Scalaz（23）－ 泛函数据结构： Zipper-游标定位

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:

   Stream(,,).toZipper                          //> res2: Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 1, <rights>))
   Stream("A","B","C").toZipper                    //> res3: Option[scalaz.Zipper[String]] = Some(Zipper(<lefts>, A, <rights>))
   Stream(Stream(,),Stream(,)).toZipper        //> res4: Option[scalaz.Zipper[scala.collection.immutable.Stream[Int]]] = Some(Z
                                                   //| ipper(<lefts>, Stream(1, ?), <rights>))
   Stream(,,).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：

 List(,,,).toZipper                          //> res0: Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 1, <rights>))
   List(List(,),List(,)).toZipper              //> res1: Option[scalaz.Zipper[List[Int]]] = Some(Zipper(<lefts>, List(1, 2), <r
                                                   //| ights>))
   NonEmptyList("A","C","E").toZipper              //> res2: scalaz.Zipper[String] = Zipper(<lefts>, A, <rights>)
   NonEmptyList(,,).zipperEnd                   //> res3: scalaz.Zipper[Int] = Zipper(<lefts>, 3, <rights>)
  

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 >  && rights.isEmpty || n <  && lefts.isEmpty) None
      else {
        if (n == ) z
        else if (n > ) move0(z flatMap ((_: Zipper[A]).next), n - )
        else move0(z flatMap ((_: Zipper[A]).previous), n + )
      }
    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：

 val zv = for {
     z <- List(,,,,,).toZipper
     s1 <- z.next
     s2 <- s1.modify{_ + }.some
   } yield s2                                      //> zv  : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 10, <rights>))

   zv.get.show          //> res8: scalaz.Cord = Zipper(Stream(2), 10, Stream(1,5,4,11))
   zv.get.toList        //> res9: List[Int] = List(2, 10, 1, 5, 4, 11)
 ...
 val zv = for {
     z <- List(,,,,,).toZipper
     s1 <- z.next
     s2 <- s1.modify{_ + }.some
     s3 <- s2.move()
     s4 <- s3.delete
   } yield s4                                      //> zv  : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 5, <rights>))

   zv.get.show       //> res8: scalaz.Cord = Zipper(Stream(10,2), 5, Stream(4,11))
   zv.get.toList     //> res9: List[Int] = List(2, 10, 5, 4, 11)
 ...
 val zv = for {
     z <- List(,,,,,).toZipper
     s1 <- z.next
     s2 <- s1.modify{_ + }.some
     s3 <- s2.move()
     s4 <- s3.delete
     s5 <- s4.findZ {_ === }
     s6 <- if (s5.focus === ) s5.delete else s2.insert().some
   } yield s6                                      //> zv  : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 12, <rights>))

   zv.get.show        //> res8: scalaz.Cord = Zipper(Stream(10,2), 12, Stream(1,5,4,11))
   zv.get.toList      //> res9: List[Int] = List(2, 10, 12, 1, 5, 4, 11)
 ...
 val zv = for {
     z <- List(,,,,,).toZipper
     s1 <- z.next
     s2 <- s1.modify{_ + }.some
     s3 <- s2.move()
     s4 <- s3.delete
     s5 <- s4.findZ {_ === }
     s6 <- if (s5.focus === ) s5.delete else s2.insert().some
     s7 <- s6.end.delete
     s8 <- s7.start.some
   } yield s8                                      //> zv  : Option[scalaz.Zipper[Int]] = Some(Zipper(<lefts>, 2, <rights>))

   zv.get.show         //> res8: scalaz.Cord = Zipper(Stream(), 2, Stream(10,12,1,5,4))
   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 >  && index < list.size -  &&
    x > list(index - ) && x > list(index + )
  }.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 +) :: 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(_ + ).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)。