Scalaz(23)- 泛函数据结构: Zipper-游标定位

来源:互联网 时间:2016-01-06

  外面沙尘滚滚一直向北去了,意识到年关到了,码农们都回乡过年去了,而我却留在这里玩弄“拉链”。不要想歪了,我说的不是裤裆拉链而是scalaz Zipper,一种泛函数据结构游标(cursor)。在函数式编程模式里的集合通常是不可变的(immutable collection),我们会发现在FP编程过程中处理不可变集合(immutable collection)数据的方式好像总是缺些什么,比如在集合里左右逐步游动像moveNext,movePrev等等,在一个集合的中间进行添加、更新、删除的功能更是欠奉了,这主要是因为操作效率问题。不可变集合只有对前置操作(prepend operation)才能获得可靠的效率,即对集合首位元素的操作,能得到相当于O(1)的速度,其它操作基本上都是O(n)速度,n是集合的长度,也就是随着集合的长度增加,操作效率会以倍数下降。还有一个原因就是编程时会很不方便,因为大多数程序都会对各种集合进行大量的操作,最终也会导致程序的复杂臃肿,不符合函数式编程要求的精简优雅表达形式。我想可能就是因为以上各种原因,scalaz提供了Zipper typeclass帮助对不可变集合操作的编程。Zipper的定义如下:scalaz/Zipper.scala

final case class Zipper[+A](lefts: Stream[A], focus: A, rights: Stream[A])

它以Stream为基础,A可以是任何类型,无论基础类型或高阶类型。Zipper的结构如上:当前焦点窗口、左边一串数据元素、右边一串,形似拉链,因而命名Zipper。或者这样看会更形象一点:

final case class Zipper[+A](

lefts: Stream[A],

focus: A,

rights: Stream[A])

scalaz提供了Zipper构建函数可以直接用Stream生成一个Zipper:

trait StreamFunctions {

...

final def toZipper[A](as: Stream[A]): Option[Zipper[A]] = as match {

case Empty => None

case h #:: t => Some(Zipper.zipper(empty, h, t))

}

final def zipperEnd[A](as: Stream[A]): Option[Zipper[A]] = as match {

case Empty => None

case _ =>

val x = as.reverse

Some(Zipper.zipper(x.tail, x.head, empty))

}

...

zipperEnd生成倒排序的Zipper:

1 Stream(1,2,3).toZipper //> res2: Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 1, <rights>))

2 Stream("A","B","C").toZipper //> res3: Option[scalaz.Zipper[String]] = Some(Zipper(<lefts>, A, <rights>))

3 Stream(Stream(1,2),Stream(3,4)).toZipper //> res4: Option[scalaz.Zipper[scala.collection.immutable.Stream[Int]]] = Some(Z

4 //| ipper(<lefts>, Stream(1, ?), <rights>))

5 Stream(1,2,3).zipperEnd //> res5: Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 3, <rights>))

scalaz也为List,NonEmptyList提供了Zipper构建函数:

trait ListFunctions {

...

final def toZipper[A](as: List[A]): Option[Zipper[A]] =

stream.toZipper(as.toStream)

final def zipperEnd[A](as: List[A]): Option[Zipper[A]] =

stream.zipperEnd(as.toStream)

...

final class NonEmptyList[+A] private[scalaz](val head: A, val tail: List[A]) {

...

def toZipper: Zipper[A] = zipper(Stream.Empty, head, tail.toStream)

def zipperEnd: Zipper[A] = {

import Stream._

tail.reverse match {

case Nil => zipper(empty, head, empty)

case t :: ts => zipper(ts.toStream :+ head, t, empty)

}

}

...

都是先转换成Stream再生成Zipper的。Zipper本身的构建函数是zipper,在NonEmptyList的Zipper生成中调用过:

trait ZipperFunctions {

def zipper[A](ls: Stream[A], a: A, rs: Stream[A]): Zipper[A] =

Zipper(ls, a, rs)

}

用这些串形结构的构建函数产生Zipper同样很简单:

1 List(1,2,3,4).toZipper //> res0: Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 1, <rights>))

2 List(List(1,2),List(2,3)).toZipper //> res1: Option[scalaz.Zipper[List[Int]]] = Some(Zipper(<lefts>, List(1, 2), <r

3 //| ights>))

4 NonEmptyList("A","C","E").toZipper //> res2: scalaz.Zipper[String] = Zipper(<lefts>, A, <rights>)

5 NonEmptyList(1,2,3).zipperEnd //> res3: scalaz.Zipper[Int] = Zipper(<lefts>, 3, <rights>)

6

有了串形集合的Zipper构建方法后我们再看看一下Zipper的左右游动函数:

final case class Zipper[+A](lefts: Stream[A], focus: A, rights: Stream[A]) {

...

/**

* Possibly moves to next element to the right of focus.

*/

def next: Option[Zipper[A]] = rights match {

case Stream.Empty => None

case r #:: rs => Some(zipper(Stream.cons(focus, lefts), r, rs))

}

/**

* Possibly moves to next element to the right of focus.

*/

def nextOr[AA >: A](z: => Zipper[AA]): Zipper[AA] =

next getOrElse z

/**

* Possibly moves to the previous element to the left of focus.

*/

def previous: Option[Zipper[A]] = lefts match {

case Stream.Empty => None

case l #:: ls => Some(zipper(ls, l, Stream.cons(focus, rights)))

}

/**

* Possibly moves to previous element to the left of focus.

*/

def previousOr[AA >: A](z: => Zipper[AA]): Zipper[AA] =

previous getOrElse z

/**

* Moves focus n elements in the zipper, or None if there is no such element.

*

* @param n number of elements to move (positive is forward, negative is backwards)

*/

def move(n: Int): Option[Zipper[A]] = {

@tailrec

def move0(z: Option[Zipper[A]], n: Int): Option[Zipper[A]] =

if (n > 0 && rights.isEmpty || n < 0 && lefts.isEmpty) None

else {

if (n == 0) z

else if (n > 0) move0(z flatMap ((_: Zipper[A]).next), n - 1)

else move0(z flatMap ((_: Zipper[A]).previous), n + 1)

}

move0(Some(this), n)

}

/**

* Moves focus to the start of the zipper.

*/

def start: Zipper[A] = {

val rights = this.lefts.reverse ++ focus #:: this.rights

this.copy(Stream.Empty, rights.head, rights.tail)

}

/**

* Moves focus to the end of the zipper.

*/

def end: Zipper[A] = {

val lefts = this.rights.reverse ++ focus #:: this.lefts

this.copy(lefts.tail, lefts.head, Stream.empty)

}

/**

* Moves focus to the nth element of the zipper, or the default if there is no such element.

*/

def moveOr[AA >: A](n: Int, z: => Zipper[AA]): Zipper[AA] =

move(n) getOrElse z

...

start,end,move,next,previous移动方式都齐了。还有定位函数:

...

/**

* Moves focus to the nearest element matching the given predicate, preferring the left,

* or None if no element matches.

*/

def findZ(p: A => Boolean): Option[Zipper[A]] =

if (p(focus)) Some(this)

else {

val c = this.positions

std.stream.interleave(c.lefts, c.rights).find((x => p(x.focus)))

}

/**

* Moves focus to the nearest element matching the given predicate, preferring the left,

* or the default if no element matches.

*/

def findZor[AA >: A](p: A => Boolean, z: => Zipper[AA]): Zipper[AA] =

findZ(p) getOrElse z

/**

* Given a traversal function, find the first element along the traversal that matches a given predicate.

*/

def findBy[AA >: A](f: Zipper[AA] => Option[Zipper[AA]])(p: AA => Boolean): Option[Zipper[AA]] = {

@tailrec

def go(zopt: Option[Zipper[AA]]): Option[Zipper[AA]] = {

zopt match {

case Some(z) => if (p(z.focus)) Some(z) else go(f(z))

case None => None

}

}

go(f(this))

}

/**

* Moves focus to the nearest element on the right that matches the given predicate,

* or None if there is no such element.

*/

def findNext(p: A => Boolean): Option[Zipper[A]] = findBy((z: Zipper[A]) => z.next)(p)

/**

* Moves focus to the previous element on the left that matches the given predicate,

* or None if there is no such element.

*/

def findPrevious(p: A => Boolean): Option[Zipper[A]] = findBy((z: Zipper[A]) => z.previous)(p)

...

操作函数如下:

...

/**

* An alias for insertRight

*/

def insert[AA >: A]: (AA => Zipper[AA]) = insertRight(_: AA)

/**

* Inserts an element to the left of focus and focuses on the new element.

*/

def insertLeft[AA >: A](y: AA): Zipper[AA] = zipper(lefts, y, focus #:: rights)

/**

* Inserts an element to the right of focus and focuses on the new element.

*/

def insertRight[AA >: A](y: AA): Zipper[AA] = zipper(focus #:: lefts, y, rights)

/**

* An alias for `deleteRight`

*/

def delete: Option[Zipper[A]] = deleteRight

/**

* Deletes the element at focus and moves the focus to the left. If there is no element on the left,

* focus is moved to the right.

*/

def deleteLeft: Option[Zipper[A]] = lefts match {

case l #:: ls => Some(zipper(ls, l, rights))

case Stream.Empty => rights match {

case r #:: rs => Some(zipper(Stream.empty, r, rs))

case Stream.Empty => None

}

}

/**

* Deletes the element at focus and moves the focus to the left. If there is no element on the left,

* focus is moved to the right.

*/

def deleteLeftOr[AA >: A](z: => Zipper[AA]): Zipper[AA] =

deleteLeft getOrElse z

/**

* Deletes the element at focus and moves the focus to the right. If there is no element on the right,

* focus is moved to the left.

*/

def deleteRight: Option[Zipper[A]] = rights match {

case r #:: rs => Some(zipper(lefts, r, rs))

case Stream.Empty => lefts match {

case l #:: ls => Some(zipper(ls, l, Stream.empty))

case Stream.Empty => None

}

}

/**

* Deletes the element at focus and moves the focus to the right. If there is no element on the right,

* focus is moved to the left.

*/

def deleteRightOr[AA >: A](z: => Zipper[AA]): Zipper[AA] =

deleteRight getOrElse z

/**

* Deletes all elements except the focused element.

*/

def deleteOthers: Zipper[A] = zipper(Stream.Empty, focus, Stream.Empty)

...

/**

* Update the focus in this zipper.

*/

def update[AA >: A](focus: AA) = {

this.copy(this.lefts, focus, this.rights)

}

/**

* Apply f to the focus and update with the result.

*/

def modify[AA >: A](f: A => AA) = this.update(f(this.focus))

...

insert,modify,delete也很齐备。值得注意的是多数Zipper的移动函数和操作函数都返回Option[Zipper[A]]类型,如此我们可以用flatMap把这些动作都连接起来。换句话说就是我们可以用for-comprehension在Option的context内实现行令编程(imperative programming)。我们可以通过一些例子来示范Zipper用法:

 1 val zv = for {

2 z <- List(2,8,1,5,4,11).toZipper

3 s1 <- z.next

4 s2 <- s1.modify{_ + 2}.some

5 } yield s2 //> zv : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 10, <rights>))

6

7 zv.get.show //> res8: scalaz.Cord = Zipper(Stream(2), 10, Stream(1,5,4,11))

8 zv.get.toList //> res9: List[Int] = List(2, 10, 1, 5, 4, 11)

9 ...

10 val zv = for {

11 z <- List(2,8,1,5,4,11).toZipper

12 s1 <- z.next

13 s2 <- s1.modify{_ + 2}.some

14 s3 <- s2.move(1)

15 s4 <- s3.delete

16 } yield s4 //> zv : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 5, <rights>))

17

18 zv.get.show //> res8: scalaz.Cord = Zipper(Stream(10,2), 5, Stream(4,11))

19 zv.get.toList //> res9: List[Int] = List(2, 10, 5, 4, 11)

20 ...

21 val zv = for {

22 z <- List(2,8,1,5,4,11).toZipper

23 s1 <- z.next

24 s2 <- s1.modify{_ + 2}.some

25 s3 <- s2.move(1)

26 s4 <- s3.delete

27 s5 <- s4.findZ {_ === 11}

28 s6 <- if (s5.focus === 12) s5.delete else s2.insert(12).some

29 } yield s6 //> zv : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 12, <rights>))

30

31 zv.get.show //> res8: scalaz.Cord = Zipper(Stream(10,2), 12, Stream(1,5,4,11))

32 zv.get.toList //> res9: List[Int] = List(2, 10, 12, 1, 5, 4, 11)

33 ...

34 val zv = for {

35 z <- List(2,8,1,5,4,11).toZipper

36 s1 <- z.next

37 s2 <- s1.modify{_ + 2}.some

38 s3 <- s2.move(1)

39 s4 <- s3.delete

40 s5 <- s4.findZ {_ === 11}

41 s6 <- if (s5.focus === 12) s5.delete else s2.insert(12).some

42 s7 <- s6.end.delete

43 s8 <- s7.start.some

44 } yield s8 //> zv : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 2, <rights>))

45

46 zv.get.show //> res8: scalaz.Cord = Zipper(Stream(), 2, Stream(10,12,1,5,4))

47 zv.get.toList //> res9: List[Int] = List(2, 10, 12, 1, 5, 4)

我在上面的程序里在for{...}yield里面逐条添加指令从而示范游标当前焦点和集合元素跟随着的变化。这段程序可以说就是一段行令程序。回到上面提到的效率和代码质量讨论。我们提过scalaz提供Zipper就是为了使集合操作编程更简明优雅,实际情况是怎样的呢?

举个例子:有一串数字,比如:List(1,4,7,9,5,6,10), 我想找出第一个高点元素,它的左边低,右边高,在我们的例子里是元素9。如果我们尝试用习惯的行令方式用索引去编写这个函数:

def peak(list: List[Int]): Option[Int] = {

list.indices.find { index =>

val x = list(index)

index > 0 && index < list.size - 1 &&

x > list(index - 1) && x > list(index + 1)

}.map(list(_))

}

哇!这东西不但极其复杂难懂而且效率低下,重复用find索引导致速度降到O(n * n)。如果用Array会把效率提高到O(n),不过我们希望用immutable方式。那么用函数式编程方式呢?

def peak_fp(list: List[Int]): Option[Int] = list match {

case x :: y :: z :: tl if y > x && y > z => Some(y)

case x :: tl => peak(tl)

case Nil => None

}

用模式匹配(pattern matching)和递归算法(recursion),这段程序好看多了,而且效率也可以提高到O(n)。

但我们再把情况搞得复杂一点:把高点值增高一点(+1)。还是用FP方式编写:

def raisePeak(list: List[Int]): Option[List[Int]] = {

def rec(head: List[Int], tail: List[Int]): Option[List[Int]] = tail match {

case x :: y :: z :: tl if y > x && y > z =>

Some((x :: head).reverse ::: ((y +1) :: z :: tl))

case x :: tl => rec(x :: head, tl) case Nil => None

}

rec(List.empty, list)

}

代码又变得臃肿复杂起来。看来仅仅用FP编程方式还不足够,还需要用一些新的数据结构什么的来帮助。scalaz的Zipper可以在这个场景里派上用场了:

def raisePeak_z(list: List[Int]): Option[List[Int]] = {

for {

zipper <- list.toZipper

peak <- zipper.positions.findNext( z =>

(z.previous, z.next) match {

case (Some(p), Some(n)) => p.focus < z.focus && n.focus < z.focus

case _ => false

})

} yield (peak.focus.modify(_ + 1).toStream.toList)

}

用Zipper来写程序表达清楚许多。这里用上了Zipper.positions:

/**

* A zipper of all positions of the zipper, with focus on the current position.

*/

def positions: Zipper[Zipper[A]] = {

val left = std.stream.unfold(this)(_.previous.map(x => (x, x)))

val right = std.stream.unfold(this)(_.next.map(x => (x, x)))

zipper(left, this, right)

}

positions函数返回类型是Zipper[Zipper[A]]符合findNext使用。我们前面已经提到:使用Zipper的成本约为O(n)。

 

 

 

 

 

 

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