Search: Heuristic

1
Search Heuristic Optimal

• Artificial Intelligence
• CMSC 25000
• January 16, 2003

2
Agenda

• Heuristic Search
• Local Hill-climbing, Beam Search
• Limitations of local heuristics
• Global Best first
• Optimal Search
• A search
• Admissible, consistent heuristics
• Dynamic programming
• Problem representation
• Configuration space

3
Searches

• Blind search Find ANY path to goal
• Know something about search space
• Most paths reach goal or terminate quickly DFS
• Low branching factor, possible long paths BFS
• Always ready to act ID-DFS

4
Heuristic Search

• A little knowledge is a powerful thing
• Order choices to explore better options first
• More knowledge gt less search
• Better search alg?? Better search space
• Measure of remaining cost to goal-heuristic
• E.g. actual distance gt straight-line distance

10.4
6.7
4.0
11.0
3.0
8.9
6.9
5
Hill-climbing Search

• Select child to expand that is closest to goal

8.9
10.4
6.9
10.4
3.0
6.7
6
Hill-climbing Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Sort new paths by estimated distance to goal
• Add new paths to FRONT of queue
• If goal foundgtsuccess else, failure

7
Beam Search

• Breadth-first search of fixed width - top w
• Guarantees limited branching factor, E.g. w2

8
Beam Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Extend all paths one step
• Reject all paths with loops
• Sort all paths in queue by estimated distance to
  goal
• Put top w in queue
• If goal foundgtsuccess else, failure

9
Best-first Search

• Expand best open node ANYWHERE in tree
• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Put in queue
• Sort all paths by estimated distance to goal
• If goal foundgtsuccess else, failure

10
Heuristic Search Issues

• Parameter-oriented hill climbing
• Make one step adjustments to all parameters
• E.g. tuning brightness, contrast, r, g, b on TV
• Test effect on performance measure
• Problems
• Foothill problem aka local maximum
• All one-step changes - worse!, but not global max
• Plateau problem one-step changes, no FOM
• Ridge problem all one-steps down, but not even
  local max
• Solution (local max) Randomize!!

11
Search Costs
12
Searches

• Heuristic search Any path, but find faster
• Estimate remaining distance to goal
• Best from current node Hill-climbing
• Best from any node Best first
• Best w at this depth Beam search

13
Optimal Search

• Find BEST path to goal
• Find best path EFFICIENTLY
• Exhaustive search
• Try all paths return best
• Optimal paths with less work
• Expand shortest paths
• Expanded shortest expected paths
• Eliminate repeated work - dynamic programming

14
Efficient Optimal Search

• Find best path without exploring all paths
• Use knowledge about path lengths
• Maintain path path length
• Expand shortest paths first
• Halt if partial path length gt complete path
  length

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Underestimates

• Improve estimate of complete path length
• Add (under)estimate of remaining distance
• u(total path dist) d(partial path)u(remaining)
• Underestimates must ultimately yield shortest
• Stop if all u(total path dist) gt d(complete path)
• Straight-line distance gt underestimate
• Better estimate gt Better search
• No missteps

16
Search with Dynamic Programming

• Avoid duplicating work
• Dynamic Programming principle
• Shortest path from S to G through I is shortest
  path from S to I plus shortest path from I to G
• No need to consider other routes to or from I

17
A Search Algorithm

• Combines good optimal search ideas
• Dynamic programming and underestimates
• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• For all paths with same terminal node, keep only
  shortest
• Add new paths to queue
• Sort all paths by total length underestimate,
  shortest first (d(partial path) u(remaining))
• If goal foundgtsuccess else, failure

18
A Search Example
13.4
12.9
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Heuristics

• A search only as good as the heuristic
• Heuristic requirements
• Admissible
• UNDERESTIMATE true remaining cost to goal
• Consistent
• h(n) lt c(n,a,n') h(n')
• Some heuristics better than others
• 0?

20
Constructing Heuristics

• Relaxation
• State problem
• Remove one or more constraints
• What is the cost then?
• Example
• 8-square Move A to B if
• 1) A B horizontally or vertically adjacent, and
• 2) B is empty
• Ignore 1) -gt Manhattan distance
• Ignore 1) 2) of misplaced squares

21
Navigation
22
Application Configuration Space

• Problem Robot navigation
• Move robot between two objects without changing
  orientation
• Possible?
• Complex search space boundary tests, etc
• First step Problem transformation
• Model robot as point
• Model obstacles by combining their perimeter
  path of robot around it
• Configuration Space simpler search

23
Navigation
24
Navigation
25
Navigation as Simple Search

• Replace funny robot shape in field of funny
  shaped obstacles with
• Point robot in field of configuration shapes
• All movement is
• Start to vertex, vertex to vertex, or vertex to
  goal
• Search Start, vertices, goal, connections
• A search yields efficient least cost path

26
Online Search

• Offline search
• Think a lot, then act once
• Online search
• Think a little, act, look, think,..
• Necessary for exploration, (semi)dynamic env
• Components Actions, step-cost, goal test
• Compare cost to optimal if env known
• Competitive ratio (possibly infinite)

27
Online Search Agents

• Exploration
• Perform action in state -gt record result
• Search locally
• Why? DFS? BFS?
• Backtracking requires reversibility
• Strategy Hill-climb
• Use memory if stuck, try apparent best neighbor
• Unexplored state assume closest
• Encourages exploration

28
BB DynProg Analysis

• Algorithm
• Select best partial path from Q
• Test for completion
• Add path extensions to Q
• Assume that we are using an Expanded list to
  implement Dynamic Prog. (implemented as a hash
  table constant access time).

Assume we have a graph with N nodes and L links.
We call a graph where nodes have O(N) links are
dense. Graphs where the nodes have a nearly
constant number of links are sparse. For dense
graphs L is O(N2).
O(N)
Paths taken from Q ?
O(N)
Cost of picking path from Q ( cleanup), using
linear scan?
O(L)
Attempts to add path to Q (many are rejected)?
O(1)
Cost of adding path extension to front of Q ?
O(N2 L)
Total cost ?
Lozano-perez, 2000
29
Cost and Performance
Searching a tree with N nodes and L links
Searching a tree with branching factor b and
depth d L N b d1
Worst case time is proportional to number of
nodes visited Worst case space is proportional to
maximal length of Q (and Expanded)
lozano-perez, 2000