Description of the Prop-Fwd-Prop-Bkwd Incremental Algorithm

INPUTS: S, an STN with a fully computed distance matrix D.
        (X,delta,Y) a new edge to be inserted into S.

OUTPUT:  D*, the distance matrix for the network S*, where
         S* contains all time-points and edges in S, together
	 with (X,delta,Y).  Or FALSE if S* is inconsistent.

If delta < -D(Y,X) return FALSE
otherwise...

First, PropFwd along SHORTEST paths emanating from Y.
    Encountered := {Y}  // Nodes encountered on shortest paths from Y
    Changed := {Y}  // Nodes whose distance from X has changed
    ToDo:  {Y}  // Nodes awaiting processing
    // order of elts in ToDo and Changed doesn't matter
    while (ToDo non-empty)
      Let V := pop first element from ToDo
      For each successor (V,d,W) of V:
        if W not in Encountered
	   and
	   D(Y,W) == D(Y,V) + d (i.e., (V,d,W) is part of a shortest
	                                 path from Y to W)
        then
	   add W to Encountered
	   if delta + D(Y,W) < D(X,W) (i.e., the new edge (X,delta,Y) is
	          part of a *shorter* path from X to W)
           then
	     set D(X,W) := delta + D(Y,W)
	     add W to Changed
	     add W to ToDo

Next, for each node W in Changed, propagate back from X.
   Encountered = {X}
   ToDo = {X}
   while (ToDo not empty)
     Let F = pop first element off ToDo
     For each predecessor edge (E,a,F) of F:
        if E not in Encountered
	   and
	   D(E,X) == a + D(F,X) (i.e., (E,a,F) is part of a shortest
	                                 path from E to X)
        then
	   add E to Encountered
	   if D(E,X) + D(X,W) < D(E,W) (i.e., the new edge (X,delta,Y) is
	          part of a *shorter* path from E to W)
           then
	     set D(E,W) := D(E,X) + D(X,W)
	     add W to ToDo

Return D (modified distance matrix)