Notes 5: Heuristic Search

1
Notes 5 Heuristic Search

• ICS 171 Summer 1999

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Summary

• Heuristic search strategies
• heuristics and heuristic estimates
• best-first search
• hill-climbing
• A optimal search using heuristics
• dynamic programmingWe will see that A can find
  optimal paths and expand relatively few nodes by
  using heuristics

3
Example of Path Costs
Optimal (minimum cost) path is S-A-D-E-F-G
4
Heuristics and Search

• in general
• a heuristic is a rule-of-thumb based on
  domain-dependent knowledge to help you solve a
  problem
• in search
• one uses a heuristic function of a state where
  h(node) estimated cost of cheapest
  path from the state for that node to a goal
  state G
• h(G) 0
• h(other nodes) gt 0
• (note we will assume all individual node-to-node
  costs are gt 0)
• How does this help in search
• we can use knowledge (in the form of h(node))to
  reduce search time
• generally, will explore more promising nodes
  before less promising ones

5
Example of Heuristic Functions
B
C
A
10.4
4.0
6.7
11.0
G
S
8.9
3.0
6.9
D
F
E
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Search Method 8 Best First Search

• Best-First
• uses estimated path cost from node to goal
  (heuristic, hcost)
• ignores actual path cost
• after each expansion
• sort nodes according to estimated path-cost from
  node to goal
• in effect, it always expands the most promising
  node on the fringe (according to the heuristic)
• e.g., finding a route to Seattle, always extend
  the path from the most northerly city among the
  existing routes
• Uniform Cost
• uses path cost from root to node
• after each expansion
• sort nodes according to path-cost from start S to
  node

7
Example of Best-First Search in action
S
10.4
8.9
D
A
10.4
A
6.9
E
3.0
F
B
6.7
0
G
Note this is not the optimal path for this
problem
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Properties of Uniform Cost and Best-First Search

• Completeness
• Both Best-first and Uniform Cost search are
  complete
• Optimality
• Best-first is not optimal in general
• why? its ordering of nodes according to estimated
  cost to goal G there is no reason this will
  produce the path which is really lowest-cost
  first
• e.g. most northerly gt route to Seattle might
  go through Denver!
• uniform cost is optimal
• Time and Space Complexity
• Both could be O(bd) in the worst case
• But in practice, best-first usually uses far
  fewer nodes

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Search Method 9 Hill-Climbing Search

• General Idea of hill-climbing
• always expand the node which appears closest to
  goal (according to the heuristic function h)
  (greedy)
• only keep 1 node in memory (on the Q) at any time
  (no backtracking)
• also known as steepest ascent or
  steepest-descent
• can be used to maximize (climb) or minimize
  (descend)
• is it complete? optimal? what is the time and
  space complexity?
• Relation to the Best-First Search Algorithm?
• it is a special case, but where the memory is
  limited to a single node at any time (i.e., no
  backtracking)
• hill-climbing with multiple restarts from
  randomly chosen initial conditions is often
  extremely simple to implement and can be quite
  effective (usually worth trying)

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Combining heuristic and path costs

• So far,
• we have only considered cost of path from start
  to current node
• or separately considered estimate of cost from
  node to goal
• We can also use both costs together
• fcost(node) d(S to node)
  h(node to G)
• Notation
• fcost(N) estimated cost from S to G via N
• d(A to B) path cost from A to B (exact)
• h(N to S) estimate of path cost from N to G
  
• If heuristic is accurate we will quickly zero in
  on goal
• But in general it is difficult to come up with a
  good heuristic

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Comments on heuristic estimation

• The estimate of the distance is called a
  heuristic
• typically it comes from domain knowledge
• e.g., the straight-line distance between 2
  points
• The closer the heuristic is to the real (unknown)
  path cost, the more effective it will be
• If the heuristic never overestimates, then the
  search procedure using this heuristic is
  admissible, i.e.,
• h(N) is less than or equal to realcost(N to
  G)
• Admissible search is optimal
• i.e., if one uses an admissible heuristic to
  order the search one is guaranteed to find the
  optimal solution

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Heuristic Underestimates of Path Cost

• Underestimates
• say we use a heuristic function h which is never
  greater than the true path-cost from a node to a
  goal G fcost(node) d(S to node) h(node
  to G)
• use branch-and-bound, ordering Q according to
  ucost
• This is optimal Why?
• consider a node N behind the optimal path in
  the Q
• if so, then fcost(G) lt fcost(N)
• on the LHS, fcost(G) realcost(G) by definition
• on the RHS fcost(N) is less than or equal to
  realcost(N) (since the heuristic underestimates,
  by assumption)
• so realcost(G) lt realcost(N)
• if we extend N further, its cost can only
  increase, i.e., so it cannot be optimal (i.e.,
  have realcost lt realcost(G)).

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The A Algorithm (Search Method 10)

• The A algorithm simply orders the nodes on the Q
  according to fcost(node) where f(cost) uses an
  admissible heuristic
• i.e., it sorts nodes on Q according to an
  admissible heuristic h
• It is like uniform-cost,
• but uses fcost(node) path-cost(S to node)
  h(node)
• rather than just path cost(S to node)
• For your project, you will be implementing A as
  defined on next overhead

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Pseudo-code for the A Search Algorithm
Initialize Let Q S While Q is not
empty pull Q1, the first element in Q if Q1 is
a goal report(success) and quit else child_node
s expand(Q1) lteliminate child_nodes which
represent loopsgt put remaining child_nodes in
Q sort Q according to ucost pathcost(S to
node) h(node) end Continue

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4
1
B
A
C
2
5
G
2
S
3
5
4
2
D
E
F
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Example of A in action
S
5 8.9 13.9
2 10.4 12..4
D
A
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Example of A Algorithm in action
S
5 8.9 13.9
2 10.4 12..4
D
A
3 6.7 9.7
D
B
4 8.9 12.9
7 4 11
8 6.9 14.9
6 6.9 12.9
E
C
E
Dead End
B
F
10 3.0 13
11 6.7 17.7
G
13 0 13
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Examples of Heuristic Functions for A

• the 8-puzzle problem
• the number of tiles in the wrong position
• is this admissible?
• the sum of distances of the tiles from their goal
  positions, where distance is counted as the sum
  of vertical and horizontal tile displacements
  (Manhattan distance)
• is this admissible?
• How can we invent admissible heuristics in
  general?
• look at relaxed problem where constraints are
  removed
• e.g., we can move in straight lines between
  cities
• e.g., we can move tiles independently of each
  other

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Effectiveness of A Search Algorithm
Average number of nodes expanded
d IDS A(h1) A(h2) 2 10 6 6 4 112 13 12
8 6384 39 25 12 364404 227 73 14 3473941 53
9 113 20 ------------ 7276 676
Average over 100 randomly generated 8-puzzle
problems h1 number of tiles in the wrong
position h2 sum of Manhattan distances
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The Dynamic Programming Principle
S
A
D
6
B
D
7
3

• S-D is cheaper (6 units) than S-A-D (7 units)
• so we can eliminate the S-A-D path from
  consideration forever
• in general, for any state in state-space
• we need only consider 1 path to that state
• this 1 path is the shortest path found so far
• this is non-trivial to implement in code
• General principle
• min distance from S to G through D sum
  mindistance(S-D) mindistance(D-G)

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Summary

• In practice we often want the goal with the
  minimum cost path
• Exhaustive search is impractical except on small
  problems
• Uniform cost search always expands the lowest
  path-cost node on the fringe
• Heuristic estimates of the path cost from a node
  to the goal can be efficient in reducing the
  search space (used in best-first search)
• The A algorithm combines both of these ideas
  with admissible heuristics (which underestimate)
  guaranteeing optimality.
• Reading
• Chapter 4 pages 92 to 104