1 search CS 331/531 Dr M M Awais REPRESENTATION METHODS Represent the information: Animals are...

CS 331/531 Dr M M Awais 1

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REPRESENTATION METHODS

Represent the information:

Animals are generally divided into birds and mammals. Birds are further classified into large birds and small birds. Small birds include sparrows, and crows. The large birds are ostriches. The mammals include whales, monkeys, and elephants. Charlie is a elephant.

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Predicate

• types(Animals, Birds,Mammals)

• types(Birds, Largebirds, Smallbirds)

• Etc…

• Unify to answer the queries

• IS THERE ANY OTHER WAY?

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Semantic Nets (Graphical Methods)

Instance

IS-A IS-A

Animal

Bird Mammal

Elephant whales

charlie

IS-A IS-A

sparrow

instance

Super Class: Animal

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Instance

IS-A IS-A

Animal

Bird Mammal

Elephant whales

charlie

IS-A IS-A

sparrow

instance

IS_A: Relation

Animal: SuperBird: Class

Mammal: Class

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Instance

IS-A IS-A

Animal

Bird Mammal

Elephant whales

charlie

IS-A IS-A

sparrow

instance

Instance: defines a specific instance of a class

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Instance

IS-A IS-A

Animal

Bird Mammal

Elephant whales

charlie

IS-A IS-A

sparrow

instance

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Instance

IS-A IS-A

Animal

Bird Mammal

Elephant whales

charlie

IS-A IS-A

sparrow

instance

bananas

Can have attribute link: likes for instance, class, superclass

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SEARCH METHODS

• Formulate a problem as a search

• Can answer queries such as:

Is CHARLIE an animal?

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Search RepresentationAnimal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

search

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

Level 1

search

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

Level 1

Level 2

search

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

Level 1

Level 2

Level 3

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Answering QueriesAnimal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

Level 1

Level 2

Level 3

Is charlie an animal?Answer:Traverse the treeIf charlie node present then answer is YES otherwise NO

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Answering QueriesAnimal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Level 0: Root node

Level 1

Level 2

Level 3

Is charlie an animal?Answer:Traverse the treeIf charlie node present then answer is YES otherwise NO

The way one traverses the tree defines SEARCH TYPES

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SEARCH METHODS

• Blind Search

•Breadth First

•Depth First

•Iterative Deepening

• Heuristic search

•Hill Climbing

•Best First

•A* Search

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Standard Terms

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Root node

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Standard Terms

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Child of Animal node

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Standard Terms

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Bird and Mammal are Siblings

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Standard Terms

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Root node

Path to a node is the list of nodes from the root to the goal node.

GOAL node

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Standard Terms

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

Root node

Path to a node is the list of nodes to the goal node (bold lines).

GOAL node

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How to Traverse the Tree to find PATH?

Animal

Bird Mammal

Elephant whales

charlie

smallLarge

sparrow

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ANALYSIS OF SEARCH STRATEGIES

Completeness: is the strategy guaranteed to find a solution where there is one?

Time Complexity: How long does it take to find a solution?

Space Complexity: How much memory does it need to perform the search?

Optimality: Does the strategy find the highest quality solution when there are several different solutions?

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Exhaustive Search

One can systematically check every state that is reachable from initial state to end state if it is a goal state.

Search Space

The set of all states is the search space

• For simple/small search space exhaustive search is applicable [BRUTE FORCE or BLIND SEARCH]

• For complex search space HEURISTIC SEARCH is used

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GRAPHS AND TREES

Graphs:-

• Consist of a set of nodes with links between them

• links can be directed / undirected

• Path is the sequence of nodes connected nodes via links.

Acyclic graphs = (Paths linking a node

with itself are absent)

Trees???

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Tree:-

A tree is a special kind of graph with only one path to each node, usually represented with a special root node at

the top

Relationship between nodes

• Parent

• Children

• Sibling

Ancestor Node, Descendant Node, Leaf Node

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Graphs VS Trees Compare the searches in the two (which

is efficient)

de f g

searchRelating State and Graph Nodes

Every physical state of a problem can be represented as a node in a graph/tree

The link between the nodes represent action required to change the states

For a navigation problem: Link: driving action Node:cities

For the Wumpus world problem: Link: movement of the agent Node:present position of the agent

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8 Puzzle Problem: states vs. nodes

A state is a (representation of) a physical configuration

A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth

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8 Puzzle Problem: states vs. nodes

A state is a (representation of) a physical configuration

A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth

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Tree search algorithms Basic idea:

offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)

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Example: Romania

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Example: Romania

On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest Formulate goal:

be in Bucharest Formulate problem:

states: various cities actions: drive between cities

Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras,

Bucharest

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Tree search example

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Tree search example

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Tree search example

Keep on expanding unless you reach the goal

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Search strategies A search strategy is defined by picking the order of

node expansion Strategies are evaluated along the following

dimensions: completeness: does it always find a solution if one

exists? time complexity: number of nodes generated space complexity: maximum number of nodes in

memory optimality: does it always find a least-cost solution?

Time and space complexity are measured in terms of b: maximum branching factor of the search tree d: depth of the least-cost solution m: maximum depth of the state space (may be ∞)

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Blind Search: Strategies

Breadth First (Breadth wise expansion)

Depth First (Depth wise expansion) Iterative Deepening (Depth Limit)

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Breadth-first search Expand shallowest unexpanded

node Implementation:

FIFO queue, i.e., new successors go at end

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Breadth-first search

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search Properties of breadth-first search

Complete? Yes (if b is finite) Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1) Space? O(bd+1) (keeps every node in memory) Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

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1. Start with queue = [initial - state] and found = FALSE

2. While queue not empty and not found do:

(a) Remove the first node n from queue

(b) if N is a goal state then found = TRUE

(c ) Find all the successor nodes of X, and put them on the end of the queue

Breath First: Algorithm

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1. Open = [A]; closed = []

• Evaluate A not goal

Goal: D

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2. Open = [B,C];closed =[A]

• Evaluate B not goal

Goal: D

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3. Open = [C,D,E];closed = [B,A]

• Evaluate C not goal

Goal: D

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3. Open = [C,D,E];closed = [B,A]

• Evaluate C not goal

4. Open = [D,E,G,F];closed = [C,B,A]

1. Evaluate D is goal: STOP

Goal: D

Path: A B D

Path Cost: The cost in reaching D from A

Path cost = 2A to B=1, B to D=1(assuming a unit cost between nodes)

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Uniform-cost search

Expand least-cost unexpanded node Implementation:

queue ordered by path cost Equivalent to breadth-first if step costs all

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Depth-first search Expand deepest unexpanded node Implementation:

LIFO queue, i.e., put successors at front

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

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Depth-first search

search Properties of depth-first search

Complete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path

complete in finite spaces

Time? O(bm): terrible if m is much larger than d but if solutions are dense, may be much faster

than breadth-first Space? O(bm), i.e., linear space! Optimal? No

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1. Start with agenda = [initial - state] and found = FALSE

2. While agenda not empty and not found do:

(a) Remove the first node N from agenda

(b) if N is not in visited then

(I) Add N to visited

(II) if N is a goal state then found = TRUE

(III) Put N’s successors on the front of the stack

Depth First

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Open=[A],closed=[] Evaluate A not a goal

Goal: C

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Open=[B,C];closed=[A] Evaluate B not a

Goal: C

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Open=[B,C];closed=[A] Evaluate B not a goal

Open=[D,E,C];closed=[B,A] Evaluate D not a goal

Goal: C

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Open=[H,I,E,C];closed=[D,B,A] Evaluate H not a goal

Goal: C

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Open=[I,E,C];closed=[H,D,B,A] Evaluate I not a goal

Goal: C

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Open=[E,C];closed=[I,H,D,B,A] Evaluate E not a goal

Goal: C

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Open=[J,K,C];closed=[E,I,H,D,B,A]

Evaluate J not a goal

Goal: C

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Open=[J,K,C];closed=[E,I,H,D,B,A] Evaluate J not a goal

Open=[K,C];closed=[K, E,I,H,D,B,A]

Evaluate K not a goal

Goal: C

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Open=[J,K,C];closed=[E,I,H,D,B,A] Evaluate J not a goal

Open=[K,C];closed=[J,E,I,H,D,B,A]

Evaluate K not a goal

Open=[C];closed=[K,E,I,H,D,B,A]

Evaluate C is goal: STOP

Goal: C

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Depth-limited search depth-first search with depth limit L nodes at depth L have no successors Do not apply depth first after a certain depth

Iterative Deepening tries almost all depth limits

Searches the tree with L=0, L=1, L=2, etc…

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Iterative deepening search L=0

For depth 0, evaluate the root node and stop

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Iterative deepening search Number of nodes generated in a depth-limited

search to depth d with branching factor b:

NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd

Number of nodes generated in an iterative deepening search to depth d with branching factor b:

NIDS = (d+1)b0 + d b1 + (d-1)b2 + … + 3bd-2 +2bd-1 + 1bd

For b = 10, d = 5, NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111 NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

Overhead = (123,456 - 111,111)/111,111 = 11%

search Properties of iterative deepening search

Complete? Yes

Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)

Space? O(bd)

Optimal? Yes, if step cost = 1

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Summary of algorithms

1 search CS 331/531 Dr M M Awais REPRESENTATION METHODS Represent the information: Animals are...

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