State Representation of State Space Searching

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State RepresentationofState Space Searching

• Alan Tam Siu Lung
• Tam_at_SiuLung.com
• 99967891
• 96397999

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Prerequisite

• State Space Search
• Pascal/C/C
• Familiar with the data types
• Mathematics

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Generic GraphSearching Algorithm

• c container of states
• insert start to c
• while c not empty
• pop an element from c (with minimum total cost)
• if found goal, return
• add/update some elements in c

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Generic Graph Searching

• Best first search choose the node with minimum
  estimated total path cost current path cost
  estimated cost to goal
• Uniform cost search estimated cost to goal 0
• Breadth first search current path cost depth
  of the node in an un-weighted graph
• Depth first search estimated total path cost
  inverse of current path cost

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Missionaries and Cannibals
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8-puzzle
4 5
6 1 8
7 3 2
? ? ?
1 2 3
4 5 6
7 8
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States Storage Problem

• Need to store many states
• Memory is limited
• Runtime is limited
• Compiler is stupid
• we need intelligence to represent states wisely

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Operation on States

• 3 operations
• Is goal?
• Are these 2 states equal?
• Find all neighbors.
• Questions
• Can some operation be slow?
• Can we pre-compute? (i.e. store redundant info)
• Can states be not uniquely represented?

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8-puzzle

• struct state
• int num9
• int space_pos
• record state
• num array1..9 of 0..8
• space_pos 1..9
• end

4 5
6 1 8
7 3 2
? ? ?
1 2 3
4 5 6
7 8
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Entropy

• Possible Different states 181440
• Space we used? 320 bits
• Using bytes? 80 bits

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Missionaries and Cannibals

• struct state
• int m2, c2
• bool b
• record state
• m, c array1..2 of 1..3
• b boolean
• end

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Entropy

• Possible Different states 16
• Space we used? 160 bits
• Using bytes? 40 bits
• One side only? 24 bits

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Be realistic

• 3, 2, 1, 02 ? L, R 32
• Store it as M C 4 B 16
• 3M3C, 3M2C, 3M1C, 3M, 2M2C, 1M1C, 3C, 2C, 1C,
  ? ? L, R 20

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Computer Organization

• x86 stores whole numbers in
• Binary Form
• x86 stores negative integers in
• 2s complement
• x86 stores real numbers in
• IEEE 754
• x86 stores alphabets in
• ASCII

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Integer Representation

• 46709394 (10)
• 00000010 11001000 10111010 10010010 (2)
• 0C C8 BA 92 (16) (0x0cc8ba92 in C/C)
• Little Endian

10010010 10111010 11001000 00000010
92 BA C8 02
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State Representation

• M C 4 B 16
• M3, C3, B0
• 00001111 (15)
• M0, C0, B1
• 00010000 (16)
• M2, C2, B1
• 00011010 (26)

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If you discovered

• M3, C3, B0
• 00001111 (15)
• M0, C0, B1
• 00010000 (16)
• M2, C2, B1
• 00011010 (26)
• This is Bitwise Representation

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Why bitwise?

• It is the native language of the computer
• It works extremely fast
• Integer Multiplication 4 cycles latency
• Moving Bits 1 cycle latency
• Packing and unpacking is easier to code and thus
  less error prone

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Bitwise Operations

• Binary
• ltlt shl
• gtgt shr
• and
• or
• xor
• Unary
• not

• shift bits left
• shift bits right
• intersection
• union
• mutual exclusion
• complement

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Examples
1 ltlt 2 4
7 gtgt 2
13 7
12 4
12 24
55 (8-bit unsigned)
-7 gtgt 2 (8-bit)
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Examples
1 ltlt 2 4
7 gtgt 2 1
13 7 5
12 4 12
12 24 20
55 (8-bit unsigned) 200
-7 gtgt 2 (8-bit) -1
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Bitmap

• Shift operators and bitwise operators can be used
  to manage a sequence of bits of ? 64 elements
• Operations
• Get(p 0..Size-1) Boolean
• Set(p 0..Size-1)
• Unset(p 0..Size-1)
• Toggle(p 0..Size-1)

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Using a 32-bit Integer

• 00000010 11001000 10111010 10010010
• 1 ltlt p a number with only bit p on
• value (1 ltlt p) ! 0 ? Get(p)
• value value (1 ltlt p) ? Set(p)
• value value (1 ltlt p) ? Unset(p)
• value value (1 ltlt p) ? Toggle(p)

Bit 31
Bit 0
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Using Bitmap

• Space-efficient
• Runtime may be faster
• Cache hit
• Less copying
• Or maybe slower
• More calculations

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Massive Calculations

• Example Count number of bits on a 16-bit integer
  v
• v (v 0x5555) ((v gtgt 1) 0x5555)
• v (v 0x3333) ((v gtgt 2) 0x3333)
• v (v 0x0f0f) ((v gtgt 4) 0x0f0f)
• v (v 0x00ff) ((v gtgt 8) 0x00ff)

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Counting bits

• 1 0 0 1 1 1 1 1 0 0 0 1 0 0 1 0
• 01 01 10 10 00 01 00 01
• 0010 0100 0001 0001
• 00000110 00000010
• 0000000000001000
• Exercise Do it for 32-bit

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Massive Calculations

• Given a bit pattern, find a bit pattern which
• only the leftmost bit of a bit group is on
• E.g.
• 1110101101111001 becomes1000101001000001

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Answer

• 1110101101111001 becomes1000101001000001
• The leftmost of a bit group means
• It is 1
• Its left is 0
• So
• value value (value gtgt 1)

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Bit Movements

• Question
• 1 1 0 1 0 1 0 0 becomes0101000100010000
• Solution
• value ((value 0x00f0) ltlt 4)) (value
  0x000f)
• value ((value 0x0c0c) ltlt 2)) (value
  0x0303)
• value ((value 0x2222) ltlt 1)) (value
  0x1111)
• Exercise
• 1 1 0 1 0 1 0 0 becomes1111001100110000
• 0 1 0 1 becomes0000111100001111

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Direct Addressing
0000 0000000000000000 1000 1111000000000000
0001 0000000000001111 1001 1111000000001111
0010 0000000011110000 1010 1111000011110000
0011 0000000011111111 1011 1111000011111111
0100 0000111100000000 1100 1111111100000000
0101 0000111100001111 1101 1111111100001111
0110 0000111111110000 1110 1111111111110000
0111 0000111111111111 1111 1111111111111111
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Complex Bit Patterns

• Othello
• 11101011 (where occupied)
• 10001001 (color)
• You are 1, where you can play?
• Solution
• 11101011 10001001 01100010
• 11101011 01100010 01001101
• 01001101 11101011 00000100
• How about the reverse direction?

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Other Uses

• Store multiple values, each one assigned a
  bit-range
• (value 32) (value 31)
• Storing sets of ? 64 elements
• Union
• Difference
• Is Subset

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Multiple Comparisons

• switch (i)
• case 1
• case 3
• case 4
• case 7
• case 8
• case 10
• f()
• break
• default
• g()

• if ((1ltlti) 0x039a)
• f()
• else
• g()

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Lexicographical Ordering

• Consider the 8 puzzle
• Store it verbatim 89 387420489
• Hash table? How large?
• Calculate a mapping from the 362880 possible
  permutations to 0..362879

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Example

• What are smaller than 530841672?
• 53084160-6? 1 ? 1
• 5308410-5?? 1 ? 2
• 530840-0??? 0 ? 6
• 53080-3???? 2 ? 24
• 5300-7????? 5 ? 120
• 53?????? 0 ? 720
• 50-2??????? 3 ? 5040
• 0-4???????? 5 ? 40320