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Chapter 4
HEURISTIC SEARCH
Contents
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Hill-climbing
Dynamic programming
Heuristic search algorithm
Admissibility, Monotonicity, and Informedness
Using Heuristics In Games
Complexity Issues
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Heuristics
Rules for choosing paths in a state
space that most likely lead to an
acceptable problem solution
Purpose
– Reduce the search space
Reasons
– May not have exact solutions, need
approximations
– Computational cost is too high
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First three levels of the tic-tac-toe state space
reduced by symmetry
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The “most wins” heuristic applied to the first
children in tic-tac-toe
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Heuristically reduced state space for tic-tac-toe
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Hill-Climbing
Analog
– Go uphill along the steepest possible path until
no farther up
Principle
– Expand the current state of the search and
evaluate its children
– Select the best child, ignore its siblings and
parent
– No history for backtracking
Problem
– Local maxima – not the best solution
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The local maximum problem for hill-climbing
with 3-level look ahead
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Dynamic Programming
Forward-backward searching
Divide-and-conquer: Divided problems into
multiple interacting and related subproblems
Address issues of reusing subproblems solutions
An example: Fibonacci series
– F(0) = 1; F(1)=1; F(n)=F(n-1)+F(n-2)
– Keep track of the computation F(n-1) and F(n-2), and
reuse their results to compute F(n)
– Compare with recursion, much more efficient
Applications:
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String matching
Spell checking
Nature language processing and understanding
planning
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Global Alignment of Strings
• Find an optimal global alignment of two character
strings
• Data structure: (n+1)(m+1) array, each element
reflects the global alignment success to that point
• Three possible costs for the current state
• If a character is shifted along in the shorter string for better
possible alignment, the cost is 1 and recorded in the column
score; and If a new character is inserted, cost is 1 and
reflected in the row score
• If the characters to be aligned are different, shift and insert,
the cost is 2
• If identical, the cost is 0
• The initialization stage and first step in completing the
array for character alignment using dynamic
programming.
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• The initialization stage and first step in completing the
array for character alignment using dynamic
programming.
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The completed array reflecting the maximum
alignment information for the strings.
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A completed backward component of the
dynamic programming example giving one (of
several possible) string alignments.
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Minimum Edit Difference
Determine the best approximate words of
s misspelling word in spelling checker
Specified as the number of character
insertion, deletion, and replacements
necessary to turn the first string into the
second
Cost: 1 for insertion and deletion, and 2
for replacement
Determine the minimum cost difference
Data structure: array
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Initialization of minimum edit difference matrix
between intention and execution
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Array elements are the costs of the minimum editing to
that point plus the minimum cost of either an insertion,
deletion or replacement
Cost(x, y) = min{ Cost(x-1, y) + 1 (insertion cost),
Cost(x-1, y-1) + 2 (replacement cost),
Cost(x, y-1) + 1 (deletion cost) }
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Complete array of minimum edit difference
between intention and execution (of several
possible) string alignments.
Intention
ntentiondelete I, cost 1
etentionreplace n with e, cost 2
exention
replace t with x, cost 2
exenution insert u, cost 1
execution replace n with c, cost 2
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The Best-First Search
Also heuristic search – use heuristic
(evaluation) function to select the
best state to explore
Can be implemented with a priority
queue
– Breadth-first implemented with a queue
– Depth-first implemented with a stack
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The bestfirst
search
algorithm
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Heuristic search of a hypothetical state
space
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A trace of the execution of best-first-search
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Heuristic search of a hypothetical state space with
open and closed states highlighted
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Implement Heuristic Evaluation Function
Heuristics can be evaluated in different
ways
8-puzzle problem
– Heuristic 1: count the tiles out of places
compared with the goal state
– Heuristic 2: sum all the distances by which the
tiles are out of pace, one for each square a tile
must be moved to reach its position in the goal
state
– Heuristic 3: multiply a small number (say, 2)
times each direct tile reversal (where two
adjacent tiles must be exchanged to be in the
order of the goal)
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The start state, first moves, and goal state for an
example-8 puzzle
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Three heuristics applied to states in the 8puzzle
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Heuristic Design
Use the limited information available in a single
state to make intelligent choices
Empirical, judgment, and intuition
Must be its actual performance on problem instances
The solution path consists of two parts: from the
starting state to the current state, and from the
current state to the goal state
The first part can be evaluated using the known
information
The second part must be estimated using unknown
information
The total evaluation can be
f(n) = g(n) + h(n)
g(n) – from the starting state to the current state n
h(n) – from the current state n to the goal state
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The heuristic f applied to states in the 8-puzzle
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State space
generated in
heuristic
search of the
8-puzzle
graph
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The successive stages of open and closed that generate
the graph are:
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Open and closed as they appear after the 3rd iteration of
heuristic search
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Heuristic Design Summary
f(n) is computed as the sum of g(n) and h(n)
g(n) is the depth of n in the search space and has
the search more of a breadth-first flavor.
h(n) is the heuristic estimate of the distance from
n to a goal
The h value guides search toward heuristically
promising states
The g value grows to determine h and force
search back to a shorter path, and thus prevents
search from persisting indefinitely on a fruitless
path
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Admissibility, Monotonicity, and
Informedness
A best-first search algorithm
guarantee to find a best path, if
exists, if the algorithm is admissible
A best-first search algorithm is
admissible if its heuristic function h is
monotone
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Admissibility and Algorithm A*
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Monotonicity and Informedness
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Comparison of state space searched using heuristic search with space
searched by breadth-first search. The proportion of the graph searched
heuristically is shaded. The optimal search selection is in bold. Heuristic used
is f(n) = g(n) + h(n) where h(n) is tiles out of place.
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Minimax Procedure
Games
– Two players attempting to win
– Two opponents are referred to as MAX
and MIN
A variant of game nim
– A number of tokens on a table between
the 2 opponents
– Each player divides a pile of tokens into
two nonempty piles of different sizes
– The player who cannot make division
losses
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Exhaustive Search
State space for a
variant of nim.
Each state
partitions the
seven matches
into one or more
piles
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Maxmin Search
Principles
– MAX tries to win by maximizing her score, moves to a
state that is best for MAX
– MIN, the opponent, tries to minimize the MAX’s score,
moves to a state that is worst for MAX
– Both share the same information
– MIN moves first
– The terminating state that MAX wins is scored 1,
otherwise 0
– Other states are valued by propagating the value of
terminating states
Value propagating rules
– If the parent state is a MAX node, it is given the
maximum value among its children
– If the parent state is a MIN state, it is given the
minimum value of its children
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Exhaustive
minimax for
the game of
nim. Bold
lines indicate
forced win
for MAX.
Each node is
marked with
its derived
value (0 or 1)
under
minimax.
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Minmaxing to Fixed Ply Depth
If cannot expand the state space to
terminating (leaf) nodes (explosive), can
use the fixed ply depth
Search to a predefined number, n, of
levels from the starting state, n-ply lookahead
The problem is how to value the nodes at
the predefined level – heuristics
Propagating values is similar
– Maximum children for MAX nodes
– Minimum children for MIN nodes
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Minimax to a hypothetical state space. Leaf
states show heuristic values; internal states
show backed-up values.
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Heuristic measuring conflict applied to states of
tic-tac-toe
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Two-ply minimax applied to the opening move of
tic-tac-toe
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Two ply minimax, and one of two possible MAX
second moves
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Two-ply minimax applied to X’s move near the end of
the game
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Alpha-Beta Procedure
Alpha-beta pruning to improve search efficiency
Proceeds in a depth-first fashion and creates two
values alpha and beta during the search
Alpha associated with MAX nodes, and never
decreases
Beta associated with MIN nodes, never increases
To begin, descend to full ply depth in a depth-first
search, and apply heuristic evaluation to a state and
all its siblings. The value propagation is the same as
minimax procedure
Next, descend to other grandchildren and terminate
exploration if any of their values is >= this beta value
Terminating criteria
– Below any MIN node having beta <= alpha of any of its MAX
ancestors
– Below any MAX node having alpha >= beta of any of its MIN
ancestors
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Alpha-beta pruning applied. States without numbers are
not evaluated.
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