ICS-171:Notes 4: 1

Notes 4: Optimal Search

ICS 171 Summer 1999

ICS-171:Notes 4: 2

Summary

• Optimal search strategies

– path costs

• examples

– now we want to search for the minimum cost path to goal G

– optimality

– exhaustive search

– uniform-cost search

ICS-171:Notes 4: 3

Searching for the Minimum Cost Path

• Previously we only cared about finding a path to any goal G

• Now we want the minimum cost path to a goal G

– Cost of a path = sum of individual transitions along path

– “best” = minimum cost path

– Note: 2 different types of cost

• 1. cost of traversing the path to goal (“path-cost”/online cost)

– cost from start S to node

– cost from node to goal G

• 2.. the space and time complexity of the search algorithm (“search-cost”/offline cost)

– Path-cost = sum of individual node-to-node costs

• individual costs assumed positive

– otherwise an algorithm could “cycle” forever adding loops

ICS-171:Notes 4: 4

Examples of Path-Cost

• Navigation

– path-cost = distance to node in miles OR the time to travel on foot or by car OR number of cities visited - ALL DIFFERENT

• minimum => minimum time, least fuel

• VLSI Design

– path-cost = length of wires between chips

• minimum => least clock/signal delay

• 8-Puzzle

– path-cost = number of pieces moved

• minimum => least time to solve the puzzle

ICS-171:Notes 4: 5

Example of Path Costs

S G

A B

D E

C

F

2

1

2

5

4

2 43

5

Optimal (minimum cost) path is S-A-D-E-F-G

ICS-171:Notes 4: 6

Completeness and Optimality

• A search procedure is complete if

– it is guaranteed to find a goal state, if one exists

• e.g., DFS is not complete in general (if repeated states exist)

• but BFS is complete

• A search procedure is optimal if

– it is guaranteed to find the minimum cost path, if multiple paths to a goal state exist

• e.g., DFS and BFS are not optimal

Note that optimal => complete, but not vice-versa

ICS-171:Notes 4: 7

Search Method 6: Exhaustive Search

• Basic idea

– find all possible paths to goal states

– choose the best one (best = minimum path-cost)

• For example,

– use BFS, modified so that it continues until all goal states are found

• but BFS scales as O(b d)!

• Conclusion

– it is certainly complete and optimal

– but exhaustive search is impractical except for very small problems

ICS-171:Notes 4: 8

The Shortest Path Principle

• We are searching for the minimum cost path to GSuppose the minimum (optimal) path to a goal state G has cost c

• Shortest Path Principle:

– we can ignore any open nodes in the search tree which have path-costs greater than or equal to c

• why ?

• because we already have a solution with path-cost c, and expanding these nodes can only increase their costs further

– but, we must still explore all other open nodes with path-cost <c

• why?

• they could produce a path to goal with path-cost < c

• We can do this simply by ordering nodes in Q according to path-cost to each node

ICS-171:Notes 4: 9

Search Method 7: Uniform Cost Search

Initialize: Let Q = {S}While Q is not empty

pull Q1, the first element in Qif Q1 is a goal report(success) and quitelse

child_nodes = expand(Q1)<eliminate child_nodes which represent loops>put remaining child_nodes in Qsort Q according to path-cost to each node

end Continue

•Uniform Cost Search

•orders the nodes on the Q according to path cost from S

•always expands the node with minimum path cost from S

ICS-171:Notes 4: 10

Example of Uniform Cost in action

S

A D2 5

ICS-171:Notes 4: 11

Example of Uniform Cost in action

S

A D

BD

C E

E

B F

G

2 5

3

8

4

6

11 10

13

7

D 10 F14

E

B F

G

7

12 11

14

14

C 15

ICS-171:Notes 4: 12

Properties of Uniform Cost Search

• Complete?

– yes

• Optimal?

– assume that all costs are greater than 0

– when we go to expand a node, we test for goal state

• if it is a goal state then, it is selected (say with path cost c)

• if selected, is this path the lowest cost path to a goal?

– all other currently open paths have higher cost (by definition)

– since all costs >0, all of them must have path costs >c (to goal)

– => the selected path is the optimal one

– this is basically Dijkstra’s algorithm (ICS 161) applied to search trees

• Complexity?

– worst O(b^d) in both space and time

– in practice, run out of space usually before you run out of time

ICS-171:Notes 4: 13

Summary

• Path Cost– in practice we often want the best path to a goal state, e.g., the

minimum cost path

• Optimality– a search algorithm which is guaranteed to find the best path to goal

• Uniform Cost Search– this is optimal – but it can have exponential complexity (especially memory) in practice

• Reading: Chapter 3, page 70 to 82

• All of the search techniques so far are “blind” in that they do not look at how far away the goal may be: next we will look at informed or heuristic search, which directly tries to minimize the distance to the goal.