DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

• The set of expansion edges X is subdivided into
a set of disjoint subsets which have the same
source node.
• A task t is connected to exactly one task by an
edge e unless e is a member of the set of
expansion edges X.
The basic principle behind the graph reduction approach
is as follows. For each type of reduced graph mapping R,
we define equivalence criteria between nodes in the graph
to find abstract nodes. An example of criteria C 1 would
be “All tasks at the same agent with parent tasks
originating from another agent.” In applying the graph
reduction algorithm, all nodes which satisfy this criteria
are aggregated into a single node.
Also, for every graph reduction mapping R, we define a
equivalence criteria for abstract edges. Abstract edges
are aggregates of equivalent subgraphs into a single
representative edge.
Continuing the example, consider a criteria C 2 that states
“Subgraphs that connect all aggregate nodes satisfying C 1
and connect to organizational assets associated with the
same agent as C 1 .” Also, we define a second node
equivalence criterion C 3 as “All tasks at the same agent
which are allocated to assets representing external
organizations.”
We apply the criteria C 1 , C 2 and C 3 to a subgraph
t 1 (→x 1 )t 2 (→x 2 ) t 3 (→l 2 )a 1 associated with agent A. This
subgraph represents a task t 1 expanded to a task t 2 which
is in turn expanded to a task t 3 and allocated to an asset
a 1 . Here, we assume that t 1 has a parent external to A. We
further assume that the asset a 1 is an organizational asset
representing agent B. The task node t 1 satisfies C 1 and
hence is aggregated into an abstract node n 1 . Similarly,
the task node t 3 satisfies C 3 and is aggregated into a node
n 2 . The subgraph (→x 2 )t 2 (→x 2 ) therefore matches C 2 and
is associated with a single edge e 1 ∈E’ which connects
the two abstract nodes n 1, , n 2, ∈ N’.
Together, the equivalence criteria for abstract nodes and
edges leads to the identification of equivalent subgraphs.
By changing the equivalence criteria for aggregated nodes
and edges, a variety of different graph reduction
mappings can be achieved.
The computational complexity of the approach described
above varies depending on the complexity of finding and
matching isomorphic subgraphs. Cougaar task graphs are
generally well structured, and for most of the equivalence
criteria that are described in the following, subgraphs can
be matched using a (worst case) O(n) graph traversal,
where n is the size of the subgraph. In the equivalence
criteria described in the next section, all subgraphs fall
within a single agent’s plan graph, thus bounding the size
of the matched subgraphs. If m is the total number of
abstract nodes discovered, then the total computational
complexity is O(m * n).
3.1 Algorithm implementation and applications
The implementation of the graph reduction algorithm
within Castellan allows generation of the task graph from
an arbitrary stream of events. Except for asset and
organizational information, it is not necessary to have a
complete plan graph to use this approach to devise
reduced graphs.
The following types of reduced graphs were found to be
useful for understanding Cougaar societies and are
implemented within the Castellan system.
Aggregate task graphs defined using an equivalence
criterion that maps tasks of the same type with the
identical verb to single nodes. Also, theis equivalence
criterion requires a strict ordering in depth between tasks
which are aggregated. Specifically, in order to map a
node n to an abstract node n’, the node n’s parents (and
all of its ancestors by implication) must map to an
abstract node n 2 ’ which is an ancestor of n’. This
requirement is imposed to prevent cycles from appearing.
Figure 2 shows a conceptual representation of task
aggregation in which multiple “similar” subgraphs are
collapsed into a single aggregate subgraph. An example
of a task aggregate plan graph is shown in Figure 3.
Asset dependency graphs consider the assets (both
organizational and physical) as abstract nodes. In this
case, no aggregation of assets is performed as all assets
are considered unique. The criteria for abstract edges are
as follows:
• All assets are mapped to abstract nodes.
(Optionally, additional asset matching criteria
can be introduced to aggregate assets.)
• In addition, all agents that generate tasks are
designed as “Source” abstract nodes. (These
serve as the roots of the reduced DAG.)
• All tasks and allocations that form plan graph
dependencies between different assets are
mapped to a single abstract edge.
Asset dependency graphs are useful for finding both
organization and physical dependencies within a
distributed plan.
Workflow graphs characterize the input/output
relationships between agents and are particularly useful