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// Rustan Leino
// 7 November 2008
// Schorr-Waite and other marking algorithms, written and verified in Dafny.
// Copyright (c) 2008, Microsoft.
class Node {
var children: seq<Node>;
var marked: bool;
var childrenVisited: int;
ghost var pathFromRoot: Path;
}
class Main {
method RecursiveMark(root: Node, ghost S: set<Node>)
requires root in S;
// S is closed under 'children':
requires (forall n :: n in S ==> n != null &&
(forall ch :: ch in n.children ==> ch == null || ch in S));
requires (forall n :: n in S ==> ! n.marked && n.childrenVisited == 0);
modifies S;
ensures root.marked;
// nodes reachable from 'root' are marked:
ensures (forall n :: n in S && n.marked ==>
(forall ch :: ch in n.children && ch != null ==> ch.marked));
ensures (forall n :: n in S ==>
n.childrenVisited == old(n.childrenVisited) &&
n.children == old(n.children));
{
RecursiveMarkWorker(root, S, {});
}
method RecursiveMarkWorker(root: Node, ghost S: set<Node>, ghost stackNodes: set<Node>)
requires root != null && root in S;
requires (forall n :: n in S ==> n != null &&
(forall ch :: ch in n.children ==> ch == null || ch in S));
requires (forall n :: n in S && n.marked ==>
n in stackNodes ||
(forall ch :: ch in n.children && ch != null ==> ch.marked));
requires (forall n :: n in stackNodes ==> n != null && n.marked);
modifies S;
ensures root.marked;
// nodes reachable from 'root' are marked:
ensures (forall n :: n in S && n.marked ==>
n in stackNodes ||
(forall ch :: ch in n.children && ch != null ==> ch.marked));
ensures (forall n: Node :: n in S && old(n.marked) ==> n.marked);
ensures (forall n :: n in S ==>
n.childrenVisited == old(n.childrenVisited) &&
n.children == old(n.children));
decreases S - stackNodes;
{
if (! root.marked) {
root.marked := true;
var i := 0;
while (i < |root.children|)
invariant root.marked && i <= |root.children|;
invariant (forall n :: n in S && n.marked ==>
n == root ||
n in stackNodes ||
(forall ch :: ch in n.children && ch != null ==> ch.marked));
invariant (forall j :: 0 <= j && j < i ==>
root.children[j] == null || root.children[j].marked);
invariant (forall n: Node :: n in S && old(n.marked) ==> n.marked);
invariant (forall n :: n in S ==>
n.childrenVisited == old(n.childrenVisited) &&
n.children == old(n.children));
{
var c := root.children[i];
if (c != null) {
RecursiveMarkWorker(c, S, stackNodes + {root});
}
i := i + 1;
}
}
}
// ---------------------------------------------------------------------------------
method IterativeMark(root: Node, ghost S: set<Node>)
requires root in S;
// S is closed under 'children':
requires (forall n :: n in S ==> n != null &&
(forall ch :: ch in n.children ==> ch == null || ch in S));
requires (forall n :: n in S ==> ! n.marked && n.childrenVisited == 0);
modifies S;
ensures root.marked;
// nodes reachable from 'root' are marked:
ensures (forall n :: n in S && n.marked ==>
(forall ch :: ch in n.children && ch != null ==> ch.marked));
ensures (forall n :: n in S ==>
n.childrenVisited == old(n.childrenVisited) &&
n.children == old(n.children));
{
var t := root;
t.marked := true;
var stackNodes := [];
ghost var unmarkedNodes := S - {t};
while (true)
invariant root.marked && t in S && t !in stackNodes;
// stackNodes has no duplicates:
invariant (forall i, j :: 0 <= i && i < j && j < |stackNodes| ==>
stackNodes[i] != stackNodes[j]);
invariant (forall n :: n in stackNodes ==> n in S);
invariant (forall n :: n in stackNodes || n == t ==>
n.marked &&
0 <= n.childrenVisited && n.childrenVisited <= |n.children| &&
(forall j :: 0 <= j && j < n.childrenVisited ==>
n.children[j] == null || n.children[j].marked));
invariant (forall n :: n in stackNodes ==> n.childrenVisited < |n.children|);
// nodes on the stack are linked:
invariant (forall j :: 0 <= j && j+1 < |stackNodes| ==>
stackNodes[j].children[stackNodes[j].childrenVisited] == stackNodes[j+1]);
invariant 0 < |stackNodes| ==>
stackNodes[|stackNodes|-1].children[stackNodes[|stackNodes|-1].childrenVisited] == t;
invariant (forall n :: n in S && n.marked && n !in stackNodes && n != t ==>
(forall ch :: ch in n.children && ch != null ==> ch.marked));
invariant (forall n :: n in S && n !in stackNodes && n != t ==>
n.childrenVisited == old(n.childrenVisited));
invariant (forall n: Node :: n in S ==> n.children == old(n.children));
invariant (forall n :: n in S && !n.marked ==> n in unmarkedNodes);
decreases unmarkedNodes, stackNodes, |t.children| - t.childrenVisited;
{
if (t.childrenVisited == |t.children|) {
// pop
t.childrenVisited := 0;
if (|stackNodes| == 0) {
return;
}
t := stackNodes[|stackNodes| - 1];
stackNodes := stackNodes[..|stackNodes| - 1];
t.childrenVisited := t.childrenVisited + 1;
} else if (t.children[t.childrenVisited] == null || t.children[t.childrenVisited].marked) {
// just advance to next child
t.childrenVisited := t.childrenVisited + 1;
} else {
// push
stackNodes := stackNodes + [t];
t := t.children[t.childrenVisited];
t.marked := true;
unmarkedNodes := unmarkedNodes - {t};
}
}
}
// ---------------------------------------------------------------------------------
function Reachable(from: Node, to: Node, S: set<Node>): bool
requires null !in S;
reads S;
{
(exists via: Path :: ReachableVia(from, via, to, S))
}
function ReachableVia(from: Node, via: Path, to: Node, S: set<Node>): bool
requires null !in S;
reads S;
decreases via;
{
match via
case Empty => from == to
case Extend(prefix, n) => n in S && to in n.children && ReachableVia(from, prefix, n, S)
}
method SchorrWaite(root: Node, ghost S: set<Node>)
requires root in S;
// S is closed under 'children':
requires (forall n :: n in S ==> n != null &&
(forall ch :: ch in n.children ==> ch == null || ch in S));
// the graph starts off with nothing marked and nothing being indicated as currently being visited:
requires (forall n :: n in S ==> ! n.marked && n.childrenVisited == 0);
modifies S;
// nodes reachable from 'root' are marked:
ensures root.marked;
ensures (forall n :: n in S && n.marked ==>
(forall ch :: ch in n.children && ch != null ==> ch.marked));
// every marked node was reachable from 'root' in the pre-state:
ensures (forall n :: n in S && n.marked ==> old(Reachable(root, n, S)));
// the structure of the graph has not changed:
ensures (forall n :: n in S ==>
n.childrenVisited == old(n.childrenVisited) &&
n.children == old(n.children));
{
var t := root;
var p: Node := null; // parent of t in original graph
ghost var path := Path.Empty;
t.marked := true;
t.pathFromRoot := path;
ghost var stackNodes := [];
ghost var unmarkedNodes := S - {t};
while (true)
invariant root.marked && t != null && t in S && t !in stackNodes;
invariant |stackNodes| == 0 <==> p == null;
invariant 0 < |stackNodes| ==> p == stackNodes[|stackNodes|-1];
// stackNodes has no duplicates:
invariant (forall i, j :: 0 <= i && i < j && j < |stackNodes| ==>
stackNodes[i] != stackNodes[j]);
invariant (forall n :: n in stackNodes ==> n in S);
invariant (forall n :: n in stackNodes || n == t ==>
n.marked &&
0 <= n.childrenVisited && n.childrenVisited <= |n.children| &&
(forall j :: 0 <= j && j < n.childrenVisited ==>
n.children[j] == null || n.children[j].marked));
invariant (forall n :: n in stackNodes ==> n.childrenVisited < |n.children|);
invariant (forall n :: n in S && n.marked && n !in stackNodes && n != t ==>
(forall ch :: ch in n.children && ch != null ==> ch.marked));
invariant (forall n :: n in S && n !in stackNodes && n != t ==>
n.childrenVisited == old(n.childrenVisited));
invariant (forall n :: n in S ==> n in stackNodes || n.children == old(n.children));
invariant (forall n :: n in stackNodes ==>
|n.children| == old(|n.children|) &&
(forall j :: 0 <= j && j < |n.children| ==>
j == n.childrenVisited || n.children[j] == old(n.children[j])));
// every marked node is reachable:
invariant old(allocated(path)); // needed to show 'path' worthy as argument to old(Reachable(...))
invariant old(ReachableVia(root, path, t, S));
invariant (forall n, pth :: n in S && n.marked && pth == n.pathFromRoot ==> old(allocated(pth)));
invariant (forall n, pth :: n in S && n.marked && pth == n.pathFromRoot ==>
old(ReachableVia(root, pth, n, S)));
invariant (forall n :: n in S && n.marked ==> old(Reachable(root, n, S)));
// the current values of m.children[m.childrenVisited] for m's on the stack:
invariant 0 < |stackNodes| ==> stackNodes[0].children[stackNodes[0].childrenVisited] == null;
invariant (forall k :: 0 < k && k < |stackNodes| ==>
stackNodes[k].children[stackNodes[k].childrenVisited] == stackNodes[k-1]);
// the original values of m.children[m.childrenVisited] for m's on the stack:
invariant (forall k :: 0 <= k && k+1 < |stackNodes| ==>
old(stackNodes[k].children)[stackNodes[k].childrenVisited] == stackNodes[k+1]);
invariant 0 < |stackNodes| ==>
old(stackNodes[|stackNodes|-1].children)[stackNodes[|stackNodes|-1].childrenVisited] == t;
invariant (forall n :: n in S && !n.marked ==> n in unmarkedNodes);
decreases unmarkedNodes, stackNodes, |t.children| - t.childrenVisited;
{
if (t.childrenVisited == |t.children|) {
// pop
t.childrenVisited := 0;
if (p == null) {
return;
}
var oldP := p.children[p.childrenVisited];
// p.children[p.childrenVisited] := t;
p.children := p.children[..p.childrenVisited] + [t] + p.children[p.childrenVisited + 1..];
t := p;
p := oldP;
stackNodes := stackNodes[..|stackNodes| - 1];
t.childrenVisited := t.childrenVisited + 1;
path := t.pathFromRoot;
} else if (t.children[t.childrenVisited] == null || t.children[t.childrenVisited].marked) {
// just advance to next child
t.childrenVisited := t.childrenVisited + 1;
} else {
// push
var newT := t.children[t.childrenVisited];
// t.children[t.childrenVisited] := p;
t.children := t.children[..t.childrenVisited] + [p] + t.children[t.childrenVisited + 1..];
p := t;
stackNodes := stackNodes + [t];
path := Path.Extend(path, t);
t := newT;
t.marked := true;
t.pathFromRoot := path;
unmarkedNodes := unmarkedNodes - {t};
}
}
}
}
datatype Path {
Empty;
Extend(Path, Node);
}
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