用于生成图中所有拓扑排序的尾递归算法

Question

给定一个图，我需要生成所有拓扑排序。例如，给定下图：

我想生成所有的拓扑排序，它们是：

2 4 7 5
2 7 4 5
2 4 5 7

因为可能存在很多拓扑排序，所以我需要懒惰地生成它们。目前，我有一个递归的工作实现并在 scala-graph 库之上工作：

import scalax.collection.Graph
import scalax.collection.GraphPredef._
import scalax.collection.GraphEdge._

import scala.collection.mutable.ArrayStack
import scala.collection.Set

def allTopologicalSorts[T](graph: Graph[T, DiEdge]): Stream[List[graph.NodeT]] = {
  val indegree: Map[graph.NodeT, Int] = graph.nodes.map(node => (node, node.inDegree)).toMap

  def isSource(node: graph.NodeT): Boolean = indegree.get(node).get == 0
  def getSources(): Set[graph.NodeT] = graph.nodes.filter(node => isSource(node))

  def processSources(sources: Set[graph.NodeT], indegrees: Map[graph.NodeT, Int], topOrder: List[graph.NodeT], cnt: Int): Stream[List[graph.NodeT]] = {
    if (sources.nonEmpty) {
      // `sources` contain all the nodes we can pick
      // --> generate all possibilities
      sources.toStream.flatMap(src => {
        val newTopOrder = src :: topOrder
        var newSources = sources - src

        // Decrease the in-degree of all adjacent nodes
        var newIndegrees = indegrees
        for (adjacent <- src.diSuccessors) {
          val newIndeg = newIndegrees.get(adjacent).get - 1
          newIndegrees = newIndegrees.updated(adjacent, newIndeg)
          // If in-degree becomes zero, add to sources
          if (newIndeg == 0) {
            newSources = newSources + adjacent
          }
        }

        processSources(newSources, newIndegrees, newTopOrder, cnt + 1)
      })
    }
    else if (cnt != graph.nodes.size) {
      throw new Error("There is a cycle in the graph.")
    }
    else {
      topOrder.reverse #:: Stream.empty[List[graph.NodeT]]
    }
  }

  processSources(getSources(), indegree, List[graph.NodeT](), 0)
}

现在，我可以生成所有（或仅几个）拓扑排序如下：

val graph: Graph[Int, DiEdge] = Graph(2 ~> 4, 2 ~> 7, 4 ~> 5)
allTopologicalSorts(graph) foreach println

如何使算法尾递归但仍然懒惰？

Answer 1

Tail recursive maximum depth method of binary tree in Scala

尝试使用scala.util.control.TailCalls

import scalax.collection.Graph
import scalax.collection.GraphPredef._
import scalax.collection.GraphEdge._

import scala.collection.Set
import scala.util.control.TailCalls.{TailRec, done, tailcall}

import cats.Monad
import cats.instances.stream._
import cats.syntax.traverse._

object App {

  implicit val tailRecMonad: Monad[TailRec] = new Monad[TailRec] {
    override def pure[A](x: A): TailRec[A] = done(x)
    override def flatMap[A, B](fa: TailRec[A])(f: A => TailRec[B]): TailRec[B] = fa.flatMap(f)
    override def tailRecM[A, B](a: A)(f: A => TailRec[Either[A, B]]): TailRec[B] = ???
  }

  def allTopologicalSorts[T](graph: Graph[T, DiEdge]): Stream[List[graph.NodeT]] = {
    val indegree: Map[graph.NodeT, Int] = graph.nodes.map(node => (node, node.inDegree)).toMap

    def isSource(node: graph.NodeT): Boolean = indegree.get(node).get == 0
    def getSources(): Set[graph.NodeT] = graph.nodes.filter(node => isSource(node))

    def processSources(sources: Set[graph.NodeT], indegrees: Map[graph.NodeT, Int], topOrder: List[graph.NodeT], cnt: Int): TailRec[Stream[List[graph.NodeT]]] = {
      if (sources.nonEmpty) {
        // `sources` contain all the nodes we can pick
        // --> generate all possibilities
        sources.toStream.flatTraverse/*flatMap*/(src => {
          val newTopOrder = src :: topOrder
          var newSources = sources - src

          // Decrease the in-degree of all adjacent nodes
          var newIndegrees = indegrees
          for (adjacent <- src.diSuccessors) {
            val newIndeg = newIndegrees.get(adjacent).get - 1
            newIndegrees = newIndegrees.updated(adjacent, newIndeg)
            // If in-degree becomes zero, add to sources
            if (newIndeg == 0) {
              newSources = newSources + adjacent
            }
          }

          tailcall(processSources(newSources, newIndegrees, newTopOrder, cnt + 1))
        })
      }
      else if (cnt != graph.nodes.size) {
        done(throw new Error("There is a cycle in the graph."))
      }
      else {
        done(topOrder.reverse #:: Stream.empty[List[graph.NodeT]])
      }
    }

    processSources(getSources(), indegree, List[graph.NodeT](), 0).result
  }

  def main(args: Array[String]): Unit = {
    val graph: Graph[Int, DiEdge] = Graph(2 ~> 4, 2 ~> 7, 4 ~> 5)
    allTopologicalSorts(graph) foreach println
  }
}

或者您可以使用 cats.free.Trampoline

http://eed3si9n.com/herding-cats/stackless-scala-with-free-monads.html

import scalax.collection.Graph
import scalax.collection.GraphEdge._
import scalax.collection.GraphPredef._

import cats.free.Trampoline
import cats.free.Trampoline.{done, defer}
import cats.instances.stream._
import cats.instances.function._
import cats.syntax.traverse._

import scala.collection.Set

object App {

  def allTopologicalSorts[T](graph: Graph[T, DiEdge]): Stream[List[graph.NodeT]] = {
    val indegree: Map[graph.NodeT, Int] = graph.nodes.map(node => (node, node.inDegree)).toMap

    def isSource(node: graph.NodeT): Boolean = indegree.get(node).get == 0
    def getSources(): Set[graph.NodeT] = graph.nodes.filter(node => isSource(node))

    def processSources(sources: Set[graph.NodeT], indegrees: Map[graph.NodeT, Int], topOrder: List[graph.NodeT], cnt: Int): Trampoline[Stream[List[graph.NodeT]]] = {
      if (sources.nonEmpty) {
        // `sources` contain all the nodes we can pick
        // --> generate all possibilities
        sources.toStream.flatTraverse(src => {
          val newTopOrder = src :: topOrder
          var newSources = sources - src

          // Decrease the in-degree of all adjacent nodes
          var newIndegrees = indegrees
          for (adjacent <- src.diSuccessors) {
            val newIndeg = newIndegrees.get(adjacent).get - 1
            newIndegrees = newIndegrees.updated(adjacent, newIndeg)
            // If in-degree becomes zero, add to sources
            if (newIndeg == 0) {
              newSources = newSources + adjacent
            }
          }

          defer(processSources(newSources, newIndegrees, newTopOrder, cnt + 1))
        })
      }
      else if (cnt != graph.nodes.size) {
        done(throw new Error("There is a cycle in the graph."))
      }
      else {
        done(topOrder.reverse #:: Stream.empty[List[graph.NodeT]])
      }
    }

    processSources(getSources(), indegree, List[graph.NodeT](), 0).run
  }

  def main(args: Array[String]): Unit = {
    val graph: Graph[Int, DiEdge] = Graph(2 ~> 4, 2 ~> 7, 4 ~> 5)
    allTopologicalSorts(graph) foreach println
  }
}

Answer 2

在不炸毁堆栈且不立即计算所有可能性的情况下实现拓扑排序的这种变体一直很痛苦。我最终得到了以下实现：

import scalax.collection.Graph
import scalax.collection.GraphPredef._
import scalax.collection.GraphEdge._
import scala.collection.Set

object test extends App {

  class TopSorter[T](val graph: Graph[T, DiEdge]) extends Iterator[List[T]] {

    final case class State[Node](indegrees: Map[Node, Int], topo: List[Node])

    sealed trait TopoRes
    final case class Res(order: List[graph.NodeT], sorter: Set[State[graph.NodeT]]) extends TopoRes
    final case object Nil extends TopoRes

    private[this] val indegs: Map[graph.NodeT, Int] = graph.nodes.map(node => (node, node.inDegree)).toMap
    private[this] var nextOrder = nextTopo(Set(State(indegs, List[graph.NodeT]())))

    override def hasNext: Boolean = nextOrder.isInstanceOf[Res]

    override def next(): List[T] = nextOrder match {
      case Res(order, sorter) => {
        nextOrder = nextTopo(sorter)
        order.map(_.value)
      }
      case Nil => throw new NoSuchElementException("next on empty iterator")
    }

    private def nextTopo(w: Set[State[graph.NodeT]]): TopoRes = {
      if (w.isEmpty) {
        Nil
      }
      else {
        w.head match {
          case State(indegrees, topo) => {
            val sources = indegrees.keySet.filter(indegrees.get(_).get == 0)
            if (sources.isEmpty) {
              Res(topo.reverse, w.tail) // The result is the order + state to compute the next order
            }
            else {
              sourcesLoop(sources, w.tail, topo, indegrees)
            }
          }
        }
      }
    }

    private def sourcesLoop(sources: Set[graph.NodeT], w: Set[State[graph.NodeT]], topo: List[graph.NodeT], indegrees: Map[graph.NodeT, Int]): TopoRes = {
      if (sources.isEmpty) {
        nextTopo(w)
      }
      else {
        val source = sources.head
        succLoop(source.diSuccessors, indegrees - source, sources, w, source, topo, indegrees)
      }
    }

    private def succLoop(succs: Set[graph.NodeT], indegrees: Map[graph.NodeT, Int], sources: Set[graph.NodeT], w: Set[State[graph.NodeT]], source: graph.NodeT, topo: List[graph.NodeT], oldIndegrees: Map[graph.NodeT, Int]): TopoRes = {
      if (succs.isEmpty) {
        sourcesLoop(sources.tail, w + State(indegrees, source :: topo), topo, oldIndegrees)
      }
      else {
        val succ = succs.head
        succLoop(succs.tail, indegrees.updated(succ, indegrees.get(succ).get - 1), sources, w, source, topo, oldIndegrees)
      }
    }
  }

  val graph: Graph[Int, DiEdge] = Graph(2 ~> 4, 2 ~> 7, 4 ~> 5)
  val it = new TopSorter(graph)

  while (it.hasNext)
    println(it.next())
}

用于生成图中所有拓扑排序的尾递归算法

Tail recursive algorithm for generating all topological orderings in a graph

scala

tail-recursion

graph

topological-sort

scala-cats