Chapter 4 Heuristic Search Continued

1
Chapter 4Heuristic SearchContinued
2
ReviewLearning Objectives

• Heuristic search strategies
• Best-first search
• A algorithm
• Heuristic functions
• Local search algorithms
• Hill-climbing
• Simulated annealing

3
ReviewHeuristic Search

• Greedy search
• Evaluation function h(n) (heuristic)
  estimate of cost from n to closest goal
• Example hSLD(n) straight-line distance from n
  to Bucharest
• Greedy search expands the node that appears to be
  closest to goal

4
ReviewHeuristic Search

• Properties of greedy search
• Complete?? No can get stuck in loops, e.g.,
• Complete in finite space with repeated-state
  checking
• Time?? O(bm), but a good heuristic can give
  dramatic improvement
• Space?? O(bm) keeps all nodes in memory
• Optimal?? No

5
ReviewHeuristic Search

• A search
• Premise - Avoid expanding paths that are already
  expansive
• Evaluation function f(n) g(n) h(n)g(n)
  cost so far to reach nh(n) estimated cost to
  goal from nf(n) estimated total cost of path
  through n to goal

6
ReviewHeuristic Search

• A search
• A search uses an admissible heuristici.e., h(n)
  ? h(n) where h(n) is the true cost from
  n.(also require h(n) ?0, so h(G) 0 for any
  goal G.)example, hSLD(n) never overestimates
  the actual road distance.

7
Heuristic Search

• A search example

8
Heuristic Search

• A search example

9
Heuristic Search

• A search example

10
Heuristic Search

• A search example

11
Heuristic Search

• A search example

12
Heuristic Search

• Properties of A
• Complete?? Yes, unless there are infinitely many
  nodes with f ? f(G)
• Time?? Exponential in relative error in
  h x length of solution.
• Space?? Keeps all nodes in memory
• Optimal?? Yes cannot expand f i1 until fi is
  finished A expands all nodes with f(n) lt C A
  expands some nodes with f(n) C A expands no
  nodes with f(n) gt C

13
Heuristic Search

• A algorithm
• Optimality of A (standard proof)
• Suppose some suboptimal goal G2 has been
  generated and is in the queue. Let n be an
  unexpanded node on a shortest path to an optimal
  goal G1.

14
Heuristic Search

• A algorithm
• f(G2) g(G2) since h(G2) 0 gt
  g(G1) since G2 is suboptimal ?
  f(n) since h is admissible
• since f(G2) gt f(n), A will never select G2 for
  expansion

15
Heuristic Functions

• Admissible heuristicexample for the
  8-puzzleh1(n) number of misplaced tilesh2(n)
  total Manhattan distance i.e. no of squares
  from desired location of each
  tileh1(S) ??h2(S) ??

16
Heuristic Functions

• Admissible heuristicexample for the
  8-puzzleh1(n) number of misplaced tilesh2(n)
  total Manhattan distance i.e. no of squares
  from desired location of each
  tileh1(S) ?? 6h2(S) ?? 40331021
  14

17
Heuristic Functions

• Dominance
• if h1(n) ? h2(n) for all n (both admissible)then
  h2 dominates h1 and is better for searchTypical
  search costs d 14 IDS 3,473,941 nodes
  A(h1) 539 nodes A(h2) 113
  nodesd 24 IDS ? 54,000,000,000 nodes
  A(h1) 39,135 nodes A(h2) 1,641
  nodes

18
Heuristic Functions

• Admissible heuristicexample for the
  8-puzzleh1(n) number of misplaced tilesh2(n)
  total Manhattan distance i.e. no of squares
  from desired location of each
  tileh1(S) ?? 6h2(S) ?? 40331021
  14

But how do you come up with a heuristic?
19
Heuristic Functions

• Relaxed problemsAdmissible heuristics can be
  derived from the exact solution cost of a relaxed
  version of the problemIf the rules of the
  8-puzzle are relaxed so that a tile can move
  anywhere, then h1(n) gives the shortest
  solutionIf the rules are relaxed so that a tile
  can move to any adjacent square, then h2(n)
  gives the shortest solutionKey point the
  optimal solution cost of a relaxed problem is no
  greater than the optimal solution cost of the
  real problem

20
Heuristic Functions

• Relaxed problems
• Well-known example traveling salesperson problem
  (TSP) find the shortest tour visiting all
  cities exactly once
• Minimum spanning tree can be computed in O(n2)
  and is a lower bound on the shortest (open) tour

21
Heuristic Functions

• Iterative improvement algorithms
• Iterative optimization problems, path is
  irrelevant the goal state itself is the solution
• Then state space set of complete
  configurations find optimal configuration e.g.
  TSPor, find configuration satisfying
  constraints, e.g. timetable
• In such cases, can use iterative improvement
  algorithms keep a single current state, try
  to improve it
• Constant space, suitable for online as well as
  offline search

22
Heuristic Functions

• Example Traveling Salesperson Problem
• Start with any complete tour, perform pairwise
  exchanges

23
Heuristic Functions

• Example n-queensPut n queens on an n x n board
  with no queens on the same row, column, or
  diagonalMove a queen to reduce number of
  conflicts

24
Local Search Algorithms

• Hill-climbing (or gradient ascent/descent)Like
  climbing Everest in thick fog with amnesia

25
Local Search Algorithms

• Hill-climbingProblem depending on initial
  state, can get stuck on local maximaIn
  continuous spaces, problems with choosing step
  size, slow convergence

26
Local Search Algorithms

• Simulated annealingEscape local maxima by
  allowing some bad moves but gradually decrease
  their size and frequency

27
Local Search Algorithms