问题
I've wrote a naïve test-bed to measure the performance of three kinds of factorial implementation: loop based, non tail-recursive and tail-recursive.
Surprisingly to me the worst performant was the loop ones («while» was expected to be more efficient so I provided both) that cost almost twice than the tail recursive alternative.
ANSWER: fixing the loop implementation avoiding the *= operator wich outperform worst with BigInt due to its internals «loops» became fastest as expected
Another «woodoo» behaviour I've experienced was the StackOverflow exception which wasn't thrown sistematically for the same input in the case of non-tail recursive implementation. I can circumvent the StackOverlow by progressively call the function with larger and larger values… I feel crazy :) Answer: JVM require to converge during startup, then behaviour is coherent and systematic
This is the code:
final object Factorial {
type Out = BigInt
def calculateByRecursion(n: Int): Out = {
require(n>0, "n must be positive")
n match {
case _ if n == 1 => return 1
case _ => return n * calculateByRecursion(n-1)
}
}
def calculateByForLoop(n: Int): Out = {
require(n>0, "n must be positive")
var accumulator: Out = 1
for (i <- 1 to n)
accumulator = i * accumulator
accumulator
}
def calculateByWhileLoop(n: Int): Out = {
require(n>0, "n must be positive")
var accumulator: Out = 1
var i = 1
while (i <= n) {
accumulator = i * accumulator
i += 1
}
accumulator
}
def calculateByTailRecursion(n: Int): Out = {
require(n>0, "n must be positive")
@tailrec def fac(n: Int, acc: Out): Out = n match {
case _ if n == 1 => acc
case _ => fac(n-1, n * acc)
}
fac(n, 1)
}
def calculateByTailRecursionUpward(n: Int): Out = {
require(n>0, "n must be positive")
@tailrec def fac(i: Int, acc: Out): Out = n match {
case _ if i == n => n * acc
case _ => fac(i+1, i * acc)
}
fac(1, 1)
}
def comparePerformance(n: Int) {
def showOutput[A](msg: String, data: (Long, A), showOutput:Boolean = false) =
showOutput match {
case true => printf("%s returned %s in %d ms\n", msg, data._2.toString, data._1)
case false => printf("%s in %d ms\n", msg, data._1)
}
def measure[A](f:()=>A): (Long, A) = {
val start = System.currentTimeMillis
val o = f()
(System.currentTimeMillis - start, o)
}
showOutput ("By for loop", measure(()=>calculateByForLoop(n)))
showOutput ("By while loop", measure(()=>calculateByWhileLoop(n)))
showOutput ("By non-tail recursion", measure(()=>calculateByRecursion(n)))
showOutput ("By tail recursion", measure(()=>calculateByTailRecursion(n)))
showOutput ("By tail recursion upward", measure(()=>calculateByTailRecursionUpward(n)))
}
}
What follows is some output from sbt console (Before «while» implementation):
scala> example.Factorial.comparePerformance(10000)
By loop in 3 ns
By non-tail recursion in >>>>> StackOverflow!!!!!… see later!!!
........
scala> example.Factorial.comparePerformance(1000)
By loop in 3 ms
By non-tail recursion in 1 ms
By tail recursion in 4 ms
scala> example.Factorial.comparePerformance(5000)
By loop in 105 ms
By non-tail recursion in 27 ms
By tail recursion in 34 ms
scala> example.Factorial.comparePerformance(10000)
By loop in 236 ms
By non-tail recursion in 106 ms >>>> Now works!!!
By tail recursion in 127 ms
scala> example.Factorial.comparePerformance(20000)
By loop in 977 ms
By non-tail recursion in 495 ms
By tail recursion in 564 ms
scala> example.Factorial.comparePerformance(30000)
By loop in 2285 ms
By non-tail recursion in 1183 ms
By tail recursion in 1281 ms
What follows is some output from sbt console (After «while» implementation):
scala> example.Factorial.comparePerformance(10000)
By for loop in 252 ms
By while loop in 246 ms
By non-tail recursion in 130 ms
By tail recursion in 136 ns
scala> example.Factorial.comparePerformance(20000)
By for loop in 984 ms
By while loop in 1091 ms
By non-tail recursion in 508 ms
By tail recursion in 560 ms
What follows is some output from sbt console (after «upward» tail recursion implementation) the world come back sane:
scala> example.Factorial.comparePerformance(10000)
By for loop in 259 ms
By while loop in 229 ms
By non-tail recursion in 114 ms
By tail recursion in 119 ms
By tail recursion upward in 105 ms
scala> example.Factorial.comparePerformance(20000)
By for loop in 1053 ms
By while loop in 957 ms
By non-tail recursion in 513 ms
By tail recursion in 565 ms
By tail recursion upward in 470 ms
What follows is some output from sbt console after fixing BigInt multiplication in «loops»: the world is totally sane:
scala> example.Factorial.comparePerformance(20000)
By for loop in 498 ms
By while loop in 502 ms
By non-tail recursion in 521 ms
By tail recursion in 611 ms
By tail recursion upward in 503 ms
BigInt overhead and a stupid implementation by me masked the expected behaviour.
Thanks guys
PS.: In the end I should re-title this post to «A lernt lesson on BigInts»
回答1:
For loops are not actually quite loops; they're for comprehensions on a range. If you actually want a loop, you need to use while. (Actually, I think the BigInt multiplication here is heavyweight enough so it shouldn't matter. But you'll notice if you're multiplying Ints.)
Also, you have confused yourself by using BigInt. The bigger your BigInt is, the slower your multiplication. So your non-tail loop counts up while your tail recursion loop counds down which means that the latter has more big numbers to multiply.
If you fix these two issues you will find that sanity is restored: loops and tail recursion are the same speed, with both regular recursion and for slower. (Regular recursion may not be slower if the JVM optimization makes it equivalent)
(Also, the stack overflow fix is probably because the JVM starts inlining and may either make the call tail-recursive itself, or unrolls the loop far enough so that you don't overflow any longer.)
Finally, you're getting poor results with for and while because you're multiplying on the right rather than the left with the small number. It turns out that the Java's BigInt multiplies faster with the smaller number on the left.
回答2:
Scala static methods for factorial(n) (coded with scala 2.12.x, java-8):
object Factorial {
/*
* For large N, it throws a stack overflow
*/
def recursive(n:BigInt): BigInt = {
if(n < 0) {
throw new ArithmeticException
} else if(n <= 1) {
1
} else {
n * recursive(n - 1)
}
}
/*
* A tail recursive method is compiled to avoid stack overflow
*/
@scala.annotation.tailrec
def recursiveTail(n:BigInt, acc:BigInt = 1): BigInt = {
if(n < 0) {
throw new ArithmeticException
} else if(n <= 1) {
acc
} else {
recursiveTail(n - 1, n * acc)
}
}
/*
* A while loop
*/
def loop(n:BigInt): BigInt = {
if(n < 0) {
throw new ArithmeticException
} else if(n <= 1) {
1
} else {
var acc = 1
var idx = 1
while(idx <= n) {
acc = idx * acc
idx += 1
}
acc
}
}
}
Specs:
class FactorialSpecs extends SpecHelper {
private val smallInt = 10
private val largeInt = 10000
describe("Factorial.recursive") {
it("return 1 for 0") {
assert(Factorial.recursive(0) == 1)
}
it("return 1 for 1") {
assert(Factorial.recursive(1) == 1)
}
it("return 2 for 2") {
assert(Factorial.recursive(2) == 2)
}
it("returns a result, for small inputs") {
assert(Factorial.recursive(smallInt) == 3628800)
}
it("throws StackOverflow for large inputs") {
intercept[java.lang.StackOverflowError] {
Factorial.recursive(Int.MaxValue)
}
}
}
describe("Factorial.recursiveTail") {
it("return 1 for 0") {
assert(Factorial.recursiveTail(0) == 1)
}
it("return 1 for 1") {
assert(Factorial.recursiveTail(1) == 1)
}
it("return 2 for 2") {
assert(Factorial.recursiveTail(2) == 2)
}
it("returns a result, for small inputs") {
assert(Factorial.recursiveTail(smallInt) == 3628800)
}
it("returns a result, for large inputs") {
assert(Factorial.recursiveTail(largeInt).isInstanceOf[BigInt])
}
}
describe("Factorial.loop") {
it("return 1 for 0") {
assert(Factorial.loop(0) == 1)
}
it("return 1 for 1") {
assert(Factorial.loop(1) == 1)
}
it("return 2 for 2") {
assert(Factorial.loop(2) == 2)
}
it("returns a result, for small inputs") {
assert(Factorial.loop(smallInt) == 3628800)
}
it("returns a result, for large inputs") {
assert(Factorial.loop(largeInt).isInstanceOf[BigInt])
}
}
}
Benchmarks:
import org.scalameter.api._
class BenchmarkFactorials extends Bench.OfflineReport {
val gen: Gen[Int] = Gen.range("N")(1, 1000, 100) // scalastyle:ignore
performance of "Factorial" in {
measure method "loop" in {
using(gen) in {
n => Factorial.loop(n)
}
}
measure method "recursive" in {
using(gen) in {
n => Factorial.recursive(n)
}
}
measure method "recursiveTail" in {
using(gen) in {
n => Factorial.recursiveTail(n)
}
}
}
}
Benchmark results (loop is much faster):
[info] Test group: Factorial.loop
[info] - Factorial.loop.Test-9 measurements:
[info] - at N -> 1: passed
[info] (mean = 0.01 ms, ci = <0.00 ms, 0.02 ms>, significance = 1.0E-10)
[info] - at N -> 101: passed
[info] (mean = 0.01 ms, ci = <0.01 ms, 0.01 ms>, significance = 1.0E-10)
[info] - at N -> 201: passed
[info] (mean = 0.02 ms, ci = <0.02 ms, 0.02 ms>, significance = 1.0E-10)
[info] - at N -> 301: passed
[info] (mean = 0.03 ms, ci = <0.02 ms, 0.03 ms>, significance = 1.0E-10)
[info] - at N -> 401: passed
[info] (mean = 0.03 ms, ci = <0.03 ms, 0.04 ms>, significance = 1.0E-10)
[info] - at N -> 501: passed
[info] (mean = 0.04 ms, ci = <0.03 ms, 0.05 ms>, significance = 1.0E-10)
[info] - at N -> 601: passed
[info] (mean = 0.04 ms, ci = <0.04 ms, 0.05 ms>, significance = 1.0E-10)
[info] - at N -> 701: passed
[info] (mean = 0.05 ms, ci = <0.05 ms, 0.05 ms>, significance = 1.0E-10)
[info] - at N -> 801: passed
[info] (mean = 0.06 ms, ci = <0.05 ms, 0.06 ms>, significance = 1.0E-10)
[info] - at N -> 901: passed
[info] (mean = 0.06 ms, ci = <0.05 ms, 0.07 ms>, significance = 1.0E-10)
[info] Test group: Factorial.recursive
[info] - Factorial.recursive.Test-10 measurements:
[info] - at N -> 1: passed
[info] (mean = 0.00 ms, ci = <0.00 ms, 0.01 ms>, significance = 1.0E-10)
[info] - at N -> 101: passed
[info] (mean = 0.05 ms, ci = <0.01 ms, 0.09 ms>, significance = 1.0E-10)
[info] - at N -> 201: passed
[info] (mean = 0.03 ms, ci = <0.02 ms, 0.05 ms>, significance = 1.0E-10)
[info] - at N -> 301: passed
[info] (mean = 0.07 ms, ci = <0.00 ms, 0.13 ms>, significance = 1.0E-10)
[info] - at N -> 401: passed
[info] (mean = 0.09 ms, ci = <0.01 ms, 0.18 ms>, significance = 1.0E-10)
[info] - at N -> 501: passed
[info] (mean = 0.10 ms, ci = <0.03 ms, 0.17 ms>, significance = 1.0E-10)
[info] - at N -> 601: passed
[info] (mean = 0.11 ms, ci = <0.08 ms, 0.15 ms>, significance = 1.0E-10)
[info] - at N -> 701: passed
[info] (mean = 0.13 ms, ci = <0.11 ms, 0.14 ms>, significance = 1.0E-10)
[info] - at N -> 801: passed
[info] (mean = 0.16 ms, ci = <0.13 ms, 0.19 ms>, significance = 1.0E-10)
[info] - at N -> 901: passed
[info] (mean = 0.21 ms, ci = <0.15 ms, 0.27 ms>, significance = 1.0E-10)
[info] Test group: Factorial.recursiveTail
[info] - Factorial.recursiveTail.Test-11 measurements:
[info] - at N -> 1: passed
[info] (mean = 0.00 ms, ci = <0.00 ms, 0.01 ms>, significance = 1.0E-10)
[info] - at N -> 101: passed
[info] (mean = 0.04 ms, ci = <0.03 ms, 0.05 ms>, significance = 1.0E-10)
[info] - at N -> 201: passed
[info] (mean = 0.12 ms, ci = <0.05 ms, 0.20 ms>, significance = 1.0E-10)
[info] - at N -> 301: passed
[info] (mean = 0.16 ms, ci = <-0.03 ms, 0.34 ms>, significance = 1.0E-10)
[info] - at N -> 401: passed
[info] (mean = 0.12 ms, ci = <0.09 ms, 0.16 ms>, significance = 1.0E-10)
[info] - at N -> 501: passed
[info] (mean = 0.17 ms, ci = <0.15 ms, 0.19 ms>, significance = 1.0E-10)
[info] - at N -> 601: passed
[info] (mean = 0.23 ms, ci = <0.19 ms, 0.26 ms>, significance = 1.0E-10)
[info] - at N -> 701: passed
[info] (mean = 0.25 ms, ci = <0.18 ms, 0.32 ms>, significance = 1.0E-10)
[info] - at N -> 801: passed
[info] (mean = 0.28 ms, ci = <0.21 ms, 0.36 ms>, significance = 1.0E-10)
[info] - at N -> 901: passed
[info] (mean = 0.32 ms, ci = <0.17 ms, 0.46 ms>, significance = 1.0E-10)
回答3:
I know everyone already answered the question, but I thought that I might add this one optimization: If you convert the pattern matching to simple if-statements, it can speed up the tail recursion.
final object Factorial {
type Out = BigInt
def calculateByRecursion(n: Int): Out = {
require(n>0, "n must be positive")
n match {
case _ if n == 1 => return 1
case _ => return n * calculateByRecursion(n-1)
}
}
def calculateByForLoop(n: Int): Out = {
require(n>0, "n must be positive")
var accumulator: Out = 1
for (i <- 1 to n)
accumulator = i * accumulator
accumulator
}
def calculateByWhileLoop(n: Int): Out = {
require(n>0, "n must be positive")
var acc: Out = 1
var i = 1
while (i <= n) {
acc = i * acc
i += 1
}
acc
}
def calculateByTailRecursion(n: Int): Out = {
require(n>0, "n must be positive")
@annotation.tailrec
def fac(n: Int, acc: Out): Out = if (n==1) acc else fac(n-1, n*acc)
fac(n, 1)
}
def calculateByTailRecursionUpward(n: Int): Out = {
require(n>0, "n must be positive")
@annotation.tailrec
def fac(i: Int, acc: Out): Out = if (i == n) n*acc else fac(i+1, i*acc)
fac(1, 1)
}
def attempt(f: ()=>Unit): Boolean = {
try {
f()
true
} catch {
case _: Throwable =>
println(" <<<<< Failed...")
false
}
}
def comparePerformance(n: Int) {
def showOutput[A](msg: String, data: (Long, A), showOutput:Boolean = true) =
showOutput match {
case true =>
val res = data._2.toString
val pref = res.substring(0,5)
val midd = res.substring((res.length-5)/ 2, (res.length-5)/ 2 + 10)
val suff = res.substring(res.length-5)
printf("%s returned %s in %d ms\n", msg, s"$pref...$midd...$suff" , data._1)
case false =>
printf("%s in %d ms\n", msg, data._1)
}
def measure[A](f:()=>A): (Long, A) = {
val start = System.currentTimeMillis
val o = f()
(System.currentTimeMillis - start, o)
}
attempt(() => showOutput ("By for loop", measure(()=>calculateByForLoop(n))))
attempt(() => showOutput ("By while loop", measure(()=>calculateByWhileLoop(n))))
attempt(() => showOutput ("By non-tail recursion", measure(()=>calculateByRecursion(n))))
attempt(() => showOutput ("By tail recursion", measure(()=>calculateByTailRecursion(n))))
attempt(() => showOutput ("By tail recursion upward", measure(()=>calculateByTailRecursionUpward(n))))
}
}
My results:
scala> Factorial.comparePerformance(20000)
By for loop returned 18192...5708616582...00000 in 179 ms
By while loop returned 18192...5708616582...00000 in 159 ms
By non-tail recursion <<<<< Failed...
By tail recursion returned 18192...5708616582...00000 in 169 ms
By tail recursion upward returned 18192...5708616582...00000 in 174 ms
By for loop returned 18192...5708616582...00000 in 212 ms
By while loop returned 18192...5708616582...00000 in 156 ms
By non-tail recursion returned 18192...5708616582...00000 in 155 ms
By tail recursion returned 18192...5708616582...00000 in 166 ms
By tail recursion upward returned 18192...5708616582...00000 in 137 ms
scala> Factorial.comparePerformance(200000)
By for loop returned 14202...0169293868...00000 in 17467 ms
By while loop returned 14202...0169293868...00000 in 17303 ms
By non-tail recursion <<<<< Failed...
By tail recursion returned 14202...0169293868...00000 in 18477 ms
By tail recursion upward returned 14202...0169293868...00000 in 17188 ms
来源：https://stackoverflow.com/questions/15138624/scala-recursion-vs-loop-performance-and-runtime-considerations