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Search in Problem Solving

ISYS 370Dr. R. Weber

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General Search Algorithm

• identify initial state• expand (generate new states)• choose option• test goal function• expand until goal not attained/no more

states to expand

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General Search Algorithm

        search-tree := initial-node         loop           if there are no leaf nodes then return FAIL           else             choose a leaf node X according to "strategy"             if X is a goal node then return X             else               expand X               add the resulting new leaf node(s) to search-tree         end loop

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Search strategies

Uninformed search strategies

• breadth-first search• uniform cost search• depth-first search• depth-limited

search• iterative deepening

search• bidirectional search

Informed /heuristic strategies

• best-first search

• A*• Heuristic

functions• memory

bounded search• simulated

annealing• Hill-climbing

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Breadth-first search

• all nodes at depth d are expanded before the nodes at depth d+1

• optimal and complete• if there are more than one, it will find the

shallowest• planning puzzles; large memory

requirements

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Depth-first search

• expands the deepest node to the deepest level of a tree• only expands nodes at shallower levels after reaching the

end• it does not guarantee a solution

(neither complete or optimal)• less memory requirements• may be faster than breadth-first, but it may get stuck

exploring long (potentially infinite) paths, when there is a solution path of less steps

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Uniform cost search

• expands the lowest-cost node• to find the best solution it requires that all operators

have positive costs• route-finding

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depth-limited search

• Improves from depth-first• imposes a bound on the maximum depth of a

path• guarantees that a solution is found if it exists

within the length of the chosen depth • Not guaranteed to be the optimal

iterative deepening search• tries to bound the search iteratively (depth 1, 2,

3, till finds a solution)• it is optimal and complete, like breadth-first

search, but has memory requirements of depth-first search.

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bidirectional search

• start searching from the origin and from the goal and stop when searches meet

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Example problems

• toy problems– used to test search algorithms

• real world problems– used to solve real problems

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Toy problems

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The 8-puzzle (Russel&Norvig)

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Cryptarithmetic (Russel&Norvig)

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Real world problems

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Applications of route finding

• Routing in computer networks• Automated travel advisory systems• Airline travel planning• Garbage collection/cleaning trucks• Food/document delivery• Does this mean that search is the only

method to perform route planning??

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Traveling Salesperson Problems

• is a category of problems• each city has to be visited once• goal is to find the shortest tour

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• Exams in university• Nurses in hospitals• Experts in call centers• Crews in different organizations• Scheduling NASA satellites• NFL games

Scheduling

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VLSI layout

• Very large scale integration• Design of silicon chip• Define the position and connections of a

million gates in a chip• Tasks: cell layout, channel routing• Goal is to min area & connection length

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Robot navigation

• Generalization of route finding problem• Any route can be chosen in a continuous space• Possible actions and states in a continuous space

are infinite• The more actions a robot can perform the more

dimensions are needed to describe states and actions

• To identify the result of an action, a robot must use vision

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Assembly sequencing

• Many industries use automatic assembly of objects

• Find an order to assemble parts of an object• Find an order to submit parts to a given process

• Design for autoclave layout

Layout Design

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General route finding algorithm 1. identify initial state as origin2. expand to all possible locations3. choose location with smallest cost/fastest

route/favor highways4. test goal function, is it the destiny?5. if yes, return location

else, return to 2

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General puzzles algorithm 1. identify initial state as origin2. expand to all possible locations3. the blank space changes place with tile on its

right4. test goal function, is it the destiny?5. if yes, return configuration

else, return to 2

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Constraint Satisfaction Problem1. Initial is given by values of variables2. Expand to all possible values3. Variables change values4. Do the variables’ values respect constraints?5. if yes, return values

else, return to 2

– Another class of problems– Design and scheduling problems– Algorithms designed specifically for CSP perform

better than general ones

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Search strategies

When do you solve a problem using a search strategy?

When there is an algorithm to solve it when:• it is guaranteed to find a solution• when it finds a solution in a timely manner• when the memory requirements are reasonable• when it can find the best solution or at least one of

the best• sometimes, one can combine searches to improve

the quality of a solution