that we often use drop in cases where the length of the input list is unknown, dropping the first element of an empty list does not necessarily indicate a program error.

Suppose drop throws an exception when called with an empty list. To avoid this situation, the program might need to determine the length of the list argument. This is inefficient, usually requiring a traversal of the entire list to count the elements.

3.3.3 Function to append lists

Consider the definition of an append (list concatenation) function. We must define the append function in terms of the constructors Nil and Cons, already defined list functions, and recursive applications of itself.

As with previous functions, we follow the type to the implementation—let the form of the data guide the form of the algorithm.

The Cons constructor takes an element as its left operand and a list as its right operand and returns a new list with the left operand as the head and the right operand as the tail.

Similarly, append must take a list as its left operand and a list as its right operand and return a new list with the left operand as the initial segment and the right operand as the final segment.

Given the definition of Cons, it seems reasonable that an algorithm for append must consider the structure of its left operand. Thus we consider the cases for nil and non-nil left operands.

• If the left operand is Nil, then the function can just return the right operand.

• If the left operand is a Cons (that is, non-nil), then the result consists of the left operand’s head followed by the append of the left operand’s tail to the right operand.

In following the type to the implementation, we use the form of the left operand in a pattern match. We define append as follows:

    def append[A](ls: List[A], rs: List[A]): List[A] = ls match {
        case Nil        => rs
        case Cons(x,xs) => Cons(x, append(xs, rs))
    }

For the recursive application of append, the length of the left operand decreases by one. Hence the left operand of an append application eventually becomes Nil, allowing the evaluation to terminate.

The number of steps needed to evaluate append(as,bs) is proportional to the length of as, the left operand. That is, it is O(n), where n is the length of list as.

Moreover, append(as,bs) only needs to copy the list as. The list bs is shared between the second operand and the result. If we did a similar function to append two (mutable) arrays, we would need to copy both input arrays to create the output array. Thus, in this case, a linked list is more efficient than arrays!

The append operation has a number of useful mathematical (algebraic) properties, for example, associativity and an identity element.

Associativity—For any finite lists xs, ys, and zs:

    append(xs,append(ys,zs)) = append(append(xs,ys),zs)

Identity—For any finite list xs:

    append(Nil,xs) = append(xs,Nil) = xs

Scala’s builtin List type uses the infix operator ++ for the “append” operation. For this operator, associativity can be stated conveniently with the equation:

    xs ++ (ys ++ zs) = (xs ++ ys) ++ zs

Mathematically, the List data type and the binary operation append form a kind of abstract algebra called a monoid. Function append is closed (i.e., it takes two lists and gives a list back), is associative, and has an identity element.

Aside: For more information on operations and algebraic structures, see the Review of Relevant Mathematics chapter [10, Ch. 80] in my book Exploring Languages with Interpreters and Functional Programming. For discussion of how to prove properties like those above, see the Proving Haskell Laws chapter [10, Ch. 25] in the same book.

3.4.1 Tail recursive function reverse

Consider the problem of reversing the order of the elements in a list.

Again we can use the structure of the data to guide the algorithm development. If the argument is a nil list, then the function returns a nil list. If the argument is a non-nil list, then the function can append the head element at the back of the reversed tail.

    def rev[A](ls: List[A]): List[A] = ls match {
        case Nil        => Nil
        case Cons(x,xs) => append(rev(xs),List(x))
    }

Given that evaluation of append terminates, the evaluation of rev also terminates because all recursive applications decrease the length of the argument by one.

How efficient is this function?

The evaluation of rev takes O(n^2) steps, where n is the length of the argument. There are O(n) applications of rev . For each application of rev there are O(n) applications of append .

The initial list and its reverse do not share data.

Function rev has a number of useful properties, for example a distribution and an inverse properties.

Distribution—For any finite lists xs and ys:

    rev(append(xs,ys)) = append(rev(ys), rev(xs))

Inverse—For any finite list xs:

    rev(rev(xs)) = xs

Can we define a function to reverse a list using a “more efficient” tail recursive solution?

As we have seen, a common technique for converting a backward linear recursive definition like rev into a tail recursive definition is to use an accumulating parameter to build up the desired result incrementally. A possible definition for a tail recursive auxiliary function is:

    def revAux[A](ls: List[A], as: List[A]): List[A] = ls match {
        case Nil        => as
        case Cons(x,xs) => revAux(xs,Cons(x,as))
    }

In this definition parameter as is the accumulating parameter. The head of the first argument becomes the new head of the accumulating parameter for the tail recursive call. The tail of the first argument becomes the new first argument for the tail recursive call.

We know that revAux terminates because, for each recursive application, the length of the first argument decreases toward the base case of Nil.

We note that rev(xs) is equivalent to revAux(xs,Nil) .

To define a single-argument replacement for rev , we can embed the definition of revAux’ as an auxiliary function within the definition of a new function reverse .

    def reverse[A](ls: List[A]): List[A] = {
        def revAux[A](rs: List[A], as: List[A]): List[A] = 
            rs match {
                case Nil        => as
                case Cons(x,xs) => revAux(xs,Cons(x,as))
            }
        revAux(ls,Nil)
    }

Function reverse(xs) returns the value from revAux(xs,Nil).

How efficient is this function?

The evaluation of reverse takes O(n) steps, where n is the length of the argument. There is one application of revAux for each element; revAux requires a single O(1) Cons operation in the accumulating parameter.

Where did the increase in efficiency come from?

Each application of rev applies append, a linear time (i.e., O(n)) function. In revAux, we replaced the applications of append by applications of Cons, a constant time (i.e., O(1)) function.

In addition, a compiler or interpreter that does tail call optimization can translate this tail recursive call into a loop on the host machine.

3.4.2 Higher-order function dropWhile

Consider a function dropWhile that removes elements from the front of a List while its predicate argument (a Boolean function) holds.

    def dropWhile [A](ls: List[A], f: A => Boolean): List[A] =
        ls match {
            case Cons(x,xs) if f(x) => dropWhile(xs, f)
            case _                  => ls
        }

This higher-order function terminates when either the list is empty or the head of the list makes the predicate false. For each successive recursive call, the list argument is one element shorter than the previous call, so the function eventually terminates.

If evaluation of function argument p is O(1), then function dropWhile has worst-case time complexity O(n), where n is the length of its first operand. The result list shares data with the input list.

3.4.3 Curried function dropWhile

We often pass anonymous functions to higher-order utility functions like dropwhile, which has the signature:

    def dropWhile[A](ls: List[A], f: A => Boolean): List[A]

When we call dropWhile with an anonymous function for f, we must specify the type of its argument, as follows:

    val xs: List[Int] = List(1,2,3,4,5)
    val ex1 = dropWhile(xs, (x: Int) => x < 4)

Even though it is clear from the first argument that higher order argument f must take an integer as its argument, the Scala type inference mechanism cannot detect this.

However, if we rewrite dropWhile in the following form, type inference can work as we want:

    def dropWhile2[A](ls: List[A])(f: A => Boolean): List[A] =
        ls match {
            case Cons(x,xs) if f(x) => dropWhile2(xs)(f)
            case _                  => ls
        }

Function dropWhile2 is written in curried form above. In this form, a function that takes two arguments can be represented as a function that takes the first argument and returns a function, which itself takes the second argument.

If we apply dropWhile2 to just the first argument, we get a function. We call this a partial application of dropWhile2.

More generally, a function that takes multiple arguments can be represented by a function that takes its arguments in groups of one or more from left to right. If the function is partially applied to the first group, it returns a function that takes the remaining groups, and so forth.

Currying and partial application are directly useful in a number of ways in our programs. Here currying is indirectly useful by assisting type inference. If a function is defined with multiple groups of arguments, the type information flows from one group to another, left to right. In dropWhile2, the first argument group binds type variable A to Int. Then this binding can be used in the second argument group.

3.5.1 Fold right

Consider the sum and product functions we defined above, ignoring the short-cut handling of the zero element in product.

    def sum(ints: List[Int]): Int = ints match {
        case Nil        => 0 
        case Cons(x,xs) => x + sum(xs)
    }

    def product(ds: List[Double]): Double = ds match {
        case Nil          => 1.0
        case Cons(x,xs)   => x * product(xs)
    }

What do sum and product have in common? What differs?

Both exhibit the same pattern of computation.

• Both take a list as input.

  But the element type differs. Function sum takes a list of Int values and product takes a list of Double values.

• Both insert a binary operator between all the consecutive elements of the list in order to reduce the list to a single value.

  But the binary operation differs. Function sum applies integer addition and product applies double-precision floating-point multiplication.

• Both group the operations from the right to the left.

• Both functions return some value for an empty list. The function extends nonempty input lists to implicitly include this value as the “rightmost” value of the input list.

  But the actual value differs.

  Function sum returns integer 0, the (right) identity element for addition.

  Function product returns 1.0, the (right) identity element for multiplication.

  In general, this value could be something other than the (right) identity element.

• Both return a value of the same element type as the input list.

  But the input type differs, as we noted above.

Both functions insert operations of type (A,A) => A between elements a list of type List[A], for some generic type A.

But these are special cases of more general operations of type (A,B) => B. In this case, the value returned must be of type B in the case of both empty and nonempty lists.

Whenever we recognize a pattern like this, we can systematically generalize the function definition as follows:

1. Do a scope-commonality-variability (SCV) analysis [5] on the set of related functions.

   That is, identify what is to be included and what not (i.e., the scope), the parts of functions that are the same (the commonalities or frozen spots), and the parts that differ (the variabilities or hot spots).

2. Leave the commonalities in the generalized function’s body.

3. Move the variabilities into the generalized function’s header—its type signature and parameter list.

   • If the part moved to the generalized function’s parameter list is an expression, then make that part a function with a parameter for each local variable accessed.

   • If a data type potentially differs from a specific type used in the set of related functions, then add a type parameter to the generalized function.

   • If the same data value or type appears in multiple roles, then consider adding distinct type or value parameters for each role.

4. Consider other approaches if the generalized function’s type signature and parameter list become too complex.

   For example, we can introduce new data or procedural abstractions for parts of the generalized function. These may be in the same “module” (i.e., object, class, etc.) as the generalized function, in an appropriately defined separate “module” that is imported, etc. A separate module may better accomplish the desired parameterization of the function.

A similar approach can be used to generalize a whole class.

Following the above guidelines, we can express the common pattern from sum and product as a new (broadly useful) polymorphic, higher-order function foldRight, which we define as follows:

    def foldRight[A,B](ls: List[A], z: B)(f: (A, B) => B): B = 
        ls match {
            case Nil        => z
            case Cons(x,xs) => f(x, foldRight(xs, z)(f))
        }

This function:

• passes in the binary operation f that combines the list elements

• passes in the element z to be returned for empty lists (often the right identity element for the operation, but this is not required)

• uses two type parameters A and B—one for the type of elements in the list and one for the type of the result

The foldRight function “folds” the list elements (of type A) into a value (of type B) by “inserting” operation f between the elements, with value z “appended” as the rightmost element. For example, foldRight(List(1,2,3),z)(f) expands to f(1,f(2,f(3,z))).

Function foldRight is not tail recursive, so it needs a new stack frame for each element of the input list. If its list argument is long or the folding function itself is expensive, then the function can terminate with a stack overflow error.

We can specialize foldRight to have the same functionality as sum and product.

    def sum2(ns: List[Int]) =
        foldRight(ns, 0)((x,y) => x + y)

    def product2(ns: List[Double]) =
        foldRight(ns, 1.0)(_ * _)

The expression (_ * _) in product2 is a concise notation for the anonymous function (x,y) => x * y. The two underscores denote two distinct anonymous variables. This concise notation can be used in a context where Scala’s type inference mechanism can determine the types of the anonymous variables.

We can construct a recursive function to compute the length of a polymorphic list. However, we can also express this computation using foldRight, as follows:

    def length[A](ls: List[A]): Int =
        foldRight(ls, 0)((_,acc) => acc + 1)

We use the z parameter to accumulate the count, starting it at 0. Higher order argument f is a function that takes an element of the list as its left argument and the previous accumulator as its right argument and returns it incremented by 1. In this application, z is not the identity element for f by a convenient beginning value for the counter.

We can construct an “append” function that uses foldRight as follows:

    def append2[A](ls: List[A], rs: List[A]): List[A] =
        foldRight(ls, rs)(Cons(_,_))

Here the list that foldRight operates on the first argument of the append. The z parameter is the entire second argument and the combining function is just Cons. So the effect is to replace the Nil at the end of the first list by the entire second list.

We can construct a recursive function that takes a list of lists and returns a “flat” list that has the same elements in the same order. We can also express this concat function in terms of foldRight, as follows:

    def concat[A](ls: List[List[A]]): List[A] =
        foldRight(ls, Nil: List[A])(append) 

Function append takes time proportional to the length of its first list argument. This argument does not grow larger because of right associativity of foldRight. Thus concat takes time proportional to the total length of all the lists.

Above, we “pass” the append function without writing an explicit anonymous function definition (i.e., function literal) such as (xs,ys) => append(xs,ys) or append(_,_).

In concat, for which Scala can infer the types of append’s arguments, the compiler can generate the needed function literal. In other cases, we would need to use partial application notation such as

    append _

or an explicit function literal such as

    (xs: List[A], ys: List[A]) => append(xs,ys)

to enable the compiler to infer the types.

Above we defined function foldRight as a backward recursive function that processes the elements of a list one by one. However, as we have seen, it is often more useful to think of foldRight as a powerful list operator that reduces the element of the list into a single value. We can combine foldRight with other operators to conveniently construct list processing programs.

3.5.2 Fold left

We designed function foldRight above as a backward linear recursive function with the signature:

    foldRight[A,B](as: List[A], z: B)(f: (A, B) => B): B

As noted:

    foldRight(List(1,2,3),z)(f) == f(1,f(2,f(3,z)))

Consider a function foldLeft such that:

    foldLeft(List(1,2,3),z)(f) == f(f(f(z,1),2),3)

This function folds from the left. It offers us the opportunity to use parameter z as an accumulating parameter in a tail recursive implementation, as follows:

    @annotation.tailrec
    def foldLeft[A,B](ls: List[A], z: B)(f: (B, A) => B): B = 
        ls match {
            case Nil        => z
            case Cons(x,xs) => foldLeft(xs, f(z,x))(f)
        }

In the first line above, we annotate function foldLeft as tail recursive using @annotation.tailrec. If the function is not tail recursive, the compiler gives an error, rather than silently generating code that does not use tail call optimization (i.e., does not convert the recursion to a loop).

We can implement list sum, product, and length functions with foldLeft, similar to what we did with foldRight.

    def sum3(ns: List[Int]) =
        foldLeft(ns, 0)(_ + _) 
        
    def product3(ns: List[Double]) =
        foldLeft(ns, 1.0)(_ * _)

Given that addition and multiplication of numbers are associative and have identity elements, sum3 and product3 use the same values for parameters z and f as foldRight.

Function length2 that uses foldLeft is like length except that the arguments of function f are reversed.

    def length2[A](ls: List[A]): Int =
        foldLeft(ls, 0)((acc,_) => acc + 1)

We can also implement list reversal using foldLeft as follows:

    def reverse2[A](ls: List[A]): List[A] =
        foldLeft(ls, List[A]())((acc,x) => Cons(x,acc))

This gives a solution similar to the tail recursive reverse function above. The z value is initially an empty list; the folding function f uses Cons to “attach” each element of the list to front of the accumulator, incrementally building the list in reverse order.

Because foldLeft is tail recursive and foldRight is not, foldLeft is usually safer and more efficient to use in than foldRight. (If the list argument is lazily evaluated or the function argument f is nonstrict in at least one of its arguments, then there are other factors to consider. We will discuss what we mean by “lazily evaluated” and “nonstrict” later in the course.)

To avoid the stack overflow situation with foldRight, we can first apply reverse to the list argument and then apply foldLeft as follows:

    def foldRight2[A,B](ls: List[A], z: B)(f: (A,B) => B): B =
        foldLeft(reverse(ls), z)((b,a) => f(a,b))

The combining function in the call to foldLeft is the same as the one passed to foldRight2 except that its arguments are reversed.

3.5.3 Map

Consider the following two functions, noting their type signatures and patterns of recursion.

The first, squareAll, takes a list of integers and returns the corresponding list of squares of the integers.

    def squareAll(ns: List[Int]): List[Int] = ns match {
        case Nil         => Nil
        case Cons(x, xs) => Cons(x*x, squareAll(xs))
    } 

The second, lengthAll, takes a list of lists and returns the corresponding list of the lengths of the element lists

    def lengthAll[A](lss: List[List[A]]): List[Int] =
        lss match {
            case Nil           => Nil
            case Cons(xs, xss) => Cons(length(xs),lengthAll(xss))
        }

Although these functions take different kinds of data (a list of integers versus a list of polymorphically typed lists) and apply different operations (squaring versus list length), they exhibit the same pattern of computation. That is, both take a list and apply some function to each element to generate a resulting list of the same size as the original.

As with the fold functions, the combination of polymorphic typing and higher-order functions allows us to abstract this pattern of computation into a higher-order function.

We can abstract the pattern of computation common to squareAll and lengthAll as the (broadly useful) function map, defined as follows:

    def map[A,B](ls: List[A])(f: A => B): List[B] = ls match {
        case Nil        => Nil
        case Cons(x,xs) => Cons(f(x),map(xs)(f))
    }

Function map takes a list of type A elements, applies function f of type A => B to each element, and returns a list of the resulting type B elements.

Thus we can redefine squareAll and lengthAll using map as follows:

    def squareAll2(ns: List[Int]): List[Int] =
        map(ns)(x => x*x)
        
    def lengthAll2[A](lss: List[List[A]]): List[Int] =
        map(lss)(length)

We can implement map itself using foldRight as follows:

    def map1[A,B](ls: List[A])(f: A => B): List[B] =
        foldRight(ls, Nil: List[B])((x,xs) => Cons(f(x),xs))

The folding function (x,xs) => Cons(f(x),xs) applies the mapping function f to the next element of the list (moving right to left) and attaches the result on the front of the processed tail.

As implemented above, function map is backward recursive; it thus requires a stack frame for each element of its list argument. For long lists, the recursion can cause a stack overflow error. Function map1 uses foldRight, which has similar characteristics. So we need to use these functions with care. However, we can use the reversal technique illustrated in foldRight2 if necessary.

We could also optimize function map using local mutation. That is, we can use a mutable data structure within the map function but not allow this structure to be accessed outside of map. The following function takes that approach, using a ListBuffer:

    def map2[A,B](ls: List[A])(f: A => B): List[B] = {
        val buf = new collection.mutable.ListBuffer[B]

        @annotation.tailrec
        def go(ls: List[A]): Unit = ls match {
            case Nil        => ()
            case Cons(x,xs) => buf += f(x); go(xs)
        }
    
        go(ls)
        List(buf.toList: _*) 
    }

A ListBuffer is a mutable list data structure from the Scala library. The operation += appends a single element to the end of the buffer in constant time. The method toList converts the ListBuffer to a Scala immutable list, which is similar to the data structure we are developing in this module.

3.5.4 Filter

Consider the following two functions.

The first, getEven, takes a list of integers and returns the list of those integers that are even (i.e., are multiples of 2). The function preserves the relative order of the elements in the list.

    def getEven(ns: List[Int]): List[Int] = ns match {
        case Nil        => Nil
        case Cons(x,xs) =>
            if (x % 2 == 0)  // divisible evenly by 2
                Cons(x,getEven(xs))
            else
                getEven(xs)
    }

The second, doublePos, takes a list of integers and returns the list of doubles of the positive integers from the input list; it preserves the order of the elements.

    def doublePos(ns: List[Int]): List[Int]  = ns match {
        case Nil        => Nil
        case Cons(x,xs) =>
            if (0 < x)
                Cons(2*x, doublePos(xs))
            else
                doublePos(xs)
    }           

We can abstract the pattern of computation common to getEven and doublePos as the (broadly useful) function filter, defined as follows:

    def filter[A](ls: List[A])(p: A => Boolean): List[A] =
        ls match {
            case Nil        => Nil
            case Cons(x,xs) =>
                val fs = filter(xs)(p)
                if (p(x)) Cons(x,fs) else fs
        }

Function filter takes a predicate p of type A => Boolean a list of type List[A] and returns a list containing those elements that satisfy p, in the same order as the input list.

Therefore, we can redefine getEven and doublePos as follows:

    def getEven2(ns: List[Int]): List[Int] =
        filter(ns)(x => x % 2 == 0)

    def doublePos2(ns: List[Int]): List[Int] =
        map(filter(ns)(x => 0 < x))(y => 2 * y)

Function doublePos2 exhibits both the filter and the map patterns of computation.

The higher-order functions map and filter allowed us to restate the definitions of getEven and doublePos in a succinct form.

We can implement filter in terms of foldRight as follows:

    def filter1[A](ls: List[A])(p: A => Boolean): List[A] =
        foldRight(ls, Nil:List[A])((x,xs) => if (p(x)) Cons(x,xs) else xs)

Above, the folding function

    (x,xs) => if (p(x)) Cons(x,xs) else xs

applies the filter predicate p to the next element of the list (moving right to left). If the predicate evaluates to true, the folding function attaches that element on the front of the processed tail; otherwise, it omits the element from the result.

3.5.5 Flat map

The higher-order function map applies its function argument f to every element of a list and returns the list of results. If the function argument f returns a list, then the result is a list of lists. Often we wish to flatten this into a single list, that is, apply a function like concat defined in Section 3.5.1.

This computation is sufficiently common that we give it the name flatMap. We can define it in terms of map and concat as

    def flatMap[A,B](ls: List[A])(f: A => List[B]): List[B] =
        concat(map(ls)(f))

or by combining map and concat into one foldRight as:

    def flatMap1[A,B](ls: List[A])(f: A => List[B]): List[B] =
        foldRight(ls, Nil: List[B])(
            (x: A, ys: List[B]) => append(f(x),ys))

Above, the function argument to foldRight applies the flatMap function argument f to each element of the list argument and then appends the resulting list in front of the result from processing the elements to the right.

We can also define filter in terms of flatMap as follows:

    def filter2[A](ls: List[A])(p: A => Boolean): List[A] =
        flatMap(ls)(x => if (p(x)) List(x) else Nil)

The function argument to flatMap generates a one-element list if the filter predicate p is true and an empty list if it is false.

3.6.1 Insertion sort and bounded generics

Consider a function to sort the elements of a list into ascending order. A simple algorithm to do this is insertion sort. To sort a non-empty list with head x and tail xs, sort the tail xs and insert the element x at the right position in the result. To sort an empty list, just return it.

If we restrict the function to integer lists, we get the following Scala functions:

    def isort(ls: List[Int]): List[Int] = ls match {
        case Nil        => Nil
        case Cons(x,xs) => insert(x,isort(xs))
    }

    def insert(x: Int, xs: List[Int]): List[Int] = xs match {
        case Nil        => List(x)
        case Cons(y,ys) =>
            if (x <= y)
                Cons(x,xs)
            else
                Cons(y,insert(x,ys))
    }

Insertion sort has a (worst and average case) time complexity of O(n^2) where n is the length of the input list. (Function isort requires $n$ consecutive recursive calls; each call uses function insert which itself requires on the order of $n$ recursive calls.)

Now suppose we want to generalize the sorting function and make it polymorphic. We cannot just add a type parameter A and substitute it for Int everywhere. Although all Scala data types support equality and inequality comparison, not all types can be compared on a total ordering (<, <=, >, and >= as well).

Fortunately, the Scala library provides a trait Ordered. Any class that provides the other comparisons can extend this trait; the standard types in the library do so. This trait adds the comparison operators as methods so that they can be called in infix form.

    trait Ordered[A] {
        def compare(that: A): Int
        def < (that: A): Boolean = (this compare that) <  0
        def > (that: A): Boolean = (this compare that) >  0
        def <=(that: A): Boolean = (this compare that) <= 0
        def >=(that: A): Boolean = (this compare that) >= 0
        def compareTo(that: a) = compare(that)
    }

We thus need to restrict the polymorphism on A to be a subtype of Ordered[A] by putting an upper bound on the type as follows:

    def isort[A <: Ordered[A]](ls: List[A]): List[A]

Note: In addition to upper bounds, we can use a lower bound. A constraint A :> T requires type A to be a supertype of type T. We can also specify both an upper and a lower bound on a type such as T1 <: A <: T2,

By using the upper bound constraint, we can sort data from any type that extends Ordered. However, the primitive types inherited from Java do not extend Ordered.

Fortunately, the Scala library defines implicit conversions between the Java primitive types and Scala’s enriched wrapper types. (This is the “type class” mechanism we discussed earlier.) We can use a weaker view bound constraint, denoted by <% instead of <:. This A to be any type that is a subtype of or convertible to Ordered[A].

    def isort1[A <% Ordered[A]](ls: List[A]): List[A] = 
        ls match {
            case Nil        => Nil
            case Cons(x,xs) => insert1(x,isort1(xs))
        }

    def insert1[A <% Ordered[A]](x: A, xs: List[A]): List[A] =
        xs match {
            case Nil        => List(x)
            case Cons(y,ys) =>
                if (x <= y)
                    Cons(x,xs)
                else
                    Cons(y,insert1(x,ys))
        }

We could define insert inside isort and avoid the separate type parameterization. But insert is separately useful, so it is reasonable to leave it external.

An alternative to use of the bound would be to pass in the needed comparison predicate, as follows:

    def isort2[A](ls: List[A])(leq: (A,A) => Boolean): List[A] =
        ls match {
            case Nil        => Nil
            case Cons(x,xs) => insert2(x,isort2(xs)(leq))(leq)
        }

    def insert2[A](x:A, xs:List[A])(leq:(A,A)=>Boolean):List[A] =
        xs match {
            case Nil        => List(x)
            case Cons(y,ys) =>
                if (leq(x,y))
                    Cons(x,xs)
                else
                    Cons(y,insert2(x,ys)(leq))
        }

Above we expressed both functions in curried form. By putting the comparison function last, we enabled the compiler to infer the argument types for the function.

If we placed the function in the first argument group, the user of the function would have to supply the types. However, putting the comparison function first might allow a more useful partial application of the isort to a comparison function.

3.6.2 Merge sort

The insertion sort given in Section 3.6.2 has an average case time complexity of O(n^2) where n is the length of the input list.

We now consider a more efficient function to sort the elements of a list: merge sort. Merge sort works as follows:

• If the list has fewer than two elements, then it is already sorted.

• If the list has two or more elements, then we split it into two sublists, each with about half the elements, and sort each recursively.

• We merge the two ascending sublists into an ascending list.

For a general implementation, we specify the type of list elements and the function to be used for the comparison of elements, giving the following implementation:

    def msort[A](less: (A, A) => Boolean)(ls: List[A]): List [A] =
    {
        def merge(as: List[A], bs: List[A]): List[A] = 
            (as,bs) match {
                case (Nil,_)                 => bs
                case (_,Nil)                 => as
                case (Cons(x,xs),Cons(y,ys)) =>
                    if (less(x,y))
                        Cons(x,merge(xs,bs))
                    else
                        Cons(y,merge(as,ys))
        }

        val n = length(ls)/2
        if (n == 0)
            ls
        else
            merge(msort(less)(take(ls,n)),
                  msort(less)(drop(ls,n)))
    }

The merge forms a tuple of the two lists and does pattern matching against that tuple. This allowed the pattern match to be expressed more symmetrically.

The above function uses a function we have not yet defined.

    def take[A](ls: List[A], n: Int): List[A]

returns the first n elements of the list; it is the dual of drop.

By nesting the definition of merge, we enabled it to directly access the parameters of msort. In particular, we did not need to pass the comparison function to merge.

The average case time complexity of msort is O(n * (log2 n)), where n is the length of the input list. (Here log2 denotes a function that computes logarithms base 2.)

• Each call level requires splitting of the list in half and merging of the two sorted lists. This takes time proportional to the length of the list argument.

• Each call of msort for lists longer than one results in two recursive calls of msort.

• But each successive call of msort halves the number of elements in its input, so there are O(log2 n) recursive calls.

So the total cost is O(n * (log2 n)). The cost is independent of distribution of elements in the original list.

We can apply msort as follows:

    msort((x: Int, y: Int) => x < y)(List(5, 7, 1, 3))

We defined msort in curried form with the comparison function first (unlike what we did with isort1). This enables us to conveniently specialize msort with a specific comparison function. For example,

    val intSort     = msort((x: Int, y: Int) => x < y) _
    val descendSort = msort((x: Int, y: Int) => x > y) _

However, we do have to give explicit type annotations for the parameters of the comparison function.

3.10.1 Description

A general tree is a hierarchical data structure in which each node of the tree has zero or more subtrees. We can define this as a Scala algebraic data type as follows:

    sealed trait GTree[+A]
    case class Leaf[+A](value: A) extends GTree[A]
    case class Gnode[+A](gnode: List[GTree[A]]) extends GTree[A]

An object of class Leaf(x) represents a leaf of the tree holding some value x of generic type A. A leaf does not have any subtrees. It has height 1.

An object of type Gnode represents an internal (i.e., non-leaf) node of the tree. It consists of a nonempty list of subtrees, ordered left-to-right. A Gnode has a height that is one more than the maximum height of its subtrees.

3.10.2 Exercise Set B

In the following exercises, write the Scala functions to operate on the GTrees. You may use functions from the extended List module as needed.

1. Write Scala function numLeavesGT that takes a GTree and returns the count of its leaves.

2. Write Scala function heightGT that takes a GTree and returns its height (i.e., the number of levels).

3. Write Scala function sumGT that takes a GTree of integers and returns the sum of the values.

4. Write Scala function findGT that takes a GTree and a value and returns true if and only if the element appears in some leaf in the tree.

5. Write Scala function mapGT that takes a GTree and a function and returns a GTree with the structure but with the function applied to all the values in the tree.

6. Write Scala function flattenGT that takes a GTree and returns a List with the values from the tree leaves ordered according to a left-to-right traversal of the tree.