Heuristic Search in Artificial Intelligence: Recent Enhancements and Applications

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Heuristic Search in Artificial Intelligence
Recent Enhancements and Applications

• Ariel Felner
• ISE Department
• Ben-Gurion University.
• felner_at_bgu.ac.il

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optimal path search algorithms

• For small graphs provided explicitly, algorithm
  such as Dijkstras shortest path, Bellman-Ford or
  Floyd-Warshal. Complexity O(n2).
• For very large graphs , which are implicitly
  defined, the A algorithm which is a best-first
  search algorithm.

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Best-first search schema

• sorts all generated nodes in an OPEN-LIST and
  chooses the node with the best cost value for
  expansion.
• generate(x) insert x into OPEN_LIST.
• expand(x) delete x from OPEN_LIST and generate
  its children.
• BFS depends on its cost (heuristic) function.
  Different functions cause BFS to expand different
  nodes..

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Open-List
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Best-first search Cost functions

• g(x) Real distance from the initial state to x
• h(x) The estimated remained distance from x to
  the goal state.
• ExamplesAir distance
• Manhattan Dinstance
• Different cost combinations of g and h
• f(x)level(x) Breadth-First Search.
• f(x)g(x) Dijkstras algorithms.
• f(x)h(x) Pure Heuristic Search (PHS).
• f(x)g(x)h(x) The A algorithm (1968).

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A

• A is a best-first search algorithm that uses
  f(n)g(n)h(n) as its cost function.
• f(x) in A is an estimation of the shortest path
  to the goal via x.
• A is admissible, complete and optimally
  effective. Pearl 84
• Result any other optimal search algorithm will
  expand at least all the nodes expanded by A

Breadth First Search
A
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How to improve search?

• Enhanced algorithms Perimeter-search, RBFS,
  Frontier-search etc, They all try to better
  explore the search tree.
• Better heuristics more parts of the search tree
  will be pruned.
• In the 3rd Millennium we have very large
  memories.
• We can build large tables.
• For enhanced algorithms large open-lists or
  transposition tables. They store nodes
  explicitly.
• A more intelligent way is to store general
  knowledge. We can do this with heuristics

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Pattern databases

• Many problems can be decomposed into subproblems
  (patterns) that must be also solved.
• The cost of a solution to a subproblem is a
  lower-bound on the cost of the complete solution
• Instead of calculating the lower bounds on the
  fly, we expand the whole pattern-space and store
  the solution to each pattern configuration in a
  pattern database

Search space
Mapping function
Pattern space
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Non-additive pattern databases

• 15 puzzle 1013 states
• Fringe pattern database Culberson Schaeffer
  1996. Has only 259 Million states.
• Improvement of a factor of 100 over Manhattan
  Distance

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Rubiks Cube (Korf 1997)

• Has 1019 States.
• PDB of the corner cubies has only 88 Million
  states.
• Korf AAAI-97 built 2 other pattern databases
  for this domain.
• The best way to combine different non-additive
  pattern databases is to take their maximum!

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Disjoint Additive Databases

• 15 and 24 puzzles Korf Felner AIJ-02,
• Felner, Korf Hanan JAIR-04

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• Values of disjoint databases can be added for the
  heuristic
• Better than maxing heuristics

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6
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Dynamically-partitioned additive databases

• Statically-partition databases do not capture
  conflicts of tiles from different patterns.
• We want to store as many pattern databases as
  possible and partition them to disjoint
  subproblems on the fly such the chosen partition
  will yield the best heuristic.
• This is called Dynamically
• Partitioning PDBs.

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1
2
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3
4
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Experimental Results15 puzzle
Fives
Sixes
SevenEight
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Results 24 puzzle.

• For the 24 puzzle we compared the SDB of sixes
  with the DDB of pairs triples on 10 random
  instances.

• The relative advantage of the SDB decreases when
  the problem scales up
• What will happen for the 6x6 35 puzzle???

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35 puzzle
We sampled 10 Billion random states and
calculated their heuristic. The table was created
by the method presented by Korf, Reid and
Edelkamp. (AIJ 129, 2001)

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Tile puzzles Summary

• The relative advantage of the SDB over DDB
  decreases over time.
• For the 15 puzzle 1/2 of the domain is stored.
• For the 24 puzzle 1/4 of the domain is stored.
• For the 35 puzzle 1/7 of the domain is stored.
• The memory needed by the DDB was 100 times
  smaller than that of the SDB!!

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4-peg Towers of Hanoi (TOH4)

• There is a conjecture about the length of optimal
  path but it was not proven.
• Systematic search is the only way to solve this
  problem or to verify the conjecture.
• There are too many cycles. IDA as a DFS will not
  prune these cycles. Therefore, A (actually
  frontier A Korf Zhang 2000) was used.

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Heuristics for the TOH

• Infinite peg heuristic (INP) Each disk moves to
  its own temporary peg.
• Additive pattern databases
• Felner, Korf Hanan, JAIR-04

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Additive PDBS for TOH4

• Partition the disks into disjoint sets
• Store the cost of the complete pattern space of
  each set in a pattern database.
• Add values from these PDBs for the heuristic
  value.
• The n-disk problem contains 4n states
• The largest database that we stored was of 14
  disks which needed 414256MB.

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10
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TOH4 results

• The difference between static and dynamic is
  covered in Felner, Korf Hanan JAIR-04

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Vertex-Cover (VC) Felner et al JAIR-04

• Given a graph we want the minimal set of vertices
  such that they cover all the edges.
• VC was one of the first problems that was proved
  to be NP-complete.
• Search tree
• At each level, either include or exclude a
  vertex.
• Improvements
• If a node is excluded, all its neighbors bust be
  included.
• Dealing with degree-0 and degree-1 vertices.

0
1
2
3
R
X0 V1,2,3
V0
V0,2 X1
V0,1
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Depth-first Baranch and Bound

• DFBnB Searches the tree from left to right.
• Expands only sub trees with costs smaller than
  the best solution found so far.
• We also used Itervative Deepening A IDA (Korf
  85)

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Heuristics for VC

• The included edges form the g part of fgh.
• We want an admissible heuristic of the free
  vertices.
• Pairwise heuristic
• A maximum-matching of the free-graph.
• For a triangle we can add two to the heuristic.
• In general, a clique of size k contributes k-1 to
  h.
• So partition the free-graph into disjoint
  cliques and sum up their heuristics.

VC EX
1
3
2
4
Free vertices
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Additive pattern databases

• Clique is NP-complete. However, in random
  graphs, cliques of size 5 and larger are rare.
  Thus, it is easy to finds small cliques
• Pattern databases Instead of finding the cliques
  on the fly we identify them before the search and
  store them in a pattern database. We stored
  cliques of size 4 or smaller.
• During the search we need to retrieve disjoint
  cliques from the pattern database.

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VC results

• The results are on random graphs of size 150 and
  an average degree of 16.
• When we added our dynamic database to the best
  proven tree search algorithm we further improved
  the running time by a fact or more than 10.

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Conclusions and Summary

• In general Additivity can be applied whenever a
  problem can be decomposed into disjoint
  subproblems such that the sum of the costs is a
  lower bound on cost of the complete problem.
• Additive databases is a special case of additive
  heuristic where we save the heuristics in a
  table.

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The Graph Partitioning ProblemFelner AMAI-2004

• Given a graph G(E,V) the problem is to partition
  the graph into two equal sized subsets of
  vertices.
• The number of edges that are crossing the
  partition should me minimized.
• The partition in the graph on the right is of
  cost 2.

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GPP as a search problem

• A sub problem in GPP is to assign a vertex to one
  of the subsets of the partition
• Each level of the search tree corresponds to a
  specific vertex of the graph.
• Each branch assigns the vertex to another subset
  of the partition.

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• Each node of the tree is a partial partition
  including some of the vertices.
• Size of the tree 2n

1,2
1
2
1
1,2,3
1,2
1,3
3
2
2,3

• Leaves of the tree are the complete partitions.
  One of them is the optimal.

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Definitions

• A node of the search tree is denoted by k while
  vertex of the graph is denoted by x.
• A vertex that is already assigned to one of the
  subsets is called an assigned vertex.
• Each of the other vertices is a free vertex.
  Free vertices are unsolved subgoals.
• Given a node k of the search tree we define
• g(k) the number of edges that already cross
  the partial partition due to assigned vertices.
• h(k) A lower bound on the number of edges
  that will cross the given partition due to free
  vertices.

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A heuristic from the free vertices

• The free vertices have many edges connected to
  them.
• Can we have an estimation on the number of such
  edges that must cross the partition?

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Free vertices
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• More definitions
• The subsets of the partial partition are A and B.
• Each of the following heuristics completes the
  partition with A and B
• We can guess about A and B
• Types of the edges
• I Edges in A A
• II Edges from A to B
• III Edges from A to B
• IV Edges from A to B

• A B
• 3
• 4

A5,6 B7,8
II
3 4 7 8
1 2 5 6
B
A
I
III
A
B
IV
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f0 Uniform Cost Search

• f0(k) g(k).
• Edges that already cross the partition. Edges of
  type II.
• Mainly for comparison reasons.

Assigned
II
3 4 7 8
1 2 5 6
A
B
Free
A
B
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f1 Adding edges of type III

• For each free vertex x we define d(x,A) as the
  number of edges from x to A and d(x,B) as the
  number of edges from to B.

A B

• An admissible heuristic for a vertex x will be
  h1(x)mind(x,A),d(x,B)
• h1(k)summing h1(x) for all free vertices x.
• f1(k)g(k)h1(k)

1 2
3 4
x
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• Results for other graphs as well as using IDA
  were very similar.
• A better heuristic solves the problem faster

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• f3 if faster than f0 by almost 10,000 for
  graphs with density of 6.
• f3 is faster than f1 by a factor of 100 for a
  graph with density of 20.

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• Graphs of size 100. Solved by f3 only.
• Once again as the density of the graph increase
  the optimal cut increases linearly and the time
  to solve the problem increases exponentially.

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Other domains

• Traveling salesman problem Korf 1996
• Number partitioning Korf 1998
• Bin-packing Korf 2002
• Rectangle packing Korf 2004.
• Multiple sequence alignment Hansen Zough 2004

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Best Usage of Memory

• Given 1 giga byte of memory, how do we best use
  it with pattern databases?
• Holte, Newton, Felner, Meshulam and Furcy,
  ICAPS-2004 showed that it is better to use many
  small databases and take their maximum instead of
  one large database.
• We will present a different (orthogonal) method
  Felner, Mushlam Holte AAAI-04.

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Compressing pattern database Felner et al
AAAI-04

• Traditionally, each configuration of the pattern
  had a unique entry in the PDB.
• Our main claim ?
• Nearby entries in PDBs are highly correlated
  !!
• We propose to compress nearby entries by storing
  their minimum in one entry.
• We show that ?
• most of the knowledge is preserved
• Consequences Memory is saved, larger patterns
  can be used ? speedup in search is obtained.

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Cliques in the pattern space

• The values in a PDB for a clique are d or d1
• In permutation puzzles cliques exist when only
  one object moves to another location.

d
G
d1
d

• Usually they have nearby entries in the PDB
• A44444

A clique in TOH4
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Compressing cliques

• Assume a clique of size K with values d or d1
• Store only one entry (instead of K) for the
  clique with the minimum d. Lose at most 1.
• A44444 A44441
• Instead of 4p we need only 4(p-1) entries.
• This can be generalized to a set of nodes with
  diameter D. (for cliques D1)
• A44444 A44411
• In general compressing by k disks reduces memory
  requirements from 4p to 4(p-k)

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TOH4 results 16 disks (142)

• Memory was reduced by a factor of 1000!!! at a
  cost of only a factor of 2 in the search effort.

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TOH4 larger versions
Memory was reduced by a factor of 1000!!! At a
cost of only a factor of 2 in the search
effort. Lossless compressing is noe efficient in
this domain.

• For the 17 disks problem a speed up of 3 orders
  of magnitude is obtained!!!
• The 18 disks problem can be solved in 5 minutes!!

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Tile Puzzles
Goal State
Clique

• Storing PDBs for the tile puzzle
• (Simple mapping) A multi dimensional array ?
  
• A1616161616 size1.04Mb
• (Packed mapping) One dimensional array ?
  A1615141312 size 0.52Mb.
• Time versus memory tradeoff !!

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15 puzzle results

• A clique in the tile puzzle is of size 2.
• We compressed the last index by two ?
• A161616168

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24 puzzle

• The same tendencies were obtained for the 24
  puzzle.
• The 6-6-6-6 partitioning is so good that adding
  another set of 6-6-6-6 did not speedup the
  search.
• We have also tried a 7-7-5-5 partitioning but it
  did not speedup the search.

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Ongoing and future work

• An item for the PDB of tiles (a,b,c,d) is in the
  form ltLa, Lb, Lc, Ldgtd
• Store the PDBs in a Trie
• A PDB of 5 tiles will have a level in the trie
  for each tile. The values will be in the leaves
  of the trie.
• This data-structure will enable flexibility and
  will save memory as subtrees of the trie can be
  pruned

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Trie pruninig
Simple (lossless) pruning Fold leaves with
exactly the same values.
No data will be lost.
2
2
2
2
2
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Trie pruninig

• Intelligent (lossy) pruning
• Fold leaves/subtrees with are correlated to each
• other (many option for this!!)
• Some data will be lost.
• Admissibility is still kept.

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2
2
2
4
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Trie Initial Results
A 5-5-5 partitioning stored in a trie with simple
folding
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Neural Networks (NN)

• We can feed a PDB into a neural network engine.
  Especially, Addition above MD
• For each tile we focus on its dx and dy from its
  goal position. (i.e. MD)
• Linear conflict
• dx1 dx2 0
• dy1 gt dy21
• A NN can learn
• these rules

2
1
dy1 2 dy20
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Neural network

• We train the NN by feeding the entire (or part of
  the) pattern space.
• For example for a pattern of 5 tiles we have 10
  features, 2 for each tile.
• During the search, given the locations of the
  tiles we look them up in the NN.

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Neural network example
dx4
Layout for the pattern of the tiles 4, 5 and 6
dy4
dx5
4
dy5
dx6
dy6
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Neural Network problems

• We face the problem of overestimating and will
  have to bias the results towards underestimating.
• We keep the overestimating values in a separate
  hash table
• Results are encouraging!!

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Ongoing and Future Work

• Dual pattern Databases
• VARIABLES versus VALUES
• In the tile puzzles locations are variables and
  tiles are the values.
• We ask Who is located in location X.
• Swap their role and ask Where is tile X
  located

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Dual pattern Databases

• In regular PDBs we asked
• Where are tiles lt1,2,3,4gt located and what
  does it take to move them to their goal postion
• We now ask
• Who is located in positions lt1,2,3,4gt and
  what does it take to distribute them to their
  goals
• The same tables answers both questions!!

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Search in Artificial Intelligence course

• Course number 37214506
• Semester B.
• Monday at 1700-1830.
• In Ben-Gurion University.
• Open for CS students.