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Jurusan Teknik Informatika – Universitas Widyatama

Heuristic Search

Pertemuan : 4

Dosen Pembina :

Sriyani Violina

Danang Junaedi

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• Deskripsi

• Heuristic Search– Simulated Annealing

– Simplified Memory Bounded A*

– Bi-Directional A*

– Modified Bi-Directional A*

– Dynamic Weighthing A*

Overview

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• Pertemuan ini mempelajari bagaimana

memecahkan suatu masalah dengan teknik

searching.

• Metode searching yang dibahas pada pertemuan

ini adalah heuristic search atau informed search

yang terdiri dari:– Simulated Annealing

– Simplified Memory Bounded A*

– Bi-Directional A*

– Modified Bi-Directional A*

– Dynamic Weighthing A*

Deskripsi

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Simulated Annealing

• A alternative to a random-restart hill-climbing when stuck

on a local maximum is to do a ‘reverse walk’ to escape

the local maximum.

• This is the idea of simulated annealing.

• The term simulated annealing derives from the roughly

analogous physical process of heating and then slowly

cooling a substance to obtain a strong crystalline

structure.

• The simulated annealing process lowers the temperature

by slow stages until the system ``freezes" and no further

changes occur.

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Search using Simulated Annealing

• Simulated Annealing = hill-climbing with non-deterministic

search

• Basic ideas:

– like hill-climbing identify the quality of the local improvements

– instead of picking the best move, pick one randomly

– say the change in objective function is δ

– if δ is positive, then move to that state

– otherwise:

• move to this state with probability proportional to δ

• thus: worse moves (very large negative δ) are executed less often

– however, there is always a chance of escaping from local maxima

– over time, make it less likely to accept locally bad moves

– (Can also make the size of the move random as well, i.e., allow

“large” steps in state space)

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Physical Interpretation of Simulated

Annealing

• A Physical Analogy:• imagine letting a ball roll downhill on the function surface

– this is like hill-climbing (for minimization)

• now imagine shaking the surface, while the ball rolls, gradually

reducing the amount of shaking

– this is like simulated annealing

• Annealing = physical process of cooling a liquid or

metal until particles achieve a certain frozen crystal

state• simulated annealing:

– free variables are like particles

– seek “low energy” (high quality) configuration

– get this by slowly reducing temperature T, which particles move

around randomly

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Simulated Annealing

• Probability of transition to higher energy

state is given by function:

– P = e –∆E/kt

Where ∆E is the positive change in the energy level

T is the temperature

K is Boltzmann constant.

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Differences

• The algorithm for simulated annealing is

slightly different from the simple-hill

climbing procedure. The three differences

are:

– The annealing schedule must be maintained

– Moves to worse states may be accepted

– It is good idea to maintain, in addition to the

current state, the best state found so far.

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Algorithm: Simulate Annealing

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

2. Initialize BEST-SO-FAR to the current state.

3. Initialize T according to the annealing schedule

4. Loop until a solution is found or until there are no new operators left to be applied in the current state.

a. Select an operator that has not yet been applied to the current state and apply it to produce a new state.

b. Evaluate the new state. Compute:• ∆E = ( value of current ) – ( value of new state)

• If the new state is a goal state, then return it and quit.

• If it is a goal state but is better than the current state, then make it the current state. Also set BEST-SO-FAR to this new state.

• If it is not better than the current state, then make it the current state with probability p’ as defined above. This step is usually implemented by invoking a random number generator to produce a number in the range [0, 1]. If the number is less than p’, then the move is accepted. Otherwise, do nothing.

c. Revise T as necessary according to the annealing schedule

5. Return BEST-SO-FAR as the answer

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Simulate Annealing: Implementation

• It is necessary to select an annealing schedule

which has three components:

– Initial value to be used for temperature

– Criteria that will be used to decide when the

temperature will be reduced

– Amount by which the temperature will be

reduced.

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• To address complex search problems, we often put constraints on memory requirements.

• We will examine two algorithms: (a) IDA* search and (b) SMA* search.

• Iterative Deepening A* Search (IDA*) uses a DFS for every iteration where an f-cost limit is used instead of a depth limit.

• Each iteration expands all the nodes that lie inside a contour for the current f-cost.

• The time complexity depends on the different values that the heuristic can take on.

• IDA* is complete, optimal, and memory-bounded.

• To improve IDA*, we often increase the f-cost limit by a fixed amount ε on each iteration. Such an approach is called ε-admissible.

Memory Bounded Search

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SIMPLIFIED MEMORY-BOUNDED A*

(SMA*)

• IDA* cannot remember its previous steps since it uses too little memory (between iterations only the f-cost limit is kept).

• If the state space is graph, IDA* often repeats itself.

• Uses all available memory for the search– drops nodes from the queue when it runs out of space

• those with the highest f-costs

– If there is a tie (equal f-values) we first delete the oldest nodes first.

– complete if there is enough memory for the shortest solution path

– often better than A* and IDA*• but some problems are still too tough

• trade-off between time and space requirements

• It is complete and optimal if the available memory allows SMA* to store the shallowest solution path.

• When enough memory is available for the whole search tree, the search is optimally efficient.

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SMA* Code

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Simple Memory-bounded A* (SMA*)

24+0=24

A

B G

C D

E F

H

J

I

K

0+12=12

10+5=15

20+5=25

30+5=35

20+0=20

30+0=30

8+5=13

16+2=18

24+0=24 24+5=29

10 8

10 10

10 10

8 16

8 8

A

12A

B

12

15

A

B G

13

15 13H

13

∞

A

G

18

13(15)

A

G

24(∞)

I

15(15)

24

A

B G

15

15 24

∞

A

B

C

15(24)

15

25

f = g+h

(Example with 3-node memory)

A

B

D

8

20

20(24)

20(∞)

Progress of SMA*. Each node is labeled with its current f-

cost. Values in parentheses show the value of the best

forgotten descendant.

Optimal & complete if enough memory

Can be made to signal when the best solution found might not be optimal

- E.g. if J=19

� = goal

Search space

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Ex: Rute Perjalanan

10

10

80

10

35

70203010040070748570608080h(n)

607050254590205302515100b(n)

MLKJHGFEDCBASn

h(n):heuristic jarak antar suatu kota dan b(n):perkiraan biaya menuju kota asal

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Solusi SMA*

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Algoritma Bi-Directional A* (BDA*)

Function BDA*(masalah) return solusi

inputs: masalah�ruang masalah,jenis solusi, start(S) dan goal(G)

variable lokal : BNs,Bng{Best node dr start & goal}

OPENs,OPENg{open dr start & goal}

CLOSEDs, CLOSEDg{Closed dar start & goal}

loop sampai goal ditemukan

BNs,OPENs,CLOSEDs A*(OPENs,CLOSEDs,S,G)

if BNs in CLOSEDs then {goal ditemukan}

vsimpul di CLOSEDg yang sama dengan BNs

g_terbaik g(G,v) {biaya sebenarnya dari G ke v melalui parent lama}

loop untuk semua suk=simpul di CLOSEDg yang merupakan suksesor v

if g(G,v) melalui suk < g_terbaik then

g_terbaik g(G,v) melalui suk

parent_baru suk

endif

endloop

g(v) g_terbaik

parent dari v parent_baru

returns solusi

endif

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Algoritma Bi-Directional A* (BDA*)

contd.BNg, OPENg, CLOSEDg A*(OPENg, CLOSEDg, G,S)

if BNg in CLOSEDg then {goal ditemukan}

usimpul di CLOSEDs yang sama dengan BNg

g_terbaik g(S,u) {biaya sebenarnya dari S ke u melalui parent lama}

loop untuk semua suk=simpul di CLOSEDg yang merupakan suksesor u

if g(S,u) melalui suk < g_terbaik then

g_terbaik g(S,u) melalui suk

parent_baru suk

endif

endloop

g(S,u) g_terbaik

parent dari u parent_baru

returns solusi

endloop

end

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Solusi Bi-Directional A*

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Modified Bi-Directional A*

• Yang dimodifikasi adalah fungsi heuristiknya menjadi

– Untuk pencarian maju

f(n)=g(S,n) + 0.5 * (h(n) - b(n))

– Untuk pencarian mundur

f(n)=g(G,n) + 0.5 * (b(n) - h(n))

Dimana

S = node asal atau initial state

G = node tujuan atau goal state

g(S,n) = jarak/biaya sebenarnya dari S ke n

g(G,n) = jarak/biaya sebenarnya dari G ke n

h(n) = heuristik jarak node n

b(n) = heuristik biaya node n

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Solusi Modified Bi-Directional A*

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Dynamic Weighting A*

• Fungsi heuristik yang digunakan

f(n)=g(S,n)+(w(n) * h(n))

Dimana

S = node asal atau initial state

G = node tujuan atau goal state

g(S,n) = jarak/biaya sebenarnya dari S ke n

h(n) = heuristik jarak node n

w(n) = bobot dari fungsi heuristik

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Solusi Dynamic Weighting A*

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Referensi

1. Suyanto.2007.”Artificial Intelligence” .Informatika. Bandung

2. -.2010. “Search[online]”. http://www-users.cselabs.umn.edu/classes/Fall-2010/csci5511/lectures/AIlecture3.ppt.Tanggal Akses : 18 Februari 2011

3. -.-.“Informed Search [online]”, http://staff.najah.edu/sites/default/files/5_Informed_Search.ppt, Tanggal Akses : 18 Februari 2011

4. Ramin Halavati.-.” Informed Search and Exploration]”. http://ce.sharif.ir/courses/84-85/2/ce417/resources/root/Slides/AI-04a-Informed%20Search%20and%20Exploration.ppt.Tanggal Akses: 18 Februari 2011

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