(*
10.3 Digraph - Sequential Unbounded Managed Iterator
In this section we provide the implementation module corresponding to
the interface given above in û10.2. The following scheme is used in
organizing this section:
* 10.3.1 Internal Representation
* 10.3.2 Exception Handling
* 10.3.3 Local Routines
* 10.3.4 Graph Constructors
* 10.3.5 Vertex Constructors
* 10.3.6 Edge Constructors
* 10.3.7 Graph Selectors
* 10.3.8 Vertex Selectors
* 10.3.9 Edge Selectors
* 10.3.10 Passive Iterators
* 10.3.11 Active Iterators
* 10.3.12 Module Initialization
*)
IMPLEMENTATION MODULE DigrSUMI;
(*==============================================================
Version : V2.01 08 December 1989.
Compiler : JPI TopSpeed Modula-2
Code size: 5988 bytes
Component: Digraph - Sequential Unbounded Managed Iterator
REVISION HISTORY
v1.00 26 May 1988 C. Lins:
Initial TML Modula-2 implementation
v1.01 06 Oct 1988 C. Lins:
Added extensive comments for book.
v1.02 07-14 Jan 1989 C. Lins:
Revised interface in conformance with new graph spec (Ch 9).
Modified implementations accordingly.
v1.03 10 Apr 1989 C. Lins:
Corrected initialization of handlers array.
v1.04 18 Apr 1989 C. Lins:
Added component id constant.
v2.00 24 Oct 1989 C. Lins
Created generic pc version
v2.01 08 Dec 1989 I.S.C. Houston.
Adapted to JPI Compiler:
Used type transfer functions instead of VAL.
Used shortened library module names for DOS and OS/2.
(C) Copyright 1989 Charles A. Lins
==============================================================*)
FROM JPIStorage IMPORT
(*--proc*) Allocate, Deallocate;
FROM Items IMPORT
(*--cons*) NullItem,
(*--type*) Item, AssignProc, DisposeProc;
FROM GraphTypes IMPORT
(*--type*) Operations, Exceptions, ComponentID;
FROM ErrorHandling IMPORT
(*--cons*) NullHandler,
(*--type*) HandlerProc,
(*--proc*) Raise;
FROM TypeManager IMPORT
(*--cons*) NullType,
(*--type*) TypeID,
(*--proc*) AssignOf, DisposeOf;
(*-------------------------*)
(*
10.3.1 Internal Representation
The internal representation of an unbounded directed graph uses a variant of a
structure known as an adjacency list (shown below in Figure 10.1). In this
representation, the set of vertices for a graph is given by a linear list. Since
an unbounded graph requires the list to grow and shrink in size dynamically, a
linked list of dynamically allocated vertex nodes is used. Then, each vertex is
associated with the set of directed edges leaving the vertex. The set of all
edges of the graph is thus the union of all the sets of edges for each of the
graph's vertices.
Following the figure is a description of the various TYPEs used for our internal
representation including the component fields of each record structure.
_Figure 10.1_
Edge: completes the opaque definition as a reference to a dynamically
allocated edge node.
Vertex: completes the opaque definition as a reference to a dynamically
allocated vertex node.
VertexNode: defines the information requirements for a single vertex of a
graph.
data: contains the label data item associated with a vertex.
next: link to the next vertex in the set of vertices for a graph. The
last vertex of the list has a 'next' of NullVertex indicating
the end of the list.
edges: link to the first directed edge leaving this vertex. If the vertex
has no edges leaving it, this field is set to the NullEdge.
indegree: is used in maintaining a count of the number of edges having this
vertex as its destination excluding self-loops (which are edges
whose source and destination vertices are the same). This count
is maintained by the constructors Link, Unlink and Assign. It
is used by the constructor Remove in detecting attempts to
delete vertices referenced as the destination of an edge. Since
the alternative technique for detecting this precondition is
a complete traversal of all the edges of a graph, we have chosen
to record this information incrementally as edges are inserted
and removed from the graph.
inGraph: contains the reference to the enclosing graph object for the vertex.
This avoids having the graph as a parameter to the vertex selectors.
In addition, simplifying the membership test between a vertex and a
graph, as well as the membership test for edges (through the
initial or final vertex references of the edge).
EdgeNode: defines the information requirements for a single edge of a graph.
initial: contains a reference to the initial (or source) vertex of the edge.
final: contains a reference to the final (or destination) vertex of the edge.
weight: contains the attribute of the edge.
next: contains the link to the next edge leaving the initial vertex.
The last edge of this list contains the NullEdge as its value
indicating the end of the list.
UnboundedGraph: describes (and holds) attributes of the graph itself.
labelType,
attrType: contain the data type ID for the vertex label and edge attribute,
respectively. These two fields are used to retrieve the procedures
accomplishing assignment and disposal of data items.
numVertices,
numEdges: contain counts of the total number of vertices and edges in the
graph, respectively. Thus, the selectors OrderOf and SizeOf are
O(1) algorithms instead of O(|V|) or O(|E|).
firstVertex: reference to the first vertex in the adjacency list for a graph.
*)
TYPE Edge = POINTER TO EdgeNode;
TYPE Vertex = POINTER TO VertexNode;
TYPE VertexNode = RECORD
inGraph : Graph; (*-- graph in which this vertex is a member *)
data : Label; (*-- data item (label) for this vertex *)
indegree : CARDINAL; (*-- # of edges ending at this vertex *)
next : Vertex; (*-- next vertex in adjacency list *)
edges : Edge; (*-- link to first edge leaving this vertex *)
END (*-- VertexNode *);
TYPE EdgeNode = RECORD
initial : Vertex; (*-- source vertex for this edge *)
final : Vertex; (*-- destination vertex for this edge *)
weight : Attribute;(*-- weight/attribute for this edge *)
next : Edge; (*-- next edge leaving this vertex *)
END (*-- EdgeNode *);
TYPE UnboundedGraph = RECORD
labelType : TypeID; (*-- vertex label data type ID *)
attrType : TypeID; (*-- edge attribute data type ID *)
numVertices: CARDINAL; (*-- current total number of vertices *)
numEdges : CARDINAL; (*-- current total number of edges *)
firstVertex: Vertex; (*-- first vertex in adjacency list *)
END (*-- UnboundedGraph *);
TYPE Graph = POINTER TO UnboundedGraph;
(*-------------------------*)
(*
10.3.2 Exception Handling
graphError holds the exception result from the most recently
invoked operation of this module. The Exceptions enumeration
constant noerr indicates successful completion of the operation and
all operations that may raise an exception assign this value to
graphError before any other processing.
The handler array holds the current exception handler for the
possible exceptions that may be raised from within this module.
Both are initialized by the module initialization (see û10.3.12).
GraphError simply returns the current exception result stored
in graphError and is used to determine whether a graph
operation completed successfully.
SetHandler makes theHandler the current exception handler for theError
by storing theHandler in the handler array.
GetHandler returns the current exception handler for theError from the
handler array.
*)
VAR graphError : Exceptions;
VAR handler : ARRAY Exceptions OF HandlerProc;
PROCEDURE GraphError () : Exceptions;
BEGIN
RETURN graphError;
END GraphError;
(*-------------------------*)
PROCEDURE SetHandler ( theError : Exceptions (*--in *);
theHandler : HandlerProc (*--in *));
BEGIN
handler[theError] := theHandler;
END SetHandler;
(*-------------------------*)
PROCEDURE GetHandler ( theError : Exceptions (*--in *))
: HandlerProc (*--out *);
BEGIN
RETURN handler[theError];
END GetHandler;
(*-------------------------*)
PROCEDURE RaiseErrIn ( theRoutine : Operations (*--in *);
theError : Exceptions (*--in *));
BEGIN
graphError := theError;
Raise(ComponentID + ModuleID, theRoutine, theError, handler[theError]);
END RaiseErrIn;
(*-------------------------*)
(*
10.3.3 Local Routines
FreeAttribute is responsible for retrieval of the edge attribute item disposal
routine and freeing the attribute when no longer needed. This occurs when
1. a graph is cleared or destroyed (Clear);
2. an edge is removed from a graph (Unlink);
3. a vertex is removed from a graph and any edges leaving it are implicitly
removed (ClearEdges); or
4. a new attribute is assigned to an edge (SetAttribute).
Complexity: O(1).
*)
PROCEDURE FreeAttribute ( theEdge : Edge (*--inout*));
VAR free : DisposeProc; (*-- attribute disposal routine, if any *)
BEGIN
WITH theEdge^ DO
free := DisposeOf(initial^.inGraph^.attrType);
free(weight);
END (*--with*);
END FreeAttribute;
(*-------------------------*)
(*
FreeLabel corresponds to FreeAttribute, above, for the clean-up of vertex labels
when they are no longer needed. The conditions are similar to those above:
1. a graph is cleared or destroyed (Clear);
2. an vertex is removed from a graph (Remove); or
3. a new label is assigned to a vertex (SetLabel).
Complexity: O(1).
*)
PROCEDURE FreeLabel ( theVertex : Vertex (*--inout*));
VAR free : DisposeProc; (*-- label disposal routine, if any *)
BEGIN
WITH theVertex^ DO
free := DisposeOf(inGraph^.labelType);
free(data);
END (*--with*);
END FreeLabel;
(*-------------------------*)
(*
ClearEdges removes all edges from a given vertex. This is necessary when
1. a graph is cleared or destroyed (Clear); or
2. a vertex is removed from a graph (Remove).
Essentially, the algorithm loops over each edge in the edge list for the vertex
removing the edge from the list, updating the number of edges in the graph,
deallocation of the edge attribute and, finally, deallocation of the edge itself.
Since we guarantee the last edge in the list has a 'next' of NullEdge, 'edges'
(the vertex's link to the first edge) is properly set to the NullEdge upon
completion of the routine. O(outdegree(v)).
*)
PROCEDURE ClearEdges ( theVertex: Vertex (*--inout*));
VAR theEdge : Edge; (*-- edge being removed from vertex *)
BEGIN
WITH theVertex^ DO
WHILE (edges # NullEdge) DO
theEdge := edges;
edges := edges^.next;
DEC(inGraph^.numEdges);
FreeAttribute(theEdge);
Deallocate(theEdge, SIZE(theEdge^));
END (*--while*);
END (*--with*);
END ClearEdges;
(*-------------------------*)
(*
NewVertex allocates and constructs a new, empty, vertex node. The vertex field
inGraph is set to the proper value (its enclosing graph object) while indegree,
edges, and next are initialized to an empty state. The caller is responsible for
adding the vertex to the adjacency list for the graph. The routine also automatically
raises the overflow exception with the appropriate parameters, if necessary. As
noted in Volume 1, the version of Allocate used here sets theVertex to NIL if the
allocation fails. Thus, we ensure that theVertex returned is the NullVertex in
case of a memory management failure. Complexity O(1).
*)
PROCEDURE NewVertex ( theGraph : Graph (*--in *);
theItem : Label (*--in *);
theRoutine : Operations (*--in *))
: Vertex (*--out *);
VAR theVertex : Vertex; (*-- newly created vertex *)
BEGIN
Allocate(theVertex, SIZE(VertexNode));
IF (theVertex = NullVertex) THEN
RaiseErrIn(theRoutine, overflow);
ELSE
WITH theVertex^ DO
inGraph := theGraph;
data := theItem;
indegree := 0;
edges := NullEdge;
next := NullVertex;
END (*--with*);
END (*--if*);
RETURN theVertex;
END NewVertex;
(*-------------------------*)
(*
NewEdge simple creates a new edge with the specified vertex endpoints and weight.
The edge is not added to any edge list, leaving this to the caller. The overflow
exception is automatically raised, if necessary, when a new edge node cannot be
allocated. The structural sharing of the edge attribute is controlled, as described
above for NewVertex, using a similar mechanism. Complexity O(1).
*)
PROCEDURE NewEdge ( fromVertex : Vertex (*--in *);
toVertex : Vertex (*--in *);
theWeight : Attribute (*--in *);
theRoutine : Operations (*--in *))
: Edge (*--out *);
VAR theEdge : Edge; (*-- newly created edge *)
BEGIN
Allocate(theEdge, SIZE(EdgeNode));
IF (theEdge = NullEdge) THEN
RaiseErrIn(theRoutine, overflow);
ELSE
WITH theEdge^ DO
initial := fromVertex;
final := toVertex;
weight := theWeight;
next := NullEdge;
END (*--with*);
END (*--if*);
RETURN theEdge;
END NewEdge;
(*-------------------------*)
(*
10.3.4 Graph Constructors
Create attempts to form a new, empty graph object with the given vertex
label and edge label data types. First, the graph descriptor is allocated and the vertex
and edge data type IDs are stored there. The number of vertices and edges are
initialized to zero. The pointer to the head of the adjacency list (firstVertex)
is initialized to the empty state (NIL). If the descriptor allocation
fails the overflow exception is raised and the NullGraph is returned,
otherwise we return the newly allocated graph. Complexity O(1).
*)
PROCEDURE Create ( labels : TypeID (*--in *);
attributes : TypeID (*--in *))
: Graph (*--out *);
VAR newGraph : Graph; (*-- temporary for new graph object *)
BEGIN
graphError := noerr;
Allocate(newGraph, SIZE(UnboundedGraph));
IF (newGraph = NullGraph) THEN
RaiseErrIn(create, overflow);
ELSE
WITH newGraph^ DO
labelType := labels;
attrType := attributes;
numVertices := 0;
numEdges := 0;
firstVertex := NullVertex;
END (*--with*);
END (*--if*);
RETURN newGraph;
END Create;
(*-------------------------*)
(*
Destroy clears theGraph and then deallocates it making theGraph undefined.
SCLStorage.Deallocate automatically releases the proper amount of space
originally allocated and alters the pointer to NIL (which is also the
value of the NullGraph). Complexity O(v+e).
*)
PROCEDURE Destroy (VAR theGraph : Graph (*--inout *));
BEGIN
Clear(theGraph);
IF (graphError = noerr) THEN
Deallocate(theGraph, SIZE(theGraph^));
END (*--if*);
END Destroy;
(*-------------------------*)
(*
Clear removes all vertices and edges from theGraph making theGraph empty.
We do this by iterating over each of the vertices and clearing all edges
leaving the vertex (ClearEdges). As a final step we ensure that the graph is left in the
empty state by resetting the head of the adjacency list to NIL and the number
of vertices and edges in the graph to zero. Complexity O(v+e).
*)
PROCEDURE Clear (VAR theGraph : Graph (*--inout *));
VAR theVertex : Vertex; (*--loop index over vertices *)
VAR oldVertex : Vertex; (*--vertex to deallocate *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(clear, undefined);
ELSE
WITH theGraph^ DO
theVertex := firstVertex;
WHILE (theVertex # NullVertex) DO
ClearEdges(theVertex);
FreeLabel(theVertex);
oldVertex := theVertex;
theVertex := theVertex^.next;
Deallocate(oldVertex, SIZE(oldVertex^));
END (*--while*);
firstVertex := NullVertex;
numVertices := 0;
numEdges := 0;
END (*--with*);
END (*--if*);
END Clear;
(*-------------------------*)
(*
Assign copies the source graph, theGraph, to the target graph, toGraph. The
main body of Assign does this by first copying all the vertices followed by
copying all the edges from the source to the destination graph.
The algorithmic complexity is O(v^2+e) due to the mapping between the vertices
of the source and target graphs while copying the edges (see the discussion
of the vertexMap following RecreateTarget).
*)
PROCEDURE Assign ( theGraph : Graph (*--in *);
VAR toGraph : Graph (*--inout *));
(*
RecreateTarget reconstructs the target graph descriptor so that the fields
defining the vertex label and edge attribute data types, and (optionally)
maximum number of vertices between the source and destination graphs are the
same. If the source and destination graphs are the same the routine returns
FALSE indicating that the postconditions for the assignment operation are
already met. The routine returns TRUE if the recreation of the target was
successful. Complexity O(v+e) where v and e are the number of vertices and
edges, respectively, in the original toGraph. When Clearing the target graph
is unnecessary (the toGraph is initially the NullGraph) the complexity falls
to O(1).
*)
PROCEDURE RecreateTarget (): BOOLEAN (*--out *);
BEGIN
IF (theGraph = NullGraph) THEN
RaiseErrIn(assign, undefined);
ELSIF (toGraph = NullGraph) THEN
WITH theGraph^ DO
toGraph := Create(labelType, attrType);
END (*--with*);
ELSIF (theGraph = toGraph) THEN
RETURN FALSE;
ELSE
Clear(toGraph);
WITH theGraph^ DO
toGraph^.labelType := labelType;
toGraph^.attrType := attrType;
END (*--with*);
END (*--if*);
RETURN (graphError = noerr);
END RecreateTarget;
(*
One thorny issue in graph assignment is how to set-up the copied edges with the
proper initial and final vertices? The edges of the source graph contain references
to the source graph's vertices, not those of the target graph. The vertex
labels cannot be used since more than one vertex can have the same label. In
this case, an edge from the second (or greater) such vertex in the target graph
would be linked incorrectly to the first vertex having that label. The solution
is in having some form of temporary mapping from the source graph's vertices to
their counterpart in the target graph. The necessary operations are add a mapping
between a vertex from the source graph and its corresponding vertex in the
target graph, and given a source graph vertex return the target graph vertex
mapped to that source vertex.
The data structure implementing our vertex mapping is an unordered array of mapping
entries, one per vertex, between the vertices of the source graph and the
target graph. This array is dynamically created on the heap based on the
number of vertices in the source graph. (The ARRAY [0..0] OF ≡ construct is
a special feature of the TML Modula-2 compiler allowing dynamic arrays.) The
variable mapExtent controls where MapVertex entries are inserted into the
array. A post-increment scheme is used so mapVertex is always one greater than
the number of entries stored in the array.
*)
TYPE MapVertex = RECORD
old : Vertex; (*-- vertex from source graph *)
new : Vertex; (*-- corresponging vertex in target graph *)
END (*--MapVertex*);
TYPE MapVertices = ARRAY [0..0] OF MapVertex;
VAR vertexMap : POINTER TO MapVertices;
VAR mapExtent : CARDINAL;
(*
CreateVertexMap allocates a dynamic array of vertex mapping entries on the
heap based on the number of vertices in the source graph. vertexMap is set to
NIL by Allocate if there isn't enough memory available to meet the request.
*)
PROCEDURE CreateVertexMap;
BEGIN
Allocate(vertexMap,
VAL(CARDINAL, SIZE(MapVertex)) * theGraph^.numVertices);
mapExtent := 0;
END CreateVertexMap;
(*
AddVertexToMap adds a mapping between the vertices of the source and target graphs.
*)
PROCEDURE AddToVertexMap ( oldVertex : Vertex (*--in *);
newVertex : Vertex (*--in *));
BEGIN
WITH vertexMap^[mapExtent] DO
old := oldVertex;
new := newVertex;
END (*--with*);
INC(mapExtent);
END AddToVertexMap;
(*
VertexInMap returns the mapping between the vertices of the source and target
graphs. Since every vertex is represented failure to find a mapping is
indicative of either a programming error in CopyVertices or a hardware/system
software error at runtime.
*)
PROCEDURE VertexInMap ( oldVertex : Vertex (*--in *))
: Vertex (*--out *);
VAR index : CARDINAL; (*-- loop index over mapping entries *)
BEGIN
FOR index := 0 TO mapExtent-1 DO
WITH vertexMap^[index] DO
IF (oldVertex = old) THEN
RETURN new;
END (*--if*);
END (*--with*);
END (*--for*);
RETURN NullVertex;
END VertexInMap;
(*
DestroyVertexMap frees up the memory used by the vertexMap when the Assign
operation is complete. Remember that Deallocate automatically
releases the proper amount of space.
*)
PROCEDURE DestroyVertexMap;
BEGIN
Deallocate(vertexMap, SIZE(vertexMap^));
END DestroyVertexMap;
(*
CopyVertices duplicates the vertices from the source graph to the destination
graph returning TRUE if every vertex was successfully copied and FALSE otherwise.
This BOOLEAN result is used by the main body of Assign to control whether the
graph assignment operation continues by copying the edges. The following local
variables are used:
1. v: indicates the current vertex being copied from the source
graph. This is also used as a 'loop index' over the vertices
of the source graph.
2. newVertex: temporary for a new vertex for the destination graph.
3. lastVertex: last vertex inserted into the destination graph. This is
used by TailInsert to add a new vertex to the end of the
destination graph's adjacency list.
4. assignItem: vertex label assignment routine.
Assignment of the vertex label presents an interesting situation. When a vertex
is added to a graph, the client module expects the given label to be copied using
the Modula-2 assignment statement (even for dynamically allocated data items)
since we simply need to store the value in the vertex object. This is known as
'structural sharing'. But when a graph is duplicated using Assign, new copies
of the vertex labels are necessary - avoiding the problems presented by structural
sharing of dynamically allocated items as described in Volume 1. CopyVertices
resolves this through the assignment procedure associated with the graph's label
data type duplicating the label as a NewVertex is created. Complexity O(v) where
v is the number of vertices in the source graph.
*)
PROCEDURE CopyVertices () : BOOLEAN;
VAR v : Vertex; (*--loop index over vertices being copied *)
VAR newVertex : Vertex; (*--new vertex in target graph *)
VAR lastVertex: Vertex; (*--last vertex added to list of vertices *)
VAR assignItem: AssignProc;
(*
TailInsert adds newVertex to the end of the target graph's adjacency list
given pointers to the first and last elements of the list. Complexity O(1).
*)
PROCEDURE TailInsert (VAR first : Vertex (*--inout *);
VAR last : Vertex (*--inout *));
BEGIN
IF (first = NullVertex) THEN
first := newVertex;
ELSE
last^.next := newVertex;
END (*--if*);
last := newVertex;
END TailInsert;
BEGIN
CreateVertexMap;
IF (vertexMap = NIL) THEN
RETURN FALSE;
END (*--if*);
assignItem := AssignOf(theGraph^.labelType);
v := theGraph^.firstVertex;
lastVertex := NullVertex;
WHILE (v # NullVertex) DO
newVertex := NewVertex(toGraph, assignItem(v^.data), assign);
IF (newVertex = NullVertex) THEN
DestroyVertexMap;
RETURN FALSE;
END (*--if*);
newVertex^.indegree := v^.indegree;
TailInsert(toGraph^.firstVertex, lastVertex);
INC(toGraph^.numVertices);
AddToVertexMap(v, newVertex);
v := v^.next;
END (*--while*);
RETURN TRUE;
END CopyVertices;
(*
CopyEdges iterates over the edges of the source graph through the adjacency
list of vertices and the edge lists for each vertex. As each edge in the
source graph is encountered, a new edge is constructed in the target graph.
This edge is then added to the target graph's vertex equivalent to the edges'
initial vertex. The initial and final vertices of the new edge are retrieved
from the vertex map created by CopyVertices. Note that NewEdge will raise the
overflow exception if necessary.
The following local variables are used:
v: indicates the current vertex from the source graph whose
edges are being copied. This is also used as a 'loop index'
over the vertices of the source graph.
e: indicates the current edge of the source graph being copied.
Also used as a 'loop index' over the edges leaving each
vertex, v.
fromVertex: the vertex of the target graph corresponding to 'v' in the
source graph. Since all edges leaving any given vertex, v,
have have 'v' as their initial vertex, fromVertex can be
derived from the current vertex of the source graph rather than
repeatedly retrieving it from the initial vertex of the edge.
newEdge: temporary for a new edge in the target graph.
lastEdge: last edge inserted into the edge list of the current vertex
(fromVertex) destination graph. This is used by TailInsert
to add a new edge to the end of fromVertex's edge list.
assignItem: edge attribute assignment routine.
So while the basic loop over the edges is linear with respect to the number of
edges, the linear search of the vertex map nested within the loop over the
source graph's vertices gives this algorithm time complexity O(v^2). A better
mapping algorithm (i.e, one of constant time O(1)) would yield a linear time
algorithm overall for the graph assignment.
*)
PROCEDURE CopyEdges;
VAR v : Vertex; (*--loop index over vertices in source graph *)
VAR e : Edge; (*--loop index over edges in source graph *)
VAR fromVertex: Vertex; (*--vertex in target graph corresponding to v *)
VAR newEdge : Edge; (*--new edge for target graph *)
VAR lastEdge : Edge; (*--last edge inserted into new list of edges *)
VAR assignItem: AssignProc; (*--attribute assignment procedure *)
(*
TailInsert adds newEdge to the end of an edge list given pointers to the first
and last elements of the list. Complexity O(1).
*)
PROCEDURE TailInsert (VAR first : Edge (*--inout *);
VAR last : Edge (*--inout *));
BEGIN
IF (first = NullEdge) THEN
first := newEdge;
ELSE
last^.next := newEdge;
END (*--if*);
last := newEdge;
END TailInsert;
BEGIN
assignItem := AssignOf(theGraph^.attrType);
v := theGraph^.firstVertex;
WHILE (v # NullVertex) DO
e := v^.edges;
lastEdge := NullEdge;
fromVertex := VertexInMap(v);
WHILE (e # NullEdge) DO
newEdge := NewEdge(fromVertex,
VertexInMap(e^.final),
assignItem(e^.weight),
assign);
IF (newEdge = NullEdge) THEN
RETURN;
END (*--if*);
TailInsert(fromVertex^.edges, lastEdge);
INC(toGraph^.numEdges);
e := e^.next;
END (*--while*);
v := v^.next;
END (*--while*);
END CopyEdges;
BEGIN (*--Assign --*)
graphError := noerr;
IF RecreateTarget() & CopyVertices() THEN
CopyEdges;
DestroyVertexMap;
END (*--if*);
END Assign;
(*-------------------------*)
(*
10.3.5 Vertex Constructors
Insert adds a vertex to the given graph labeling the vertex with the given item.
The first step is to allocate a new vertex node which, if successful,
is followed by adding the vertex at the head of the adjacency list. If we cannot
create a new vertex node the overflow exception is raised and the Insert operation
aborted. Complexity O(1).
*)
PROCEDURE Insert (VAR theGraph : Graph (*--inout*);
theItem : Label (*--in *);
VAR theVertex : Vertex (*--out *));
BEGIN
graphError := noerr;
theVertex := NullVertex;
IF (theGraph = NullGraph) THEN
RaiseErrIn(insert, undefined);
ELSE
theVertex := NewVertex(theGraph, theItem, insert);
IF (theVertex # NullVertex) THEN
WITH theGraph^ DO
theVertex^.next := firstVertex;
firstVertex := theVertex;
INC(numVertices);
END (*--with*);
END (*--if*);
END (*--if*);
END Insert;
(*-------------------------*)
(*
Remove deletes the given vertex from the specified graph. If no such vertex
can be found in the graph the novertex exception is raised and the routine
aborted. Deleting a vertex that is the head of an edge would leave dangling
edges (pointing to a non-existent vertex), and so, if the vertex is referenced
by an edge (other than a self-loop) the exception references is raised and
Remove is aborted. After we have checked that no exceptions can occur we remove
all edges leaving the vertex, remove the vertex from the adjacency list,
release any dynamically allocated memory used by the vertex label, release the
vertex itself, and update the count of vertices in the graph. Complexity
O(outdegree(v)).
*)
PROCEDURE Remove (VAR theGraph : Graph (*--inout*);
VAR theVertex : Vertex (*--inout*));
VAR loopVertex : Vertex; (*--loop index over vertices *)
VAR priorVertex: Vertex; (*--immediate predecessor of theVertex *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(remove, undefined);
ELSIF (theVertex = NullVertex) THEN
RaiseErrIn(remove, nullvertex);
ELSIF (theVertex^.inGraph # theGraph) THEN
RaiseErrIn(remove, novertex);
ELSIF (theVertex^.indegree > 0) THEN
RaiseErrIn(remove, referenced);
ELSE
loopVertex := theGraph^.firstVertex;
priorVertex := NullVertex;
WHILE (loopVertex # theVertex) DO
priorVertex := loopVertex;
loopVertex := loopVertex^.next;
END (*--while*);
ClearEdges(theVertex);
IF (priorVertex = NullVertex) THEN
theGraph^.firstVertex := theVertex^.next;
ELSE
priorVertex^.next := theVertex^.next;
END (*--if*);
FreeLabel(theVertex);
Deallocate(theVertex, SIZE(theVertex^));
DEC(theGraph^.numVertices);
END (*--if*);
END Remove;
(*-------------------------*)
(*
SetLabel assigns a new label to the given vertex of the graph. Prior to
assigning a new vertex label we must release any dynamically allocated memory
used by the old vertex label. Complexity O(1).
*)
PROCEDURE SetLabel ( theVertex : Vertex (*--inout*);
theItem : Label (*--in *));
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(setlabel, nullvertex);
ELSE
FreeLabel(theVertex);
theVertex^.data := theItem;
END (*--if*);
END SetLabel;
(*-------------------------*)
(*
10.3.6 Edge Constructors
Link adds a directed edge between the from and to vertices labeling the
edge with the given weight attribute. The new edge is linked to the front
of the fromVertex's edge list. Thus, edges appear in reverse order to their
order of insertion. Complexity O(1).
*)
PROCEDURE Link (VAR theGraph : Graph (*--inout*);
fromVertex : Vertex (*--in *);
toVertex : Vertex (*--in *);
theWeight : Attribute (*--in *);
VAR theEdge : Edge (*--out *));
BEGIN
graphError := noerr;
theEdge := NullEdge;
IF (theGraph = NullGraph) THEN
RaiseErrIn(link, undefined);
ELSIF (fromVertex = NullVertex) OR (toVertex = NullVertex) THEN
RaiseErrIn(link, nullvertex);
ELSIF (fromVertex^.inGraph # theGraph) OR
(toVertex^.inGraph # theGraph) THEN
RaiseErrIn(link, novertex);
ELSE
theEdge := NewEdge(fromVertex, toVertex, theWeight, link);
IF (theEdge # NullEdge) THEN
theEdge^.next := fromVertex^.edges;
fromVertex^.edges := theEdge;
IF (fromVertex # toVertex) THEN
INC(toVertex^.indegree);
END (*--if*);
INC(theGraph^.numEdges);
END (*--if*);
END (*--if*);
END Link;
(*-------------------------*)
(*
Unlink removes a directed edge between the two vertices, fromVertex and
toVertex. Complexity O(d) where d is the out-degree of the from vertex (derived
from the initial vertex of theEdge).
*)
PROCEDURE Unlink (VAR theGraph : Graph (*--inout*);
VAR theEdge : Edge (*--inout*));
VAR e : Edge; (*--pointer to edge (v,w), if any *)
VAR f : Edge; (*--pointer to edge preceeding (v,w) in adjacency list *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(unlink, undefined);
ELSIF (theEdge = NullEdge) THEN
RaiseErrIn(unlink, nulledge);
ELSIF (theEdge^.initial^.inGraph # theGraph) THEN
RaiseErrIn(unlink, noedge);
ELSE
(*
The following loop searches for the edge being removed in the list of edges
associated with the edges' initial vertex. In addition, we need the prior
edge in the list for the actual deletion of the edge from the list. This is
a standard singly-linked list technique for the deletion of list nodes.
This loop is guaranteed to succeed as Link ensures that the edge is placed in
the initial vertex's list of edges. Complexity O(outdegree(v)).
*)
e := theEdge^.initial^.edges;
f := NullEdge;
WHILE (e # theEdge) DO
f := e;
e := e^.next;
END (*--while*);
WITH theEdge^ DO
(*
Update the edge list of the initial vertex by removing the edge.
*)
IF (f = NullEdge) THEN
initial^.edges := next;
ELSE
f^.next := next;
END (*--if*);
(*
Self-loops are not included in the final vertex's indegree count since such
edges may be freely unlinked and do not effect the removal of the vertex. In
any case, we must maintain the proper count of edges in the graph. As a final
step, the edge attribute and the edge itself may be deallocated. In case a
client module has stored a copy of the edge somewhere, we insure that these
duplicates are invalid by clearing their inital vertex (used to determine if
an edge is an element of a graph) and cut their link to any other edges.
*)
IF (initial # final) THEN
DEC(final^.indegree);
END (*--if*);
DEC(initial^.inGraph^.numEdges);
FreeAttribute(theEdge);
initial := NullVertex;
next := NullEdge;
Deallocate(theEdge, SIZE(theEdge^));
END (*--with*);
END (*--if*);
END Unlink;
(*-------------------------*)
(*
SetAttribute assigns a new edge labelling to the given edge. Prior to assigning
a new edge attribute we must release any dynamically allocated memory
used by the old edge attribute. Complexity O(1).
*)
PROCEDURE SetAttribute ( theEdge : Edge (*--inout*);
theWeight : Attribute (*--in *));
BEGIN
graphError := noerr;
IF (theEdge = NullEdge) THEN
RaiseErrIn(setattr, nulledge);
ELSE
FreeAttribute(theEdge);
theEdge^.weight := theWeight;
END (*--if*);
END SetAttribute;
(*-------------------------*)
(*
10.3.7 Graph Selectors
IsDefined verifies to the best of its ability whether theGraph has been
created and is still an active object. Complexity: O(1).
*)
PROCEDURE IsDefined ( theGraph : Graph (*--in *))
: BOOLEAN (*--out *);
BEGIN
RETURN (theGraph # NullGraph);
END IsDefined;
(*-------------------------*)
(*
IsEmpty returns True if theGraph is in the empty state, as indicated by
the number of vertices being zero, and False otherwise. As per the
specification (û9.3) undefined graphs are considered empty. Complexity: O(1).
*)
PROCEDURE IsEmpty ( theGraph : Graph (*--in *))
: BOOLEAN (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(isempty, undefined);
RETURN TRUE;
END (*--if*);
RETURN theGraph^.numVertices = 0;
END IsEmpty;
(*-------------------------*)
(*
TypeOf simply returns the vertex label and edge attribute data type IDs for
the given graph. Undefined graphs, as always, raise the undefined exception
and return a reasonable value, in this case the NullType. Complexity O(1).
*)
PROCEDURE TypeOf ( theGraph : Graph (*--in *);
VAR labelType : TypeID (*--out *);
VAR attrType : TypeID (*--out *));
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(typeof, undefined);
labelType := NullType;
attrType := NullType;
ELSE
labelType := theGraph^.labelType;
attrType := theGraph^.attrType;
END (*--if*);
END TypeOf;
(*-------------------------*)
(*
OrderOf returns the number of vertices in the graph, or zero for an undefined
graph. Complexity O(1).
*)
PROCEDURE OrderOf ( theGraph : Graph (*--in *))
: CARDINAL (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(orderof, undefined);
RETURN 0;
END (*--if*);
RETURN theGraph^.numVertices;
END OrderOf;
(*-------------------------*)
(*
OrderOf returns the number of edges in the graph, or zero for an undefined
graph. Complexity O(1).
*)
PROCEDURE SizeOf ( theGraph : Graph (*--in *))
: CARDINAL (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(sizeof, undefined);
RETURN 0;
END (*--if*);
RETURN theGraph^.numEdges;
END SizeOf;
(*-------------------------*)
(*
10.3.8 Vertex Selectors
InDegree returns the number of edges entering the given vertex. Complexity O(1).
*)
PROCEDURE InDegree ( theVertex : Vertex (*--in *))
: CARDINAL (*--out *);
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(indegree, nullvertex);
ELSE
RETURN theVertex^.indegree;
END (*--if*);
RETURN 0;
END InDegree;
(*-------------------------*)
(*
OutDegree returns the number of edges leaving the given vertex. We do this by
simply iterating over the edges of the vertex counting them along the way.
Complexity O(outdegree(v)).
*)
PROCEDURE OutDegree ( theVertex : Vertex (*--in *))
: CARDINAL (*--out *);
VAR theEdge : Edge; (*-- loop index over edges of the vertex *)
VAR edgeCount : CARDINAL; (*--running count of edges leaving this vertex *)
BEGIN
graphError := noerr;
edgeCount := 0;
IF (theVertex = NullVertex) THEN
RaiseErrIn(outdegree, nullvertex);
ELSE
theEdge := theVertex^.edges;
WHILE (theEdge # NullEdge) DO
INC(edgeCount);
theEdge := theEdge^.next;
END (*--while*);
END (*--if*);
RETURN edgeCount;
END OutDegree;
(*-------------------------*)
(*
LabelOf returns the vertex label associated with the given vertex.
If the vertex is undefined the NullItem is returned. Complexity O(1).
*)
PROCEDURE LabelOf ( theVertex : Vertex (*--in *))
: Label (*--out *);
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(labelof, nullvertex);
ELSE
RETURN theVertex^.data;
END (*--if*);
RETURN NullItem;
END LabelOf;
(*-------------------------*)
(*
Since we have stored a copy of the graph object associated with each vertex
as a field of the vertex itself, IsVertex simply needs to compare the given
graph with its own local state. This saves us from having to search the graph.
Complexity O(1).
*)
PROCEDURE IsVertex ( theGraph : Graph (*--in *);
theVertex : Vertex (*--in *))
: BOOLEAN (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(isvertex, undefined);
ELSIF (theVertex = NullVertex) THEN
RaiseErrIn(isvertex, nullvertex);
ELSE
RETURN theVertex^.inGraph = theGraph;
END (*--if*);
RETURN FALSE;
END IsVertex;
(*-------------------------*)
(*
GraphOf simply returns its copy of the enclosing graph or the NullGraph if the
vertex is undefined. Complexity O(1).
*)
PROCEDURE GraphOf ( theVertex : Vertex (*--in *))
: Graph (*--out *);
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(graphof, nullvertex);
ELSE
RETURN theVertex^.inGraph;
END (*--if*);
RETURN NullGraph;
END GraphOf;
(*-------------------------*)
(*
10.3.9 Edge Selectors
AttributeOf returns the edge attribute associated with the given edge.
If the edge is undefined the NullItem is returned. Complexity O(1).
*)
PROCEDURE AttributeOf ( theEdge : Edge (*--in *))
: Attribute (*--out *);
BEGIN
graphError := noerr;
IF (theEdge = NullEdge) THEN
RaiseErrIn(attrof, nulledge);
ELSE
RETURN theEdge^.weight;
END (*--if*);
RETURN NullItem;
END AttributeOf;
(*-------------------------*)
(*
Given an edge, InitialOf returns the vertex that is the origin of
the directed edge or the NullVertex if the edge is undefined. Complexity O(1).
*)
PROCEDURE InitialOf ( theEdge : Edge (*--in *))
: Vertex (*--out *);
BEGIN
graphError := noerr;
IF (theEdge = NullEdge) THEN
RaiseErrIn(initialof, nulledge);
ELSE
RETURN theEdge^.initial;
END (*--if*);
RETURN NullVertex;
END InitialOf;
(*-------------------------*)
(*
Given an edge, FinalOf returns the vertex that is the destination of
the directed edge or the NullVertex if the edge is undefined. Complexity O(1).
*)
PROCEDURE FinalOf ( theEdge : Edge (*--in *))
: Vertex (*--out *);
BEGIN
graphError := noerr;
IF (theEdge = NullEdge) THEN
RaiseErrIn(finalof, nulledge);
ELSE
RETURN theEdge^.final;
END (*--if*);
RETURN NullVertex;
END FinalOf;
(*-------------------------*)
(*
IsEdge returns true if there the given directed edge is an edge of the given
graph and false otherwise. An advantage of having each vertex identify its
enclosing graph object is use of this field in testing whether the edge is
part of a specified graph. This saves use from having to search every edge in
the graph. Complexity O(1).
*)
PROCEDURE IsEdge ( theGraph : Graph (*--in *);
theEdge : Edge (*--in *))
: BOOLEAN (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(isedge, undefined);
ELSIF (theEdge = NullEdge) THEN
RaiseErrIn(isedge, nulledge);
ELSIF (theEdge^.initial = NullVertex) THEN
RaiseErrIn(isedge, nullvertex);
ELSE
RETURN theEdge^.initial^.inGraph = theGraph;
END (*--if*);
RETURN FALSE;
END IsEdge;
(*-------------------------*)
(*
10.3.10 Passive Iterators
LoopVertices simply iterates over the vertices of the given graph until every
vertex has been examined or the process procedure parameter returns FALSE,
whichever occurs first. Complexity O(|V|).
*)
PROCEDURE LoopVertices ( theGraph : Graph (*--in *);
process : VertexLoopProc (*--in *));
VAR theVertex : Vertex; (*--loop index over vertices *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(loopvertices, undefined);
ELSE
theVertex := theGraph^.firstVertex;
WHILE (theVertex # NullVertex) & process(theVertex) DO
theVertex := theVertex^.next;
END (*--while*);
END (*--if*);
END LoopVertices;
(*-------------------------*)
(*
LoopEdges loops over the vertices of the given graph to access the edges associated
with each vertex. Once the process procedure parameter returns FALSE, we exit
both WHILE statements through the use of a RETURN which exits the procedure.
Complexity O(|E|).
*)
PROCEDURE LoopEdges ( theGraph : Graph (*--in *);
process : EdgeLoopProc (*--in *));
VAR theVertex : Vertex; (*--loop index over vertices *)
VAR theEdge : Edge; (*--loop index over edges of a vertex *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(loopedges, undefined);
ELSE
theVertex := theGraph^.firstVertex;
WHILE (theVertex # NullVertex) DO
theEdge := theVertex^.edges;
WHILE (theEdge # NullEdge) DO
IF ~process(theEdge) THEN
RETURN;
END (*--if*);
theEdge := theEdge^.next;
END (*--while*);
theVertex := theVertex^.next;
END (*--while*);
END (*--if*);
END LoopEdges;
(*-------------------------*)
(*
LoopIterate simply loops over the edges leaving a specified vertex. Complexity
O(outdegree(v)).
*)
PROCEDURE LoopIterate ( theVertex : Vertex (*--in *);
process : EdgeLoopProc (*--in *));
VAR theEdge : Edge; (*--loop index over edges of the vertex *)
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(loopiterate, nullvertex);
ELSE
theEdge := theVertex^.edges;
WHILE (theEdge # NullEdge) & process(theEdge) DO
theEdge := theEdge^.next;
END (*--while*);
END (*--if*);
END LoopIterate;
(*-------------------------*)
(*
TravVertices simply iterates over every vertex in the graph. Complexity O(|V|).
*)
PROCEDURE TravVertices ( theGraph : Graph (*--in *);
process : VertexProc (*--in *));
VAR theVertex : Vertex; (*--loop index over vertices *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(travvertices, undefined);
ELSE
theVertex := theGraph^.firstVertex;
WHILE (theVertex # NullVertex) DO
process(theVertex);
theVertex := theVertex^.next;
END (*--while*);
END (*--if*);
END TravVertices;
(*-------------------------*)
(*
TravEdges simply iterates over every edge in the graph. Since the only way
to get at all the edges is through the vertices, we iterate over all the
vertices and over each edge leaving each vertex. Complexity O(v+e).
*)
PROCEDURE TravEdges ( theGraph : Graph (*--in *);
process : EdgeProc (*--in *));
VAR theVertex : Vertex; (*--loop index over vertices *)
VAR theEdge : Edge; (*--loop index over edges of a vertex *)
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(travedges, undefined);
ELSE
theVertex := theGraph^.firstVertex;
WHILE (theVertex # NullVertex) DO
theEdge := theVertex^.edges;
WHILE (theEdge # NullEdge) DO
process(theEdge);
theEdge := theEdge^.next;
END (*--while*);
theVertex := theVertex^.next;
END (*--while*);
END (*--if*);
END TravEdges;
(*-------------------------*)
(*
Iterate simply loops over every edge leaving the specified vertex of a graph.
Complexity O(outdegree(v)).
*)
PROCEDURE Iterate ( theVertex : Vertex (*--in *);
process : EdgeProc (*--in *));
VAR theEdge : Edge; (*--loop index over edges of the vertex *)
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(iterate, nullvertex);
ELSE
theEdge := theVertex^.edges;
WHILE (theEdge # NullEdge) DO
process(theEdge);
theEdge := theEdge^.next;
END (*--while*);
END (*--if*);
END Iterate;
(*-------------------------*)
(*
10.3.11 Active Iterators
Each of the active iterators are essentially selectors for the underlying
representation of the adjacency list. Their complexity is O(1).
*)
PROCEDURE FirstVertex ( theGraph : Graph (*--in *))
: Vertex (*--out *);
BEGIN
graphError := noerr;
IF (theGraph = NullGraph) THEN
RaiseErrIn(firstvertex, undefined);
ELSE
RETURN theGraph^.firstVertex;
END (*--if*);
RETURN NullVertex;
END FirstVertex;
(*-------------------------*)
PROCEDURE NextVertex ( theVertex : Vertex (*--in *))
: Vertex (*--out *);
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(nextvertex, nullvertex);
ELSE
RETURN theVertex^.next;
END (*--if*);
RETURN NullVertex;
END NextVertex;
(*-------------------------*)
PROCEDURE FirstEdge ( theVertex : Vertex (*--in *))
: Edge (*--out *);
BEGIN
graphError := noerr;
IF (theVertex = NullVertex) THEN
RaiseErrIn(firstedge, nullvertex);
ELSE
RETURN theVertex^.edges;
END (*--if*);
RETURN NullEdge;
END FirstEdge;
(*-------------------------*)
PROCEDURE NextEdge ( theEdge : Edge (*--in *))
: Edge (*--out *);
BEGIN
graphError := noerr;
IF (theEdge = NullEdge) THEN
RaiseErrIn(nextedge, nulledge);
ELSE
RETURN theEdge^.next;
END (*--if*);
RETURN NullEdge;
END NextEdge;
(*-------------------------*)
(*
10.3.12 Module Initialization
The module's local variables are initialized to known states. graphError is
used to fill the handler array with a routine that does nothing when an exception
is raised (saving the declaration of a special loop control variable for this
purpose). Applying MIN and MAX to cover all exceptions ensures that this
initialization will be unaffected by any future changes to the number of
Exceptions or their order of declaration within the enumeration. Since a FOR
loop control variable is undefined following the loop, graphError must be
set to indicate that an error has not yet occurred.
*)
BEGIN
FOR graphError := MIN(Exceptions) TO MAX(Exceptions) DO
SetHandler(graphError, NullHandler);
END (*--for*);
SetHandler(noerr, NullHandler);
graphError := noerr;
NullGraph := NIL;
NullVertex := NIL;
NullEdge := NIL;
END DigrSUMI.