Tail Recursion in Scala

Motivation

I love learning new things, and this year might finally be the year of Functional Programming for me. I’ve worked on Scala on-and-off and have finally decided to dive right into the Functional Programming in Scala book. One of the things that caught my attention when I first started writing code in Scala was Tail Recursion. It took me a while to wrap my head around the concept, and found that all the material that I read just spoke about what tail recursive functions were, but not how to make your function tail recursive. Maybe I wasn’t looking hard enough either.

When I finally got around to understanding it, it was simple enough. When I encountered it in the FP in Scala book again, I thought I’d write a quick post about it.

Getting right to it

I’ll explain the concepts of tail recursion by picking up a simple example: Factorial of a number

How do we calculate the factorial of a number?

Let’s break down what a factorial method does:

  1. Take a number n
  2. If the number is less than or equal to 1, the factorial of the number is 1
  3. If the number is greater than 1, the factorial is equal to the number multiplied by the factorial of the number one less than itself.

What are recursive functions?

You’ve heard this a few times short of a billion - Recursive functions are functions that call themselves. Let’s see how the factorial of a number can be written as a recursive function (BigInt, cause this method generates big ints!):

  def factorial(n: BigInt): BigInt = {
    if (n <= 1)
      1
    else
      n * factorial(n - 1)
  }

The recursive method is a direct translation of the algorithm above, which can be represented mathematically as the series below:

factorial(n) = 1 * 2 * ... * n-1 * n

Do I need recursive functions?

Not always. You can convert the above recursive factorial method into a non-recursive method that uses loops:

  def fac(n: BigInt): BigInt = {
    var i: BigInt = n
    var result: BigInt = 1
    while (i > 1) {
      result = result * i
      i -= 1
    }
    result
  }

This method uses vars. A more functional version can be found at the bottom of this blog.

Recursive functions are sometimes easier to reason about, and simpler to understand in code - if you can break your problem into sub-problems, and you see that the sub-problem is basically the problem itself with a different starting point, you can easily convert your solution into a recursive solution.

Recursion sounds great, what’s the problem then?

The problem with recursion

When you call a method in a language such as Scala or Java, the state of the system is stored in a Stack. When your method calls another method, the state of the current method is stored in the stack, and the called method gets control. When the called method complets, the old state is popped out of the stack and the caller’s state is restored. This can be made more clear here:

  val x = 10
  println(x)

  def someMethod() = {
    val x = 12
    println(x)
  }

  someMethod()
  println(x)

This prints:

10
12
10

The value of x is pushed onto the stack when someMethod is called, and then restored when someMethod completes. The value of x that someMethod sees is the value of x defined inside it.

When a recursive method is called, it basically stores its own state onto the stack before calling itself. Let’s look at what the factorial method is doing:

factorial(10):
= 10 * factorial(9)
= 10 * (9 * factorial(8))
= 10 * (9 * (8 * factorial(7)))
...

As each factorial call calls the factorial for the previous integer, the number of layers in the call stack is equal to n. Stacks have limits, and like all sensible stacks, when you go over that limit, they throw a StackOverflowError. Your stack-trace (The stack in the stack trace is exactly this stack) looks something like this:

Exception in thread "main" java.lang.StackOverflowError
	at scala.math.BigInt$.int2bigInt(BigInt.scala:97)
	at com.caffinc.fp.Experiments$.factorial(Experiments.scala:11)
	at com.caffinc.fp.Experiments$.factorial(Experiments.scala:14)
	at com.caffinc.fp.Experiments$.factorial(Experiments.scala:14)
	at com.caffinc.fp.Experiments$.factorial(Experiments.scala:14)
	at com.caffinc.fp.Experiments$.factorial(Experiments.scala:14)
	...

A whole lot of the same line calling itself. Sometimes you might encounter stack overflows due to extremely complicated layering of methods calling other methods which call the first method, etc. (Think poorly written language parsers). The above stacktrace can be reproduced by running the recursive version of factorial with an n = 10000. However, the iterative version that doesn’t use recursive calls but instead uses vars works just fine. Should you sacrifice your cool recursive solution for one that uses vars? Nope. This is where tail recursion comes in.

So what the heck are tail recursive functions?

A function is tail recursive if all the computation branches end either in values or a call to itself. Let’s look at why our recursive factorial is not tail recursive:

  def factorial(n: BigInt): BigInt = {
    if (n <= 1)
      1 // Constant, looks fine
    else
      n * factorial(n - 1) // This branch actually needs to return back to this point to multiply n with the result of factorial(n - 1), so this branch is not tail-recursive
  }

So Bob, can we fix it? Yes we can! Let’s look at a tail recursive version of factorial:

  def factorial(n: BigInt): BigInt = {
    def go(acc: BigInt, n: BigInt): BigInt = {
      if (n <= 1)
        acc
      else
        go(acc * n, n - 1)
    }
    go(1, n)
  }

Here, the inner def go is the replacement of the original factorial method. Look at the body of the method - the first branch is the value acc. The second branch is a call to itself which does not have to return back to this point to do any further calculations. So essentially the caller of go can fire-and-forget. The compiler can now optimize this tail call into a while loop which eliminates the need for a stack to store the current state.

So how exactly did we turn a recursive method using a stack to a tail call method that can be translated into a while loop? We use the acc, or an accumulator value. Each successive recursive call passes its own result down, thereby eliminating the need to use the stack to hold this result. You might have a couple of things to consider while converting your recursive call to a tail call:

  1. How to restructure the code to “accumulate” results instead of holding it in the stack and using it when the recursive call returns.
  2. In the base condition, where a default value is returned, you will return the “accumulated” value instead.

If the compiler did not perform any optimization, the stack can still blow up, but you can verify that it indeed performs tail call optimization by calling this new factorial method:

  println(factorial(10000))

This prints:

28462596809170545189064132121198688901480514017027992307941799942744113400037644437729907867577847758158840621423175288300423399401535187390524211613827161748198241998275924182892597878981242531205946599625986706560161572036032397926328736717055741975962099479720346153698119897092611277500484198845410475544642442136573303076703628825803548967461117097369578603670191071512730587281041158640561281165385325968425825995584688146430425589836649317059251717204276597407446133400054194052462303436869154059404066227828248371512038322178644627183822923899638992827221879702459387693803094627332292570555459690027875282242544348021127559019169425429028916907219097083690539873747452483372899521802363282741217040268086769210451555840567172555372015852132829034279989818449313610640381489304499621599999359670892980190336998484404665419236258424947163178961192041233108268651071354516845540936033009607210346944377982349430780626069422302681885227592057029230843126188497606560742586279448827155956831533440534425446648416894580425709461673613187605234982286326452921529423479870603344290737158688499178932580691483168854251956006172372636323974420786924642956012306288720122652952964091508301336630982733806353972901506581822574295475894399765113865541208125788683704239208764484761569001264889271590706306409661628038784044485191643790807186112370622133415415065991843875961023926713276546986163657706626438638029848051952769536195259240930908614471907390768585755934786981720734372093104825475628567777694081564074962275254993384112809289637516990219870492405617531786346939798024619737079041868329931016554150742308393176878366923694849025999607729684293977427536263119825416681531891763234839190821000147178932184227805135181734921901146246875769835373441456013122615221391178759688367364087207937002992038279198038702372078039140312368997608152840306051116709484722224870389199993442071395836983063962232079115624044250808919914319837120445598344047556759489212101498152454543594285414390843564419984224855478532163624030098442855331829253154206551237079705816393460296247697010388742206441536626733715428700789122749340684336442889847100840641600093623935261248037975293343928764398316390312776450722479267851700826669598389526150759007349215197592659192708873202594066382118801988854748266048342256457705743973122259700671936061763513579529821794290797705327283267501488024443528681645026165662837546519006171873442260438919298506071515390031106684727360135816706437861756757439184376479658136100599638689552334648781746143243573224864326798481981458432703035895508420534788493364582482592033288089025782388233265770205248970937047210214248413342465268206806732314214483854074182139621846870108359582946965235632764870475718351616879235068366271743711915723361143070121120767608697851559721846485985918643641716850899625516820910793570231118518174775010804622585521314764897490660752877082897667514951009682329689732000622392888056658036140311285465929084078033974900664953205873164948093883816198658850827382468034897864757116679890423568018303504133875731972630897909435710687797301633918087868474943633533893373586906405848417828065196275826434429258058422212947649402948622670761832988229004072390403733168207417413251656688443079339447019208905620788387585342512820957359307018197708340163817638278562539516825426644614941044711579533262372815468794080423718587423026200264221822694188626212107297776657401018376182280136857586442185863011539843712299107010094061929413223202773193959467006713695377097897778118288242442920864816134179562017471831609687661043140497958198236445807368209404022211181530051433387076607063149616107771117448059552764348333385744040212757031851527298377435921878558552795591028664457917362007221858143309977294778923720717942857756271300923982397921957581197264742642878266682353915687857271620146192244266266708400765665625807109474398740110772811669918806268726626565583345665007890309050656074633078027158530817691223772813510584527326591626219647620571434880215630815259005343721141000303039242866457207328473481712034168186328968865048287367933398443971236735084527340196309427697652684170174990756947982757825835229994315633322107439131550124459005324702680312912392297979030417587823398622373535054642646913502503951009239286585108682088070662734733200354995720397086488066040929854607006339409885836349865466136727880748764700702458790118046518296111277090609016152022111461543158317669957060974618085359390400067892878548827850938637353703904049412684618991272871562655001270833039950257879931705431882752659225814948950746639976007316927310831735883056612614782997663188070063044632429112260691931278881566221591523270457695867512821990938942686601963904489718918597472925310322480210543841044325828472830584297804162405108110326914001900568784396341502696521048920272140232160234898588827371428695339681755106287470907473718188014223487248498558198439094651708364368994306189650243288353279667190184527620551085707626204244509623323204744707831190434499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If that isn’t a BigInt, I don’t know what is.

Your IDE might show you a hint that your method is tail recursive:

Tail Call

You can also add an annotation in Scala to ensure that your method is indeed tail recursive:

  @scala.annotation.tailrec

The compiler will complain if the method is not tail recursive.

Conclusion

Tail calls are simple to do. If your method is recursive, try to use an accumulator to pass the results back into the methor itself so that all the branches of your recursive method either end in a call to itself or a value, without requiring any type of calculations after the recursive call.

I hope this was useful!

A shorter, more functional factorial

Can we be more functional and use foldLeft or something to write the factorial method?

  def factorial(n: BigInt): BigInt = (BigInt(1) to n).foldLeft(BigInt(1))(_ * _)

Can I go smaller?

  def factorial(n: BigInt): BigInt = (BigInt(1) to n).product

The foldLeft and product methods are tail-recursion optimized already, so they solve the problem with recursion without leaking their detals to the caller.