SetA

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SetA

• An Efficient BDD-Based Heuristic Search Algorithm

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BDD-based Search

• Binary Decision Diagram
• A BDD is a directed acyclic graph representing a
  Boolean function
• Each state is encoded as a bit vector

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G
c
h0 (0,1)
(1,1) h0
b
d
a
h1 (0,0)
(1,0) h1
i
4-tuple represents search problems (S, T, i,
G) S set of states T transition
function i initial state G goal states
T(so, s1, s?0, s?1) ?s0 ? ? s1 ? s?0 ? ?
s?1 ? ?s0 ? ? s1 ? ? s?0 ? s?1 ? ?s0 ? s1 ?
s?0 ? s?1 ? s0 ? s1 ? s?0 ? ? s?1
We want to get from the start state, i, to the
goal state, G
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Problems with BDD Search

• Intermediate BDDs tend to be large compared to
  the BDD representing the result
• Can be solved with disjunctive partitioning

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SetA

• Generalization of weighted A
• f(1-w)g wh, w ? 0,1
• Assumes finite search domain and unit-cost
  transitions
• Expands a set of states instead of just a single
  state
• Input is an improvement partitioning (i.e.) the
  h-value is reduced by the same amount

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SetA Data Structures

• Priority Queue and Reach Structure
• Each node in the queue contains a BDD
  representing the states with particular g and h
  values
• Lowest f-value has highest priority
• Lowest h-value breaks ties
• Upper bound limits the size nodes
• Reach structure is for loop detection

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SetA Algorithm

• Insert initial state onto queue
• Compute h-value on initial state
• Loop until queue is empty or reached goal
• Pop node of queue
• Find the image of it and the Improvement function
• Prune states seen before
• Path is extracted by applying transitions
  backwards

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Function SetA(IP, init, goal, u,
w) Q.initialize(u,w,goal) g lt- 0 h lt-
h(init) Q.insert(init, g, h) R.update(init,
g) while ?Q.empty() and ?Q.topAtGoal() top lt-
Q.pop() for j 0 to IP next lt- image(top,
IPj) R.prune(next) g lt- top.g 1 h lt-
top.h impr(IPj) Q.insert(next, g,
h) R.update(next, g) if Q.empty() then
NoPathExists else R.extractPath()
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Weight

• For w 0.5 SetA behaves like A
• For w 0.0 SetA behaves like best-first search
• For w 1.0 SetA behaves like breadth-first
  search
• Can be used to compensate for a bad heuristic

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Improvement Partitioning

• A disjunctive partitioning where the transitions
  of a partition will reduce the h-value by the
  same amount
• The improvement partitioning is non-trival to
  compute
• Only needs to be computed once, prior to the
  search

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Intro to IGk problem

• A state is a set of facts and an action is a
  triple sets of facts
• Actions are (pre, add, del)
• Solution is non-trivial to find since the
  heuristic gives no information to guide the
  search for the first k steps

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IGK
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Intro to DxVyMz

• A set of sliders are moved between the corner
  positions of hyercubes
• A corner can be occupied by at most one slider
• Dimension of the hyercube is y
• There are z sliders
• x sliders are on the same cube

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Example D5V3M7
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DxV4M15
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Gripper Experiment
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Logistics Experiment
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Conclusion

• Successfully combined BDD-based search and
  heuristic search
• Several order of magnitude faster than BDD based
  breadth-first search and A
• However, success may be due to an inherent
  simplicity of the benchmark domains when using
  the right heuristics