Global Constraints for Strategy Graphs

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Global Constraints for Strategy Graphs

• Gary Gengo
• COM 3220
• June 9, 1998

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Outline

• Global constraints on strategies
• Traversal algorithm
• Checking constraints algorithm
• Demo

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Global Constraints on Strategies

• Given class graph CG, strategy graph SG, source
  s,target t, constraint X lt Y
• Question Does the following constraint hold for
  all paths P in CG corresponding to some path from
  s to t in SG?
• for every vertex Y in the path P, there is a
  vertex X in P that comes before Y

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Constraint Example 1
Class Graph
A lt B true B lt A false A lt X true X lt A
false B lt X true X lt B false
Strategy Graph
Note A lt B even if B can come before A (e.g.
path ABXABX)
Traversal Graph
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Constraint Example 2
Class Graph
P lt Q true Q lt P true
Strategy Graph
Since neither P nor Q are part of any path from
source to target, P lt Q and Q lt P
Traversal Graph
source node
target node
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Constraint Example 3
Class Graph
Dlt C false (ABCABDE) C lt D false (ABDE)
Strategy Graph
Traversal Graph
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Constraint Example 4
Class Graph
Dlt C true C lt D false
Strategy Graph
Traversal Graph
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Traversal Graph Algorithm TG(CG,SG,s,t)

• Step1For each SG edge (u,v), make a copy Guv of
  CG
• Step 2For each vertex v in SG, and for each
  combination (u,v) (v,w) edges in SG add
  intercopy edges of the form (pred(v) in Guv , v
  in Gvw)
• Step 3Eliminate edges not reachable from some
  source s and target t

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Traversal AlgorithmStep 1
Class Graph
Strategy Graph
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Traversal AlgorithmStep 2
Class Graph
Strategy Graph
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Traversal AlgorithmStep 3
Class Graph
Strategy Graph
Traversal Graph
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Traversal Algorithm Property

• Each node in the traversal graph can reach the
  target.
• This occurs in Step 3 of the algorithm

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Alternate Algorithm

• You might compute traversal graph by
• First computing the TG for each edge
• Then add intercopy edges
• HOWEVER, property mentioned before holds ONLY if
  the strategy graph does not have any useless
  edges
• Previous algorithm is better in this sense
• We use previous algorithm in checking constraints
  algorithm

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Algorithm to check constraint XltY

• Step 1 Compute traversal graph TG1
  TG(CG,SG, s, t)
• Step 2 Remove all edges with X as a vertex from
  TG1
• Step 3 For each source s and each Y inTG1
• compute newTG TG(TG1,newSG,s,Y) where newSG is
  
• if newTG is not empty, constraint is violated

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If you use Alternate Traversal Algorithm

• Change Step 3 to For each source s, each target
  t, and each Y inTG1
• compute newTG TG(TG1,newSG,s,t) where newSG is
  
• if newTG is not empty, constraint is violated
• change necessary because there might not be a
  path from Y to t in TG

i
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Checking Constraint Example 1
check constraint P lt Q
Class Graph
compute traversal graph for class graph
strategy graph
Strategy Graph
Traversal Graph
traversal graph is empty so constraint is
satisfied
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Checking Constraint Example 2
check constraint C lt D
Class Graph
compute traversal graph for class graph
strategy graph
Strategy Graph
Traversal Graph
traversal graph is not empty so constraint is
not satisfied
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Checking Constraint Example 3
check constraint C lt D
Class Graph
Strategy Graph
Traversal Graph
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Checking Constraint Example 3(contd)
compute traversal graph for class graph
check constraint C lt D
strategy graph
Ai
for i,j 1 to 2
traversal graph non-empty for i1 and j1
constraint C lt D is not satisfied
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Implementation Details

• command line
• java graph Cgfile SGfile source target X Y
• check constraint XltY on CG,SG
• graphs stored as text files
• first line is list of vertices
• all other lines give adjacency list
• no error checking on graph input
• need ltreturngt at end of each line