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Chapter 7 in FP in Scala deals with creating a purely functional library for handling concurrency. To that end, it defines a type
type Par[A] = ExecutorService => Future[A]
and a set of useful functions such as fork
def fork[A] (a: => Par[A]): Par[A] =
es => es.submit(new Callable[A] {
def call = a(es).get
})
One of the exercises is about the function sequence with the following signature
def sequence[A](ps: List[Par[A]]): Par[List[A]]
The solution using foldRight is straightforward. However the authors included two other versions as answers, one of which states the following
// This implementation forks the recursive step off to a new logical thread,
// making it effectively tail-recursive. However, we are constructing
// a right-nested parallel program, and we can get better performance by
// dividing the list in half, and running both halves in parallel.
// See `sequenceBalanced` below.
def sequenceRight[A](as: List[Par[A]]): Par[List[A]] =
as match {
case Nil => unit(Nil)
case h :: t => map2(h, fork(sequenceRight(t)))(_ :: _)
}
I am not quite sure what is meant by "effectively tail recursive". From the definition of fork it is clear that it accepts a by name parameter and so map2(h, fork(sequenceRight(t)))(_ :: _) will not evaluate the second parameter (until the executor service is provided). But that doesn't tell me how and why it is "effectively tail recursive".
Let's take some List(a, b, c). After passing it into sequenceRight it will turn into:
map2(
a,
fork(
map2(
b,
fork(
map2(
c,
fork(unit(Nil)
)(_ :: _)
)
)(_ :: _)
)
)(_ :: _)
This isn't tail recursive at all and compiler cannot treat it as one. However, when you would evaluate how it would be executed:
fork would make whatever you pass to it async, so it would return immediately,
map2 implementation will not block the execution until fork is executed to apply the function passed to map2, instead it would asynchronously transform the result calculated in fork to prepend the value
since recursion is done asynchronously, posting things to ExecutorService and appending operation to Future let you treat ExecutorService+Future like a trampoline
As a result what actually happens is:
sequenceRight(List(a, b, c)) call `map2(a, fork(sequenceRight(List(b, c))(_ :: _)
a will complete when it will complete but we can hold it as value even now
fork(sequenceRight(List(b, c)) is scheduled, but we aren't waiting until it complete, we can pass it around already
we can create a Future that will combine the result of the 2 above (and return it) without waiting for any of them completing!
as a result, this Future is returned immediately! It still runs, but this one call is finished!
same is true for recursively created Futures
once c and fork(unit(Nil)) completes, rResult :: Nil is computed
this allows completion of bResult :: cResult :: Nil
and this finally allows computation of the final result
In other words tail-recursive refers to recursive calls being rewritten into while loop to avoid stack overflow. But stack overflow is only an issue if recursive calls are being made synchronously. If you are returning something async, then the backtracing is shoved to ExecutionServices and Futures' observers, so they are hidden in a heap. From that point of view they solve the same issue of stack-overflow as tail-recursive calls, so "in spirit" they could be considered somewhat similar.
This is certainly not tail-recursion, and I think, they know it – that's why they added "effectively" :).
What they must mean is that this implementation does not create additional frames on stack for recursive invocations, which is true, since those invocations happen asynchronously, as you noted.
Now, whether this situation is even deserves to be called a "recursion" at all is a good question. I don't think there is a single accepted answer to that. I personally lean toward "yes", because the definition of recursion as "referencing itself" definitely includes this situation, and I don't really know how else to define it so that async invocations are excluded, but tail-recursion is not.
BTW, I am not much of an expert in Javascript, but I hear that the term "asynchronous recursion" is used fairly widely there.
Here is a minimal code that raise the compilation error 'Recursive call not in tail position'. However, I'm using an #inline and the recursive call is in tail position. The reason why I'm using this #inline is that I have the code pf the original reccall duplicated twice.
import scala.annotation._
object Test {
#tailrec private def test(i: Int): Int = {
#inline def reccall(i: Int): Int = test(i-1)
i match {
case 0 => 0
case i => reccall(i)
}
}
}
I've looked at the answers Recursive call not in tail position #tailrec why does this method not compile with 'contains a recursive call not in tail position'? but they do not apply to my case. Using Scala 2.12
It appears, the way #inline is implemented is that it still passes the parameter via stack. The jump is eliminated, by inserting the code inline, but the stack is still used for the arguments. This makes it impossible to be in a tail position, because the stack needs to be cleaned up after the call is completed.
Besides, annotating a function with #inline does not guarantee that the optimizer will inline it, just that it will "try especially hard".
Well, the mechanism of how tail recursion is actualized in JVM is explained in following way:
Scala, in the case of tail recursion, can eliminate the creation of a
new stack frame and just re-use the current stack frame. The stack
never gets any deeper, no matter how many times the recursive call is
made.
So in your case it cannot reuse the current stack frame belonging to the test method since it MUST create a new stack frame for the reccall method anyway.
Recursive call is implicit in this case, made from another method. So I believe you cannot really have tail recursion implemented for such case.
You may just remove the reccall method altogether and write case i => test(i-1) and then compiler will not complain.
NOTE: also I believe #inline has nothing to do here and is not essential in this example, since if I remove it - compiler still complains the same reason.
The issue here is that #inline is strictly advisory: it doesn't guarantee that the compiler will inline the function. Since #tailrec only works if it's absolutely guaranteed that the tail-calls can be eliminated, this means that using #tailrec has to assume no inlining.
I have been looking into FP languages (off and on) for some time and have played with Scala, Haskell, F#, and some others. I like what I see and understand some of the fundamental concepts of FP (with absolutely no background in Category Theory - so don't talk Math, please).
So, given a type M[A] we have map which takes a function A=>B and returns a M[B]. But we also have flatMap which takes a function A=>M[B] and returns a M[B]. We also have flatten which takes a M[M[A]] and returns a M[A].
In addition, many of the sources I have read describe flatMap as map followed by flatten.
So, given that flatMap seems to be equivalent to flatten compose map, what is its purpose? Please don't say it is to support 'for comprehensions' as this question really isn't Scala-specific. And I am less concerned with the syntactic sugar than I am in the concept behind it. The same question arises with Haskell's bind operator (>>=). I believe they both are related to some Category Theory concept but I don't speak that language.
I have watched Brian Beckman's great video Don't Fear the Monad more than once and I think I see that flatMap is the monadic composition operator but I have never really seen it used the way he describes this operator. Does it perform this function? If so, how do I map that concept to flatMap?
BTW, I had a long writeup on this question with lots of listings showing experiments I ran trying to get to the bottom of the meaning of flatMap and then ran into this question which answered some of my questions. Sometimes I hate Scala implicits. They can really muddy the waters. :)
FlatMap, known as "bind" in some other languages, is as you said yourself for function composition.
Imagine for a moment that you have some functions like these:
def foo(x: Int): Option[Int] = Some(x + 2)
def bar(x: Int): Option[Int] = Some(x * 3)
The functions work great, calling foo(3) returns Some(5), and calling bar(3) returns Some(9), and we're all happy.
But now you've run into the situation that requires you to do the operation more than once.
foo(3).map(x => foo(x)) // or just foo(3).map(foo) for short
Job done, right?
Except not really. The output of the expression above is Some(Some(7)), not Some(7), and if you now want to chain another map on the end you can't because foo and bar take an Int, and not an Option[Int].
Enter flatMap
foo(3).flatMap(foo)
Will return Some(7), and
foo(3).flatMap(foo).flatMap(bar)
Returns Some(15).
This is great! Using flatMap lets you chain functions of the shape A => M[B] to oblivion (in the previous example A and B are Int, and M is Option).
More technically speaking; flatMap and bind have the signature M[A] => (A => M[B]) => M[B], meaning they take a "wrapped" value, such as Some(3), Right('foo), or List(1,2,3) and shove it through a function that would normally take an unwrapped value, such as the aforementioned foo and bar. It does this by first "unwrapping" the value, and then passing it through the function.
I've seen the box analogy being used for this, so observe my expertly drawn MSPaint illustration:
This unwrapping and re-wrapping behavior means that if I were to introduce a third function that doesn't return an Option[Int] and tried to flatMap it to the sequence, it wouldn't work because flatMap expects you to return a monad (in this case an Option)
def baz(x: Int): String = x + " is a number"
foo(3).flatMap(foo).flatMap(bar).flatMap(baz) // <<< ERROR
To get around this, if your function doesn't return a monad, you'd just have to use the regular map function
foo(3).flatMap(foo).flatMap(bar).map(baz)
Which would then return Some("15 is a number")
It's the same reason you provide more than one way to do anything: it's a common enough operation that you may want to wrap it.
You could ask the opposite question: why have map and flatten when you already have flatMap and a way to store a single element inside your collection? That is,
x map f
x filter p
can be replaced by
x flatMap ( xi => x.take(0) :+ f(xi) )
x flatMap ( xi => if (p(xi)) x.take(0) :+ xi else x.take(0) )
so why bother with map and filter?
In fact, there are various minimal sets of operations you need to reconstruct many of the others (flatMap is a good choice because of its flexibility).
Pragmatically, it's better to have the tool you need. Same reason why there are non-adjustable wrenches.
The simplest reason is to compose an output set where each entry in the input set may produce more than one (or zero!) outputs.
For example, consider a program which outputs addresses for people to generate mailers. Most people have one address. Some have two or more. Some people, unfortunately, have none. Flatmap is a generalized algorithm to take a list of these people and return all of the addresses, regardless of how many come from each person.
The zero output case is particularly useful for monads, which often (always?) return exactly zero or one results (think Maybe- returns zero results if the computation fails, or one if it succeeds). In that case you want to perform an operation on "all of the results", which it just so happens may be one or many.
The "flatMap", or "bind", method, provides an invaluable way to chain together methods that provide their output wrapped in a Monadic construct (like List, Option, or Future). For example, suppose you have two methods that produce a Future of a result (eg. they make long-running calls to databases or web service calls or the like, and should be used asynchronously):
def fn1(input1: A): Future[B] // (for some types A and B)
def fn2(input2: B): Future[C] // (for some types B and C)
How to combine these? With flatMap, we can do this as simply as:
def fn3(input3: A): Future[C] = fn1(a).flatMap(b => fn2(b))
In this sense, we have "composed" a function fn3 out of fn1 and fn2 using flatMap, which has the same general structure (and so can be composed in turn with further similar functions).
The map method would give us a not-so-convenient - and not readily chainable - Future[Future[C]]. Certainly we can then use flatten to reduce this, but the flatMap method does it in one call, and can be chained as far as we wish.
This is so useful a way of working, in fact, that Scala provides the for-comprehension as essentially a short-cut for this (Haskell, too, provides a short-hand way of writing a chain of bind operations - I'm not a Haskell expert, though, and don't recall the details) - hence the talk you will have come across about for-comprehensions being "de-sugared" into a chain of flatMap calls (along with possible filter calls and a final map call for the yield).
Well, one could argue, you don't need .flatten either. Why not just do something like
#tailrec
def flatten[T](in: Seq[Seq[T], out: Seq[T] = Nil): Seq[T] = in match {
case Nil => out
case head ::tail => flatten(tail, out ++ head)
}
Same can be said about map:
#tailrec
def map[A,B](in: Seq[A], out: Seq[B] = Nil)(f: A => B): Seq[B] = in match {
case Nil => out
case head :: tail => map(tail, out :+ f(head))(f)
}
So, why are .flatten and .map provided by the library? Same reason .flatMap is: convenience.
There is also .collect, which is really just
list.filter(f.isDefinedAt _).map(f)
.reduce is actually nothing more then list.foldLeft(list.head)(f),
.headOption is
list match {
case Nil => None
case head :: _ => Some(head)
}
Etc ...
listening to Scala courses and explanations I often hear: "but in real code we are not using recursion, but tail recursion".
Does it mean that in my Real code I should NOT use recursion, but tail recursion that is very much like looping and does not require that epic phrase "in order to understand recursion you first need understand recursion" .
In reality, taking into account your stack.. you more likely would use loop-like tail recursion.
Am I wrong? Is that 'classic' recursion is good only for education purposes to make your brain travel back to the university-past?
Or, for all that, there is place where we can use it.. where the depth of recursion calls is less than X (where X your stack-overflow limit). Or we can start coding from classic-recursion and then, being afraid of your stack blowing one day, apply couple of refactorings to make it tail-like to make use even stronger on refactoring field?
Question: Some real samples that you would use / have used 'classic head' recursion in your real code, which is not refactored yet into tail one, maybe?
Tail Recursion == Loop
You can take any loop and express it as tail-recursive call.
Background: In pure FP, everything must result in some value. while loop in scala doesn't result in any expression, only side-effects (e.g. update some variable). It exists only to support programmers coming from imperative background. Scala encourages developers to reconsider replacing while loop with recursion, which always result in some value.
So according to Scala: Recursion is the new iteration.
However, there is a problem with previous statement: while "Regular" Recursive code is easier to read, it comes with a performance penalty AND carries an inherent risk of overflowing the stack. On the other hand, tail-recursive code will never result in stack overflow (at least in Scala*), and the performance will be the same as loops (In fact, I'm sure Scala converts all tail recursive calls to plain old iterations).
Going back to the question, nothing wrong with sticking to the "Regular" recursion, unless:
The algorithm you are using in calculating large numbers (stack overflow)
Tail Recursion brings a noticeable performance gain
There are two basic kinds of recursion:
head recursion
Tail recursion
In head recursion, a function makes its recursive call and then performs some more calculations, maybe using the result of the recursive call, for example. In a tail recursive function, all calculations happen first and the recursive call is the last thing that happens.
The importance of this distinction doesn’t jump out at you, but it’s extremely important! Imagine a tail recursive function. It runs. It completes all its computation. As its very last action, it is ready to make its recursive call. What, at this point, is the use of the stack frame? None at all. We don’t need our local variables anymore because we’re done with all computations. We don’t need to know which function we’re in because we’re just going to re-enter the very same function. Scala, in the case of tail recursion, can eliminate the creation of a new stack frame and just re-use the current stack frame. The stack never gets any deeper, no matter how many times the recursive call is made. That’s the voodoo that makes tail recursion special in Scala.
Let's see with the example.
def factorial1(n:Int):Int =
if (n == 0) 1 else n * factorial1(n -1)
def factorial2(n:Int):Int = {
def loop(acc:Int,n:Int):Int =
if (n == 0) 1 else loop(acc * n,n -1)
loop(1,n)
}
Incidentally, some languages achieve a similar end by converting tail recursion into iteration rather than by manipulating the stack.
This won’t work with head recursion. Do you see why? Imagine a head recursive function. First it does some work, then it makes its recursive call, then it does a little more work. We can’t just re-use the current stack frame when we make that recursive call. We’re going to NEED that stack frame info after the recursive call completes. It has our local variables, including the result (if any) returned by the recursive call.
Here’s a question for you. Is the example function factorial1 head recursive or tail recursive? Well, what does it do? (A) It checks whether its parameter is 0. (B) If so, it returns 1 since factorial of 0 is 1. (C) If not, it returns n multiply by the result of a recursive call. The recursive call is the last thing we typed before ending the function. That’s tail recursion, right? Wrong. The recursive call is made, and THEN n is multiplied by the result, and this product is returned. This is actually head recursion (or middle recursion, if you like) because the recursive call is not the very last thing that happens.
For more info please refer the link
The first thing one should look at when developing software is the readability and maintainability of the code. Looking at performance characteristics is mostly premature optimization.
There is no reason not to use recursion when it helps to write high quality code.
The same counts for tail recursion vs. normal loops. Just look at this simple tail recursive function:
def gcd(a: Int, b: Int) = {
def loop(a: Int, b: Int): Int =
if (b == 0) a else loop(b, a%b)
loop(math.abs(a), math.abs(b))
}
It calculates the greatest common divisor of two numbers. Once you know the algorithm it is clear how it works - writing this with a while-loop wouldn't make it clearer. Instead you would probably introduce a bug on the first try because you forgot to store a new value into one of the variables a or b.
On the other side see these two functions:
def goRec(i: Int): Unit = {
if (i < 5) {
println(i)
goRec(i+1)
}
}
def goLoop(i: Int): Unit = {
var j = i
while (j < 5) {
println(j)
j += 1
}
}
Which one is easier to read? They are more or less equal - all the syntax sugar you gain for tail recursive functions due to Scalas expression based nature is gone.
For recursive functions there is another thing that comes to play: lazy evaluation. If your code is lazy evaluated it can be recursive but no stack overflow will happen. See this simple function:
def map(f: Int => Int, xs: Stream[Int]): Stream[Int] = f -> xs match {
case (_, Stream.Empty) => Stream.Empty
case (f, x #:: xs) => f(x) #:: map(f, xs)
}
Will it crash for large inputs? I don't think so:
scala> map(_+1, Stream.from(0).takeWhile(_<=1000000)).last
res6: Int = 1000001
Trying the same with Scalas List would kill your program. But because Stream is lazy this is not a problem. In this case you could also write a tail recursive function but generally this not easily possible.
There are many algorithms which will not be clear when they are written iteratively - one example is depth first search of a graph. Do you want to maintain a stack by yourself just to save the values where you need to go back to? No, you won't because it is error prone and looks ugly (beside from any definition of recursion - it would call a iterative depth first search recursion as well because it has to use a stack and "normal" recursion has to use a stack as well - it is just hidden from the developer and maintained by the compiler).
To come back to the point of premature optimization, I have heard a nice analogy: When you have a problem that can't be solved with Int because your numbers will get large and it is likely that you get an overflow then don't switch to Long because it is likely that you get an overflow here as well.
For recursion it means that there may be cases where you will blow up your stack but it is more likely that when you switch to a memory only based solution you will get an out of memory error instead. A better advice is to find a different algorithm that doesn't perform that badly.
As conclusion, try to prefer tail recursion instead of loops or normal recursion because it will for sure not kill your stack. But when you can do better then don't hesitate to do it better.
If you're not dealing with a linear sequence, then trying to write a tail-recursive function to traverse the entire collection is very difficult. In such cases, for the sake of readability/maintainability, you usually just use normal recursion instead.
A common example of this is a traversal of a binary tree data structure. For each node you might need to recur on both the left and right child nodes. If you were to try to write such a function recursively, where first the left node is visited and then the right, you'd need to maintain some sort of auxiliary data structure to track all the remaining right nodes that need to be visited. However, you can achieve the same thing just using the stack, and it's going to be more readable.
An example of this is the iterator method from Scala's RedBlack tree:
def iterator: Iterator[(A, B)] =
left.iterator ++ Iterator.single(Pair(key, value)) ++ right.iterator
I was reading about fold techniques in Programming in Scala book and came across this snippet:
def reverseLeft[T](xs:List[T]) = (List[T]() /: xs) {
(y,ys) => ys :: y
}
As you can see, it was done using foldLeft or /: operator. Curious how it would look like if I did it using :\, I came up with this:
def reverseRight[T](xs:List[T]) = (xs :\ List[T]()) {
(y,ys) => ys ::: List(y)
}
As I understand it, ::: doesn't seem to be as fast as :: and has a linear cost depending on the size of the operand list. Admittedly, I don't have a background in CS and no prior FP experience. So my questions are:
How do you recognise/distinguish between foldLeft/foldRight in problem approaches?
Is there a better way of doing this without using :::?
Since foldRight on List in the standard library is strict and implemented using linear recursion, you should avoid using it, as a rule. An iterative implementation of foldRight would be as follows:
def foldRight[A,B](f: (A, B) => B, z: B, xs: List[A]) =
xs.reverse.foldLeft(z)((x, y) => f(y, x))
A recursive implementation of foldLeft could be this:
def foldLeft[A,B](f: (B, A) => B, z: B, xs: List[A]) =
xs.reverse.foldRight(z)((x, y) => f(y, x))
So you see, if both are strict, then one or the other of foldRight and foldLeft is going to be implemented (conceptually anyway) with reverse. Since the way lists are constructed with :: associates to the right, the straightforward iterative fold is going to be foldLeft, and foldRight is simply "reverse then foldLeft".
Intuitively, you might think that this would be a slow implementation of foldRight, since it folds the list twice. But:
"Twice" is a constant factor anyway, so it's asymptotically equivalent to folding once.
You have to go over the list twice anyway. Once to push computations onto the stack and again to pop them off the stack.
The implementation of foldRight above is faster than the one in the standard library.
Operations on a List are intentionally not symmetric. The List data structure is a singly-linked list where each node (both data and pointer) are immutable. The idea behind this data structure is that you perform modifications on the front of the list by taking references to internal nodes and adding new nodes that point to them -- different versions of the list will share the same nodes for the end of the list.
The ::: operator which appends a new element on to the end of the list has to create a new copy of the entire list, because otherwise it would modify other lists that share nodes with the list you're appending to. This is why ::: takes linear time.
Scala has a data structure called a ListBuffer that you can use instead of the ::: operator to make appending to the end of a list faster. Basically, you create a new ListBuffer and it starts with an empty list. The ListBuffer maintains a list completely separate from any other list that the program knows about, so it's safe to modify it by adding on to the end. When you're finished adding on to the end, you call ListBuffer.toList, which releases the list into the world, at which point you can no longer add on to the end without copying it.
foldLeft and foldRight also share a similar assymmetry. foldRight requires you to walk the entire list to get to the end of the list, and keep track of everywhere you've visited on the way there, so that you an visit them in reverse order. This is usually done recursively, and it can lead to foldRight causing stack overflows on large lists. foldLeft on the other hand, deals with nodes in the order they appear in the list, so it can forget the ones it's visited already and only needs to know about one node at a time. Though foldLeft is also usually implemented recursively, it can take advantage of an optimization called tail recursion elimination, in which the compiler transforms the recursive calls into a loop because the function doesn't do anything after returning from the recursive call. Thus, foldLeft doesn't overflow the stack even on very long lists. EDIT: foldRight in Scala 2.8 is actually implemented by reversing the list and running foldLeft on the reversed list -- so the tail recursion issue is not an issue -- both data structures optimize tail recursion correctly, and you could choose either one (You do get into the issue now that you're defining reverse in terms of reverse -- you don't need to worry if you're defining your own reverse method for the fun of it, but you wouldn't have the foldRight option at all if you were defining Scala's reverse method.)
Thus, you should prefer foldLeft and :: over foldRight and :::.
(In an algorithm that would combine foldLeft with ::: or foldRight with ::, then you need to make a decision for yourself about which is more important: stack space or running time. Or you should use foldLeft with a ListBuffer.)