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

⟫ × "$& 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

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 ()

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 *+, ( ≈ "

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 ' (

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 ')( (

⟫ 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

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 # %

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

/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