src/Pure/General/graph.scala

towards showing " in the tutorial

/*  Title:      Pure/General/graph.scala
    Module:     PIDE
    Author:     Makarius

Directed graphs.
*/

package isabelle


import scala.collection.immutable.{SortedMap, SortedSet}
import scala.annotation.tailrec


object Graph
{
  class Duplicate[Key](x: Key) extends Exception
  class Undefined[Key](x: Key) extends Exception
  class Cycles[Key](cycles: List[List[Key]]) extends Exception

  def empty[Key, A](implicit ord: Ordering[Key]): Graph[Key, A] =
    new Graph[Key, A](SortedMap.empty(ord))

  def string[A]: Graph[String, A] = empty(Ordering.String)
  def int[A]: Graph[Int, A] = empty(Ordering.Int)
  def long[A]: Graph[Long, A] = empty(Ordering.Long)
}


final class Graph[Key, A] private(rep: SortedMap[Key, (A, (SortedSet[Key], SortedSet[Key]))])
{
  type Keys = SortedSet[Key]
  type Entry = (A, (Keys, Keys))

  def ordering: Ordering[Key] = rep.ordering
  def empty_keys: Keys = SortedSet.empty[Key](ordering)


  /* graphs */

  def is_empty: Boolean = rep.isEmpty

  def entries: Iterator[(Key, Entry)] = rep.iterator
  def keys: Iterator[Key] = entries.map(_._1)

  def dest: List[(Key, List[Key])] =
    (for ((x, (_, (_, succs))) <- entries) yield (x, succs.toList)).toList

  override def toString: String =
    dest.map(p => p._1.toString + " -> " + p._2.map(_.toString).mkString("{", ", ", "}"))
      .mkString("Graph(", ", ", ")")

  private def get_entry(x: Key): Entry =
    rep.get(x) match {
      case Some(entry) => entry
      case None => throw new Graph.Undefined(x)
    }

  private def map_entry(x: Key, f: Entry => Entry): Graph[Key, A] =
    new Graph[Key, A](rep + (x -> f(get_entry(x))))


  /* nodes */

  def get_node(x: Key): A = get_entry(x)._1

  def map_node(x: Key, f: A => A): Graph[Key, A] =
    map_entry(x, { case (i, ps) => (f(i), ps) })


  /* reachability */

  /*nodes reachable from xs -- topologically sorted for acyclic graphs*/
  def reachable(next: Key => Keys, xs: List[Key]): (List[List[Key]], Keys) =
  {
    def reach(reached: (List[Key], Keys), x: Key): (List[Key], Keys) =
    {
      val (rs, r_set) = reached
      if (r_set(x)) reached
      else {
        val (rs1, r_set1) = ((rs, r_set + x) /: next(x))(reach)
        (x :: rs1, r_set1)
      }
    }
    def reachs(reached: (List[List[Key]], Keys), x: Key): (List[List[Key]], Keys) =
    {
      val (rss, r_set) = reached
      val (rs, r_set1) = reach((Nil, r_set), x)
      (rs :: rss, r_set1)
    }
    ((List.empty[List[Key]], empty_keys) /: xs)(reachs)
  }

  /*immediate*/
  def imm_preds(x: Key): Keys = get_entry(x)._2._1
  def imm_succs(x: Key): Keys = get_entry(x)._2._2

  /*transitive*/
  def all_preds(xs: List[Key]): List[Key] = reachable(imm_preds, xs)._1.flatten
  def all_succs(xs: List[Key]): List[Key] = reachable(imm_succs, xs)._1.flatten

  /*strongly connected components; see: David King and John Launchbury,
    "Structuring Depth First Search Algorithms in Haskell"*/
  def strong_conn: List[List[Key]] =
    reachable(imm_preds, all_succs(keys.toList))._1.filterNot(_.isEmpty).reverse


  /* minimal and maximal elements */

  def minimals: List[Key] =
    (List.empty[Key] /: rep) {
      case (ms, (m, (_, (preds, _)))) => if (preds.isEmpty) m :: ms else ms }

  def maximals: List[Key] =
    (List.empty[Key] /: rep) {
      case (ms, (m, (_, (_, succs)))) => if (succs.isEmpty) m :: ms else ms }

  def is_minimal(x: Key): Boolean = imm_preds(x).isEmpty
  def is_maximal(x: Key): Boolean = imm_succs(x).isEmpty


  /* node operations */

  def new_node(x: Key, info: A): Graph[Key, A] =
  {
    if (rep.isDefinedAt(x)) throw new Graph.Duplicate(x)
    else new Graph[Key, A](rep + (x -> (info, (empty_keys, empty_keys))))
  }

  def default_node(x: Key, info: A): Graph[Key, A] =
  {
    if (rep.isDefinedAt(x)) this
    else new_node(x, info)
  }

  private def del_adjacent(fst: Boolean, x: Key)(map: SortedMap[Key, Entry], y: Key)
      : SortedMap[Key, Entry] =
    map.get(y) match {
      case None => map
      case Some((i, (preds, succs))) =>
        map + (y -> (i, if (fst) (preds - x, succs) else (preds, succs - x)))
    }

  def del_node(x: Key): Graph[Key, A] =
  {
    val (preds, succs) = get_entry(x)._2
    new Graph[Key, A](
      (((rep - x) /: preds)(del_adjacent(false, x)) /: succs)(del_adjacent(true, x)))
  }

  def restrict(pred: Key => Boolean): Graph[Key, A] =
    (this /: entries){ case (graph, (x, _)) => if (!pred(x)) graph.del_node(x) else graph }


  /* edge operations */

  def is_edge(x: Key, y: Key): Boolean =
    try { imm_succs(x)(y) }
    catch { case _: Graph.Undefined[_] => false }

  def add_edge(x: Key, y: Key): Graph[Key, A] =
    if (is_edge(x, y)) this
    else
      map_entry(y, { case (i, (preds, succs)) => (i, (preds + x, succs)) }).
      map_entry(x, { case (i, (preds, succs)) => (i, (preds, succs + y)) })

  def del_edge(x: Key, y: Key): Graph[Key, A] =
    if (is_edge(x, y))
      map_entry(y, { case (i, (preds, succs)) => (i, (preds - x, succs)) }).
      map_entry(x, { case (i, (preds, succs)) => (i, (preds, succs - y)) })
    else this


  /* irreducible paths -- Hasse diagram */

  private def irreducible_preds(x_set: Keys, path: List[Key], z: Key): List[Key] =
  {
    def red(x: Key)(x1: Key) = is_edge(x, x1) && x1 != z
    @tailrec def irreds(xs0: List[Key], xs1: List[Key]): List[Key] =
      xs0 match {
        case Nil => xs1
        case x :: xs =>
          if (!(x_set(x)) || x == z || path.contains(x) ||
              xs.exists(red(x)) || xs1.exists(red(x)))
            irreds(xs, xs1)
          else irreds(xs, x :: xs1)
      }
    irreds(imm_preds(z).toList, Nil)
  }

  def irreducible_paths(x: Key, y: Key): List[List[Key]] =
  {
    val (_, x_set) = reachable(imm_succs, List(x))
    def paths(path: List[Key])(ps: List[List[Key]], z: Key): List[List[Key]] =
      if (x == z) (z :: path) :: ps
      else (ps /: irreducible_preds(x_set, path, z))(paths(z :: path))
    if ((x == y) && !is_edge(x, x)) List(Nil) else paths(Nil)(Nil, y)
  }


  /* maintain acyclic graphs */

  def add_edge_acyclic(x: Key, y: Key): Graph[Key, A] =
    if (is_edge(x, y)) this
    else {
      irreducible_paths(y, x) match {
        case Nil => add_edge(x, y)
        case cycles => throw new Graph.Cycles(cycles.map(x :: _))
      }
    }

  def add_deps_cyclic(y: Key, xs: List[Key]): Graph[Key, A] =
    (this /: xs)(_.add_edge_acyclic(_, y))

  def topological_order: List[Key] = all_succs(minimals)
}