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Scalaz（23）－ 泛函数据结构： Zipper-游标定位

定位数据结构 23 游标 Scalaz 泛函
2023-09-14 08:57:17 时间

  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 ))

 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 ))

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 ))

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 ))

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)。



C语言|数据结构——树的定义、存储与遍历 基本概念 1.有且只有一个称为根的节点； 2.有若干个互不相交的子树，这些子树本身也是一棵树； 3.由节点和边组组成的； 4.每个节点只有一个父节点，可以有无数个子节点（除了根节点）。 |一般树。任意一个子节点个数不受限制，可以是有序树也可以是无序树。 |二叉树。任意一个节点最大度为2，二叉树是有序树，左右节点不能随意互换。 | 一般二叉树 |满二叉树。每一层节点都是满的。 |完全二叉树。除最后一层外，每一层节点都是满的，最后一层节点一定从左向右连续排列。 |森林。n个互不相交的树的集合，可以是互不相连的几个树 一些专业术语：

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