lec 4 heuristic search methods 1

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Generate and Test (Contd..)

Example Cubes puzzle problemFour Cubes in a puzzle, each cube has

six faces.Each face is painted with one of four colours.GOAL : To arrange cubes in a row such that on all four sides, one block face of each colour is

showing.The Heuristic is to use as few of the faces with the

same colour as possible as outside faces.

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Generate and Test (Contd..)

Generate-and-test can be combined with other techniques.

Example : Dendral uses Plan-Generate-Test • Planning process uses constrained

satisfaction techniques to create a list of recommended substructures.

• Generate-and-Test is applied only on a limited substructures.




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HEURISTIC SEARCH METHODS(Contd..)

Hill-Climbing• Variation of Generate-and-Test• Feedback from test procedure is used• Used when a good Heuristic function is available

for evaluating states.Example : Getting downtown in an unfamiliar city

without a map.Heuristic Function : Distance between current

location and location of tall buildings.Desirable States: Distance is minimized.




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Algorithm : Simple Hill Climbing

1. Evaluate the initial state. If it is a goal state return and quit. Otherwise continue with initial state as current state.

2. Repeat until a solution is found or there are no new operators available to apply to current state.a. Select an operator and apply it to

produce a new state.




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Algorithm : Simple Hill Climbing(Contd..)

b. Evaluate the new state(i) If it is a goal state, return and quit(ii) If not, but better than the current

state, make it the current state.(iii) If it is not better than the current

state, continue in the loop.Continue in the LoopHeuristic Function : Sum of the number of different

colours on each of the four sides.




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Steepest Ascent Hill Climbing

• It is a variation of simple hill climbing.

• All the moves from the current state are considered and the best one is selected as the next state.




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Steepest Ascent Hill Climbing (Contd..)

IS CS

NS1 NS2 NS3 NS4Best one stored in Succ.




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Algorithm - Steepest Ascent Hill Climbing

1. Evaluate the initial state. If it is a goal state return and quit. Otherwise continue with initial state as current state.

2. Loop until a solution is found or until a complete iteration produces no change to current state.(a) Let succ be a state such that any successor of current state (cs) will be better than Succ. (Succ is replaced by better state).




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Algorithm - Steepest Ascent Hill Climbing (Contd..)

(b) For each operator that applies to CS do: (i) Apply-the operator and generate NS. (ii) If NS is a goal state return and quit. If not compare NS to Succ. If NS is better than Succ, set Succ to NS, If not leave Succ same.

(c) If Succ is better than current state, then set current state to Succ.




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Differences in the two hill-climbing algorithms

Time to select a move is longer

Time required to select a move is smaller

Less number of moves for a solution

The number of moves to get to a solution is more

Steepest-Ascent Hill-Climbing

Basic Hill-Climbing




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Hill-Climbing Methods may fail when the program has reached one of the following

• Local Maximum• Plateau ( a high flat land) or a• Ridge (a long narrow hill top, a line of

junction of two surfaces sloping upwards towards each other).




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Local Maximum• Is a state better than all its neighbors, but not

better than states far away.• Occurs with in the sight of a solution• Also named as FoothillHow to deal with Local Maximum ?• Backtrack to some earlier node and try going in

a different direction.• Implementation requires to maintain a list of

paths.




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Plateau• A flat area in which a set of neighbours

have the same value.• It is not possible to find best direction to

move in.How to deal with a Plateau ?• Make a big jump in some direction to get a

new search space.• Apply single small steps several times to

make a big Jump.




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RIDGE• Is a special kind of local maximum.• Is higher than surrounding areas and has

a slope.• Is impossible to traverse a ridge by single

moves.How to deal with Ridges ?• Apply two or more rules before doing the

test. This corresponds to moving in several directions at once.




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Drawbacks of Hill-Climbing

• Local Method – It looks only at immediate consequences for a new state.

• Lacks the guarantee to provide a solutionAdvantage : Less combinatorially explosiveExample to Demonstrate Hill-Climbing

failure due to local searching.




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Blocks World ProblemA H Rules/OperatorsH G i) Pick one Block and put itG F on the table.F E ii) Pick one Block and put itE D on another one.D CC BB A

Initial GoalState State




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Blocks World Problem(Contd..)

• Local (Heuristic Function) : Add one point for every block that is resting on the thing that it is suppose to be resting on.

• Subtract one point for every block that is sitting on the wrong thing.




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• Initial state has function value or score= -1+1+1+1+1+1+1-1 = -2+6 = 4

• Goal state Score = 1+1+1+1+1+1+1+1=8

The possible move(s) from initial state is only one.




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H This is accepted as currentG state.FE Three possible moves fromD current state are (a),(b),(c)CB AScore = - 1+6+1 = 6 > 4




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A G G H F F Score (a)= -1+6-1=4G E E Score (b)= -1+5+1-1=4F D D Score (c)= -1+5+1-1=4E C CD B BC HB A A H(a) (b) (c)




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• The process has reached a local maximum that is not global maximum.

• By changing the Heuristic Function, a solution may be found.




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Global (Heuristic Function)• For each Block that has the current

support structure. (i.e. the complete structure underneath it is exactly as it should be) Add One Point in the Support Structure

• For each Block that has an incorrect support structure, subtract one point for every Block in the existing support structure.




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Global (Heuristic Function)(Contd..)

A HH G Score for initial stateG F = -1-2-3-4-5-6-7F E = - (1+2 +..+7)E D = -(7 x 8/2) = -28D C Score for Goal State C B = 1+2+3+4+5+6+7 = 28B A

Initial Goal B ( A B has correct support structure, +1

C has correct structure C B 1 + 1 = 2A




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H The possible move fromG initial state isFE Score = -1-2-3-4-5-6D = -(6x7/2) = -21CB ACurrent state - 21 > -28




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A G G The three possible movesH F F from CS areG E E Score a = -28F D D Score b = -16E C C Score c = -15D B B = (-1-2-3-4-5)C H State (c) is chosen as current one

B A A H(a) (b) (c)




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• The New Heuristic Function takes correct structures and leaves incorrect structures.

• Due to lack of knowledge about the problem, It is not always possible to construct such a perfect Heuristic Function.

• In the Extreme case even if perfect knowledge is available, it may not be computationally tractable to use

• Hill Climbing is useful when combined with other methods.




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Simulated Annealing• It is a variation of Hill Climbing.It lowers the chances of getting caught at a local

maximum, a plateau or ridge.In this method enough exploration of the whole

space is done early so that the final solution is relatively insensitive to initial state

Annealing Is a Physical Process – In which physical substances as metals are melted and gradually cooled until desired state is reached.




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Simulated Annealing (Contd..)

Viewing Objective Function as Energy Level, this is like valley Descending

as the Goal is to produce a minimal energy final state.

• Physical substances usually move from high energy state to lower one.

• But there is a probability that lower to higher can occur.




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P = - E/KTe

E is positive change in energy level T is the temperature, k is Boltzmann’s constant.

The Revised probability




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Simulated Annealing (Contd..)P1 = - E/T, E is change in Heuristic

e Function

We need to choose a schedule of values for TAnnealing Schedule – Rate at which system is

cooledTo compute annealing schedule

– initial value of T is required-Criteria to change the value of T




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As T approaches zero, the probability of accepting a move to a worse state becomes zero, and simulated annealing becomes hill climbing

- The values of T be scaled so that the ratio be meaningful ; for example, T could be initialized to a value such that for an average E, p1 = 0.5




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Algorithm : Simulated Annealing

1. Evaluate the initial state. If it is a Goal state, return and quit. Otherwise continue with initial state as current state (CS)

2. Initialize Best –so-far to CS3. Initialize T4. Loop until a solution is found or there are

no new operators left to apply to CS.




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Algorithm : Simulated Annealing(Contd..)

(a) Select an operator and apply it to CS to get NS.

(b) Evaluate NS. Compute E = (value of current) – (value of new)

• If NS is GS, return and quit.• If NS is not GS, but is better than CS,

make it CS. Also set Best-So-Far to this state.




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• If NS is not better than CS, make it CS with probability p1. Involve a random number r in [ 0 1]. If r is less than p1 then the state is accepted.

(c) Revise T as necessary according to annealing schedule.

5. Return Best-So-Far as the answer.Simulated-Annealing is often used to solve problems in which the number of moves from a given state is very large




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Annealing Schedule• A schedule of values for T is known as Annealing

Schedule• Initial value of T is selected such that for average

change in heuristic function, the probability is 0.5• The optimal annealing schedule for each annealing

problem must be discovered empirically• The best way to select an annealing schedule is by

trying several alternatives and observing the effect on both the quality of the solution and the rate at which the process converges




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Best – So – Far is storedBest – So – Far is not stored

Annealing schedule is to be maintained

Annealing scheduled is not required

Moves to worse states may be accepted

Moves to worse states are not accepted

Simulated AnnealingSimple Hill Climbing




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Best – First Search (BFS)BFS – Combines advantages of Breadth-First

search and Depth First Search.Breadth First – Does not get trapped at

dead ends.Depth First – Allows a solution to be found

without expanding other branches.Best-First Search follows a single path at a time,

but switches paths when some competing path is more promising than the current one.




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Best – First Search (BFS)(Contd..)

Heuristic Function – Cost of getting to a solution from a given node.

The successors of a node are generated.If one of them is a solution, quit;otherwise the nodes are added to the list of

nodes generated so farThe most promising node is selected and

the process continues.




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Best – First Search (BFS)(Contd..)

Step – 1 Step -2 Step – 3 A A A

B C D B C D (1)(3) (5) (1) (3) (5)

E F(4) (6)




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Best – First Search (BFS) (Contd..)

Step – 4 Step – 5A A

B C D B C D

G H E F G H E F(6) (5) (4) (6)

I J(2) (1)




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• It is important to search a graph (not tree) so that duplicate paths will not be pursued.

• Two lists of nodes are needed for implementation.

Open – Nodes that have been generated, but not yet been examined. Open is a priority queue with the most promising Heuristic value node having highest priority.

Closed – Nodes that have been examined; need to keep them in memory to check duplicate nodes.




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Each Node in a Directed graph contains –

• The problem state• Priority• Parent Node• A List of nodes generated from it.




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Algorithm : Best First Search1. Start with open containing just initial

node.2. Until Goal is found or there are no nodes

left on open DO(a) Pick the best node on open(b) Generate its successors(c) For each successor DO




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(i) If it has not been generated before, evaluate it, add it to open, record its parent.

(ii) If it has been generated before, change the parent if this new path is better than the previous one. Also update the cost of getting to this node and to any successors that this node may already have.




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A* Algorithm

A* AlgorithmBest – First search is a simplification of A* the Heuristic function in A* is given by

f1 = g + h1

g is a measure of the cost of getting from initial to current node.h1 is an estimate of getting from current to goal node.




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A* Algorithm (Contd..)

In BFS, the Heuristic function is h1 only; BFS examines nodes closer to goal, so it reaches goal with fewer steps.

A* given highest priority to nodes that are on the shortest path from start node to goal node.




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Algorithm – A*1. Place the start node S on open.2. If open is empty stop and return failure.3. Remove from open the node n that has

the smallest value of f1 (n). If n is a goal node, return and stop otherwise go to step 4.




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4. Expand n generating all of its successors n1 and place n on closed. For every successor n1, if n1 is not already on open or closed, attach a back pointer to n, compute f1(n1) and place it on open.

5. Each n1 that is already on open or closed should be attached to back pointers which reflect the lowest g(n1) path.

6. Return to step 2.




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Observations about A* algorithm1. Role of g function

– This allows to choose a node based on how good the path to the node was.

– If the requirement is only to get a solution some how, define g = 0.

– If the requirement is to find a path with fewest number of steps, set the cost of going from a node to successor to 1.

– If the requirement is to find the cheapest path set the cost of going from one node to another to reflect the costs.




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A* can be used to find a minimal cost overall path or any path as quickly as possible.

2. Role of h1

- If h1 is perfect A* will converge to goal with no search- If h1 = 0, the search is controlled by g.- If h1 = 0, g = 0, the search is random- If h1 = 0, g = 1, the search will be breadth- first.- Suppose h1 ≠ 0, and h1 never over estimates h. (i.e. estimated value of h ≤ actual value of h)

A* is guaranteed to find an optional path to Goal




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A* Algorithm (Contd..)A A

B C D B C D(3+1) (4+1) (5+1) (3+1) (4+1) (5+1)

E E(2+2) (2+2) 1 = h1+g

F F(1+3) (3+3)

G (0+4) Fig. 4-a Fig. 4-b

h1Overestimates h h1 Underestimates h




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Fig. 4-a• Suppose there is a directed path from D to

a solution.• By overestimating h1(D) we may find

worse solution without ever expanding D.Fig. 4-b - 1(F) = 6 > 1 (C), so C is expanded

again; by underestimating h1(B), some effort is wasted; but optimal path is traced.




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3. The Third Observation : A* is stated as applied to Graphs. It can be simplified to apply to trees.

• Then there is no need to check whether a new node is already on open or closed.

• Faster to generate nodes but the same search may be conducted many times.




lec 4 heuristic search methods 1

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