PROBLEM SOLVING BY SEARCHING (1) CAP 492 Dr. Souham Meshoul.

PROBLEM SOLVING BY SEARCHING PROBLEM SOLVING BY SEARCHING ((11))

CAP 492Dr. Souham Meshoul

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Objectives

Learn how to formulate a search problem.

Learn the different algorithms used to solve problems by searching.

Learn how to assess the performance of a search algorithm.

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Introduction To solve a particular problem, we need to do define the

elements of the problem and its solution:

1. Define the problem precisely.

2. Isolate and represent the task knowledge that is necessary to solve the problem.

3. Choose and apply the best problem-solving technique(s) to the particular problem.

PROBLEM SOLVING BY SEARCHING (1)

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Introduction

Search is a necessity to determine solutions to an enormous range of problems.

Search is a method that can be used by computers to examine a huge problem space to find a goal,

e.g searching for a contact lens on a football field.

Challenge: how to find the goal as quickly as possible or without using too many resources.

PROBLEM SOLVING BY SEARCHING (1)

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Introduction

start

end

PROBLEM SOLVING BY SEARCHING (1)PROBLEM SOLVING BY SEARCHING (1)

A problem can be considered to consist of a goal and a set of actions That can be taken to lead to the goal.

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PROBLEM SOLVING BY SEARCHING(1)PROBLEM SOLVING BY SEARCHING(1)

Example Problems

Toy problems

Vacuum world

The 8-Puzzle

The N-Queen...

Real world Problems

Touring problems

Route Finding

VLSI Layout

Robot navigation

Internet searching

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The 8-puzzle belongs to the family of Sliding-block puzzles.

The 8-puzzle has 9!/2=181,440 reacheable states.

The 8-puzzle has around 1.3 trillion states………..

Rubik’s cube: Rubik 1974.

Example Problems

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Problem solving as search

Search can be performed without any additional information about how to reach a solution: BLIND SEARCH.

Search can be performed using heuristics: INFORMED SEARCH.

To successfully operate, blind search should satisfy some properties:

It must be complete: it must generate every possible solution otherwise it might miss a suitable solution.

It must be non redundant: it should not generate the same solution twice.

Search can be data driven (starting from an initial state) or goal driven (starting fom goal).

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Problem solving as search

Key questions to be addressed:

1. What goal do we need to achieve?

2. How to know when a goal is reached?

3. What knoweldge we need?

4. What actions do we need to do?

Goal formulation: is the first step in problem solving. It takes into account current situation and some performance measure.

A goal can be described as: To know if a goal is achieved

1. a task to be accomplished. Need to define a goal test

2. a situation to be reached.

3. a set of properties to be acquired.

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Problem formulation

Search= Process of deciding what actions and states to consider given a goal.

Solution: sequence of actions that help achieving the goal.

Search: process of looking for a solution. Takes input (problem) and returns solution.

Execution: to carry out the actions recommended by the solution.

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Problem definition: a problem can be formally defined by1. Initial state: starting state2. Successor function: Given a particular state x,

fsuccessor(x)={<action,new-state>} where action is one of the legal actions in state x and new-state is the state that can be reached from x by applying action.

Allows together with the initial state to define the STATE SPACE that is the set of all states reachable from the initial state.

Problem formulation

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Problem formulation

A path in the state space is a sequence of states connected by a sequence of actions.

3. Goal test : determines whether a given state is a goal state.

4. Path cost funtion: assigns a cost to each path. Should recflect the performance measure.

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Arad

Zerind Sibiu Timisoara

Oradea Fagaras Rimnicu VilceaArad

Sibiu Bucharest

Solution definition: A potential solution is a path from the initial state to a goal state. Solution quality is measured by the path cost function and an optimal solution has the lowest path cost among all solutions.

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Touring problem: Given a set of n cities, the touring problem consists in visiting cities at least once starting and ending in the same city.

States: specified by the current city and the set of cities already visited.

Initial state: Any state can be designed as the initial state.

Successor function: generates the next city to visit according to the current state.

Goal test: Ending city reached and all cities have been visited.

Path cost: sum of all step costs.

Problem formulation

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Searching for solutions

To solve the problem, we need to define a search strategy that allows to explore efficiently the state space.

How to represent the state space? Tree Graph

Each state is represented by a node.

Expansion operation: A node is expanded if the successor function for the corresponding state generates new states.

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Informal description of Tree-Search

Function Tree-Search(problem,strategy) returns a solution or a failure

Initialize the search tree using the initial state of problem

Loop do

If there are no candidates for expansion then return failure

Choose a leaf node for expansion according to strategy

If the node contains a goal state then return the corresponding solution

Else expand the node and add the resulting node to the search tree

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Node definition:Data structure with 5

components: State: representation of a

physical configuration. Parent node: The node in the

search tree that generated this node.

Action: action applied to the parent node to generate this node.

Path cost: cost of the path from the initial state to the node.

Depth: The number of steps along the path from the initial state.

2 1 3

4 7

5 8 6

Node

Parent Node

Action: rightDepth= 3Path-cost=3

State

Towards a formal description

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How to deal with non expanded nodes?

Fringe : set of nodes generated but not yet expanded. Each node is a leaf node.

What is the suitable representation of fringe?

Set of nodes: Simplest way but could be computationally expensive.

Queue: best implementation of the fringe

oTowards a formal description

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What are the operations to be performed on the queue?

Make-Node(state,parent,action,depth,cost) : creates a node with the given parameters.

Empty?(queue) : returns TRUE only if there are no more nodes in the queue.

First(queue) : returns the first node of the queue. Remove-First(queue) : returns First(queue) and removes

it from the queue. Insert(element, queue) : inserts a node into the queue

and returns the resulting queue. Insert-All(elements, queue) : inserts a set of nodes into

the queue and returns the resulting queue.

Towards a formal description

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Function Tree-Search(problem,fringe) returns a solution or a failure

fringe ← Insert (Make-Node(Initial-state[problem],NULL,NULL,d,c),fringe)

Loop do If Empty?(fringe) then return failure node ← Remove-First(fringe) If Goal-Test[ problem] applied to State[node]

succeeds then return Solution(node) fringe ← Insert-all( Expand (node,problem),fringe)

Formal description of Tree-Search algorithm

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PROBLEM SOLVING BY SEARCHINGPROBLEM SOLVING BY SEARCHING

Function Expand (node,problem) returns a set of nodes

successors ← empty set For each <action,result> in

fsuccessor(state(node)) doPath-cost ← path-cost[node] + step-

cost(State[node],action,result)Depth ← Depth[node]+1S ← Make-node (result, node, action, Path-cost,

Depth)Add s to successorsreturn successors

Formal description of Expand algorithm

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Completeness: is it guaranteed to find a solution when there is one.

Optimality: Does the strategy find the optimal solution.

Time complexity: how long does it take to find a solution? It can be measured by the number of generated nodes.

Space complexity: how much memory is needed to perform the search? It can be measured in terms of the maximum number of nodes stored in memory.

Performance Evaluation:

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Time and space complexity are measured in terms of:

b : The branching factor: maximum number of successors of any node.

d : The depth of the shallowest goal node.

m : The maximum length of any path in the state space.

Performance Evaluation

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Uninformed search Informed search

►Breadth-first ►Greedy best first

► Uniform cost ►A*

► Depth first ►Local search

► Depth limited ►……..► Iterative deepening depth first….

Basic Search Algorithms

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Strategies that order nodes without using any domain specific information (only problem definition is provided).

Generate successors and simply differentiate between a goal state and a non goal state.

Blind search strategies differ by the order of node expansion.

Uninformed search (blind search)

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Main idea: Nodes at depth i are expanded before nodes at depth (i+1).

Implementation: use of a First-In-First-Out queue (FIFO). Nodes visited first are expanded first.

Shallow nodes are expanded before deeper nodes.

Breadth first search (BFS)

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Completeness: yes if the branching factor b is finite.

Optimality: shallowest goal is not necessarily the optimal one. It is optimal if all actions have the same cost.

Time complexity: At the worst case, BFS expands every node (except goal node) thus taking time as follows:

1 +b + b2 + b3 +….+ bd + (bd+1 – b)=O(bd+1)

Memory complexity: BFS keeps every node in memory. Space is a big problem.

Breadth first search Evaluation

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In case of equal step costs, Breadth First search finds the optimal solution.

For any step-cost function, Uniform Cost search expands the node n with the lowest path cost.

UCS takes into account the total cost.

UCS is guided by path costs rather than depths. Nodes are ordered according to their path cost.

Uniform cost search (UCS)

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Uniform cost search

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Uniform cost search

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Uniform cost search

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DFS expands the deepest node in the current fringe of the search tree.

As nodes are expanded, they are dropped from the fringe so then the search « backs up » to the next shallowest node that still has unexplored successors.

DFS strategy is implemented using a Last-In_First-Out (LIFO) queue or stack.

Another alternative is to implement DFS with a recursive function that calls itself on each of its children in turn.

Depth first search (DFS)

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Completeness: Incomplete in case of unbounded depth containing no solution.

Optimality: does not provide always optimal solutions.

Time complexity: At the worst case, DFS generates about O(bm) nodes in tha search tree.

Memory complexity: DFS requires less memory than BFS. It needs to store only a single path from the root to a leaf node along with the remaining unexpanded sibling nodes for each node in the path. The required storage is (bm+1) where b is the branching factor and m maximium depth.

Depth first search Evaluation

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It is simply DFS with a depth bound. Searching is not permitted beyond the depth

bound.

Works well if we know what the depth of the solution is.

Termination is guaranteed.

If the solution is beneath the depth bound, the search cannot find the goal (hence this search algorithm is incomplete).

Otherwise use Iterative deepening search (IDS).

Depth-Limited Search (DLS)

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Iterative Deepening Depth-First Search (IDS)

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Key idea: Iterative deepening search (IDS) applies DLS repeatedly with increasing depth. It terminates when a solution is found or no solutions exists.

IDS combines the benefits of BFS and DFS: Like DFS the memory requirements are very modest (O(bd)). Like BFS, it is complete when the branching factor is finite.

The total number of generated nodes is :

N(IDS)=(d)b + (d-1) b2 + …+(1)bd

In general, iterative deepening is the preferred uninformed search method when there is a large search space and the depth of the solution is not known.

Iterative Deepening Depth-First Search (IDS)

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Problem: Possibility of wasting time by expanding states that have already been encountered and expanded before.

Occurs when actions are reversible.

A solvable problem can become unsolvable if the algorithm does not detect them systematically.

For detection purposes, a comparison operation is needed.

For DFS, looping paths can be discarded immediately.

Need to keep more nodes in memory: tradeoff between time and space.

Keep a list to store every expanded node.

Avoiding repeated states

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Function Graph-Search(problem,fringe) returns a solution or a failure

closed ← empty;fringe ← Insert (Make-Node(Initial-

state[problem],NULL,NULL,d,c),fringe)

Loop do If Empty?(fringe) then return failure node ← Remove-First(fringe) If Goal-Test[ problem] applied to State[node] succeeds then

return Solution(node) If state [node] is not in closed then add State [node] to closed fringe ← Insert-all( Expand (node,problem),fringe)

Solution: Graph Search instead of tree search

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When the world is not fully observable, the agent must reason about sets of states that might get to rather than single states.

Search is performed in the belief state space instead of state space.

Initial state is a belief state.

Searching with partial information

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Vaccum world example:

initial belief state = set of possible states


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GO Right action on the belief state leads to:

Go Left action on the belief state leads to:

Suck action on the belief state leads to:


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Search: process of constructing sequences of actions that achieve a goal given a problem.

The studied methods assume that the environment is observable, deterministic, static and completely known.

Goal formulation is the first step in solving problems by searching. It facilitates problem formulation.

Formulating a problem requires specifying four components: Initial states, successor function, goal test and path cost function. Environment is represented as a state space.

A solution is a path from the initial state to a goal state.

Search algorithms are judged on the basis of completeness, optimality, time complexity and space complexity.

Several search strategies: BFS, DFS, DLS, IDS,…

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PROBLEM SOLVING BY SEARCHING (1) CAP 492 Dr. Souham Meshoul.

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