C++ Reference: ebert_graph
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A few variations on a theme of the "star" graph representation by Ebert, as described in J. Ebert, "A versatile data structure for edgeoriented graph algorithms." Communications of the ACM 30(6):513519 (June 1987). http://portal.acm.org/citation.cfm?id=214769
In this file there are three representations that have much in common. The general one, called simply EbertGraph, contains both forward and backwardstar representations. The other, called ForwardEbertGraph, contains only the forwardstar representation of the graph, and is appropriate for applications where the reverse arcs are not needed.
The point of including all the representations in this one file is to capitalize, where possible, on the commonalities among them, and those commonalities are mostly factored out into base classes as described below. Despite the commonalities, however, each of the three representations presents a somewhat different interface because of their different underlying semantics. A quintessential example is that the AddArc() method, very natural for the EbertGraph representation, cannot exist for an inherently static representation like ForwardStaticGraph.
Many clients are expected to use the interfaces to the graph objects directly, but some clients are parameterized by graph type and need a consistent interface for their underlying graph objects. For such clients, a small library of class templates is provided to give a consistent interface to clients where the underlying graph interfaces differ. Examples are the AnnotatedGraphBuildManager<> template, which provides a uniform interface for building the various types of graphs; and the TailArrayManager<> template, which provides a uniform interface for applications that need to map from arc indices to arc tail nodes, accounting for the fact that such a mapping has to be requested explicitly from the ForwardStaticGraph and ForwardStarGraph representations.
There are two base class templates, StarGraphBase, and EbertGraphBase; their purpose is to hold methods and data structures that are in common among their descendants. Only classes that are leaves in the following hierarchy tree are eligible for freestanding instantiation and use by clients. The parentheses around StarGraphBase and EbertGraphBase indicate that they should not normally be instantiated by clients:
(StarGraphBase)  / \  / \  / \  / \  (EbertGraphBase) ForwardStaticGraph  / \  / \  EbertGraph ForwardEbertGraph In the general EbertGraph case, the graph is represented with three arrays. Let n be the number of nodes and m be the number of arcs. Let i be an integer in [0..m1], denoting the index of an arc.
* head_[i] contains the endnode of arc i,
* head_[i1] contains the startnode of arc i. Note that in two'scomplement arithmetic, i1 = ~i.
Consequently:
* head_[~i] contains the endnode of the arc reverse to arc i,
* head_[i] contains the startnode of the arc reverse to arc i.
Note that if arc (u, v) is defined, then the data structure also stores
(v, u).
Arc ~i thus denotes the arc reverse to arc i.
This is what makes this representation useful for undirected graphs and for
implementing algorithms like bidirectional shortest paths.
Also note that the representation handles multigraphs. If several arcs
going from node u to node v are added to the graph, they will be handled as
separate arcs.
Now, for an integer u in [0..n1] denoting the index of a node:
* first_incident_arc_[u] denotes the first arc in the adjacency list of u.
* going from an arc i, the adjacency list can be traversed using
j = next_adjacent_arc_[i].
The EbertGraph implementation has the following benefits:
* It is able to handle both directed or undirected graphs.
* Being based on indices, it is easily serializable. Only the contents
of the head_ array need to be stored. Even so, serialization is
currently not implemented.
* The node indices and arc indices can be stored in 32 bits, while
still allowing to go a bit further than the 4gigabyte
limitation (48 gigabytes for a pure graph, without capacities or
costs.)
* The representation can be recomputed if edges have been loaded from
* The representation can be recomputed if edges have been loaded from
external memory or if edges have been reordered.
* The memory consumption is: 2 * m * sizeof(NodeIndexType)
+ 2 * m * sizeof(ArcIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant.
The EbertGraph implementation differs from the implementation described in
[Ebert 1987] in the following respects:
* arcs are represented using an (i, ~i) approach, whereas Ebert used
(i, i). Indices for direct arcs thus start at 0, in a fashion that is
compatible with the index numbering in C and C++. Note that we also tested
a (2*i, 2*i+1) storage pattern, which did not show any speed benefit, and
made the use of the API much more difficult.
* because of this, the 'nil' values for nodes and arcs are not 0, as Ebert
first described. The value for the 'nil' node is set to 1, while the
value for the 'nil' arc is set to the smallest integer representable with
ArcIndexSize bytes.
* it is possible to add arcs to the graph, with AddArc, in a much simpler
way than described by Ebert.
* TODO(user) although it is already possible, using the
GroupForwardArcsByFunctor method, to group all the outgoing (resp.
incoming) arcs of a node, the iterator logic could still be improved to
allow traversing the outgoing (resp. incoming) arcs in O(out_degree(node))
(resp. O(in_degree(node))) instead of O(degree(node)).
* TODO(user) it is possible to implement arc deletion and garbage collection
in an efficient (relatively) manner. For the time being we haven't seen an
application for this.
The ForwardEbertGraph representation is like the EbertGraph case described
above, with the following modifications:
* The part of the head_[] array with negative indices is absent. In its
place is a pointer tail_ which, if assigned, points to an array of tail
nodes indexed by (nonnegative) arc index. In typical usage tail_ is NULL
and the memory for the tail nodes need not be allocated.
* The array of arc tails can be allocated as needed and populated from the
adjacency lists of the graph.
* Representing only the forward star of each node implies that the graph
cannot be serialized directly nor rebuilt from scratch from just the head_
array. Rebuilding from scratch requires constructing the array of arc
tails from the adjacency lists first, and serialization can be done either
by first constructing the array of arc tails from the adjacency lists, or
by serializing directly from the adjacency lists.
* The memory consumption is: m * sizeof(NodeIndexType)
+ m * sizeof(ArcIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant when the array of arc tails is absent. Allocating
the arc tail array adds another m * sizeof(NodeIndexType).
The ForwardStaticGraph representation is restricted yet farther
than ForwardEbertGraph, with the benefit that it provides higher performance to those applications that can use it.
* As with ForwardEbertGraph, the presence of the array of arc
tails is optional.
* The outgoing adjacency list for each node is stored in a
contiguous segment of the head_[] array, obviating the
next_adjacent_arc_ structure entirely and ensuring good locality
of reference for applications that iterate over outgoing
adjacency lists.
* The memory consumption is: m * sizeof(NodeIndexType)
+ n * sizeof(ArcIndexType)
plus a small constant when the array of arc tails is absent. Allocating
the arc tail array adds another m * sizeof(NodeIndexType).
Classes 


ArcFunctorOrderingByTailAndHead  
GraphBuilderFromArcs  
PermutationIndexComparisonByArcHead  
StarGraphBase  
TailArrayManager 