EliasFano

10th StringBeginnersでの発表資料

Transcript

1. EliasFano
   
   10th StringBeginners
   
   
   → Kanda → K and A → K & A → K ampersand A →
   

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2. 
   ⟫
   × "$& n
   '% S[0,n)
   × i.e., S[i-1] ≤ S[i] for each 0 < i < n
   ⟫ !
   × Access / Predecessor / Successor (
   × #(
   2
   0 1 2 3 4 5 6 7
   S 2 7 18 28 42 43 44 59
   Access(4) = 42
   Predecessor(10) = 7 Successor(43) = 44
   

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3. EliasFano
   3
   S
   2
   7
   18
   28
   42
   43
   44
   59
   000
   000
   010
   011
   101
   101
   101
   111
   
   
   n = 8
   u
   log $ log %
   &
   010
   111
   010
   100
   010
   011
   100
   011
   2
   1
   3
   1
   1
   001
   100
   110
   0
   0
   0
   log
   H = 110 0 10 10 0 1110 0 10
   2 0 1 1 0 0 1
   3 ( )
   L = 010 111 010 100 010 011 100 011
   ()
   

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4. 2" + " log '
   (
   
   4
   L = 010 111 010 100 010 011 100 011
   H = 110 0 10 10 0 1110 0 10
   0 1 2 3 4 5 6 7
   S 2 7 18 28 42 43 44 59
   EliasFano
   log
   )
   "
   " log '
   (
   
   "
   2"
   
   2 *+, ( ≈ "
   

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5. 2" + " log '
   (
   "6
   ⟫ [0,u) 2 n (,/ (i.e. 4)
   -)#.
   %$*0+3 6
   5
   S 2 4 5 8 9 0010110011
   2 45 89
   n 5'& 2 u
   '
   (
   1
   5!5
   
   3 log '
   (
   
   

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6. 2" + " log '
   (
   4
   ⟫ [0,u) 1 n *-+&
   
   !, #"'.(24
   6
   S 2 4 4 5 8 9 0010011010001010
   2 5
   ')(
   (
   /
   33
   (0))
   44 8 9
   
   n 3%$1 u+n
   Less than half a bit per element away (Quasi-succinct)
   2 log ')(
   (
   ≈ " log ')(
   (
   
   

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7. 
   ⟫ Access(i) = S[i]
   ⟫ Predecessor(x) = max{S[i] : S[i] ≤ x}
   ⟫ Successor(x) = min{S[i] : S[i] > x}
   7
   O(1)
   O(log $
   %
   )
   0 1 2 3 4 5 6 7
   S 2 7 18 28 42 43 44 59
   Access(4) = 42
   Predecessor(10) = 7 Successor(43) = 44
   

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8. Access&O(1) $
   ⟫ &Access(4) = 42 = 101 0102
   8
   L = 010 111 010 100 010 011 100 011
   H = 110 0 10 10 0 1110 0 10
   ① !"# $
   %
   i !
   
   Select1
   (4) – 4 = 9 – 4 = 5 = 1012
   ② !"# % Select1
   (i) – i
   H Select " o(n)
   Selectb
   (H, i)&H i ! b #
   %
   

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9. SuccessorO(log $
   %
   )
   ⟫ Successor(43) = 44 = 101 1002
   9
   L = 010 111 010 100 010 011 100 011
   H = 110 0 10 10 0 1110 0 10
   Select0
   (4) = 8 Select0
   (5) = 12
   log $
   %
   = 3
   3 × (Select0
   (4) – 4) = 12 3 × (Select0
   (5) – 5) = 21
   101 0112
   5
   ②
   O(&'( )
   *
   )
   ① &'( *
   Select
   

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10. /6 EliasFano7)
    ⟫ # (;+'AA
    × 10%?*&
    × =98."
    × TRIE2!
    × 3<5,4!@:->
    10
    
    
    
    10
    7
    5
    0
    1 4 8
    6 9
    3
    a t
    e
    t
    a
    a t e
    c c
    2
    $A
    

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