An Optimal Ancestry Labeling Scheme Amos Korman 1 Pierre Fraigniaud - PowerPoint PPT Presentation

An Optimal Ancestry Labeling Scheme Amos Korman 1 Pierre Fraigniaud CNRS and University Paris Diderot 1 Speaker 1 / 31

Outline Informative Labeling Scheme Why should we fight for constants? Optimal ancestry-labeling scheme Small universal posets Conclusion 2 / 31

Informative Labeling scheme Graph representations: ◮ traditional: names given to the nodes serve merely as pointers to entries in a data structure ◮ informative labeling: mechanism for assigning short, yet informative, names to nodes (Kannan, Naor, Rudich [STOC ’88]) General objective To assign labels to nodes in such a way that allows one to infer information regarding any two nodes directly from their labels . Main quality measure Label size = number of bits used to form the labels 3 / 31

Example 1: adjacency in trees Input: tree T 4 / 31

Example 1: adjacency in trees Input: tree T 1. Give distinct IDs to the nodes, between 1 and n 2. Root T at an arbitrary vertex L ( u ) = ( ID ( u ) , ID ( parent ( u )) u and v are adjacent ⇐ ⇒ u = parent ( v ) or v = parent ( u ) Label size = 2 ⌈ log 2 n ⌉ bits 4 / 31

Informative Labeling Scheme Let P be a boolean predicate defined on pairs of vertices for graphs in F Encoder (or marker) M Given G ∈ F , M ( G ) = L where L : V ( G ) → { 0 , 1 } ∗ Decoder D D : { 0 , 1 } ∗ × { 0 , 1 } ∗ → { true, false } For any G ∈ F , and any ( u , v ) ∈ V ( G ) × V ( G ) , P ( u , v ) = true ⇐ ⇒ D ( L ( u ) , L ( v )) = true Can be generalized to various types of functions (distance, connectivity, etc.), or tasks (e.g., routing). 5 / 31

Outline Informative Labeling Scheme Why should we fight for constants? Optimal ancestry-labeling scheme Small universal posets Conclusion 6 / 31

Adjacency in trees Definition A graph U is universal for a graph family F if any G ∈ F is isomorphic to an induced subgraph of U . Theorem ( Kannan, Naor, Rudich [STOC ’88] ) There exists an adjacency labeling scheme for F with labels of at most k bits if and only if there exists a universal graph for F of order at most 2 k . 7 / 31

Adjacency in trees Definition A graph U is universal for a graph family F if any G ∈ F is isomorphic to an induced subgraph of U . Theorem ( Kannan, Naor, Rudich [STOC ’88] ) There exists an adjacency labeling scheme for F with labels of at most k bits if and only if there exists a universal graph for F of order at most 2 k . Adjacency: State of the art 2 log n (Kannan, Naor, and Rudich [STOC ’88]) log n + O ( log ∗ n ) (Alstrup and Rauhe [FOCS ’02]) ⇒ universal graph of order n 2 log ∗ n 7 / 31

Example 2: ancestry in trees Input: rooted tree 8 / 31

Example 2: ancestry in trees Input: rooted tree Give distinct DFS numbers to the nodes, between 1 and n L ( u ) = ( DFS ( u ) , DFS ( u max )) where u max is the node with largest DFS number in the subtree rooted at u . u is an ancestor of v ⇐ ⇒ DFS ( v ) ∈ [ DFS ( u ) , DFS ( u max )] Label size = 2 ⌈ log 2 n ⌉ bits 8 / 31

XML trees < art > < book > < Sutter’s Gold > < author > Blaise Cendrars < /author > art < Release > 1925 < /Release > < /Sutter’s Gold > book movie < /book > < movie > < Citizen Kane > Sutter's Gold Citizen Kane Once Upon a Time < direct > Orson Wells < /direct > in the West < Release > 1941 < /Release > < /Citizen Kane > author Release director director Release Release < Once Upon a Time in the West > date date date < direct > Sergio Leone < /direct > < Release > 1968 < /Release > < /Once Upon a Time in the West > < /movie > < /art > • Answer queries using the index labels only, without accessing the actual documents. • A small improvement in the label size ⇒ significant improvement in the performances of XML search engines. 9 / 31

State of the art: ancestry in trees Ancestry 2 log n (Kannan, Naor, and Rudich [STOC ’88]) 3 2 log n + O ( log log n ) (Abiteboul, Kaplan, and Milo [SODA ’01]) log n + O ( log n / log log n ) (Thorup and Zwick [SPAA ’01]) � log n + O ( log n ) (Alstrup and Rauhe [SODA ’02]) log n + Ω( log log n ) (Alstrup, Bille and Rauhe [SODA ’03]) log n + 2 log ( depth ) + O ( 1 ) (Fraigniaud and Korman, [SODA ’10]) log n + O ( log log n ) (Fraigniaud and Korman, [STOC ’10]) 10 / 31

Outline Informative Labeling Scheme Why should we fight for constants? Optimal ancestry-labeling scheme Small universal posets Conclusion 11 / 31

Interval containment v ancestor of u ⇐ ⇒ I ( u ) ⊆ I ( v ) 2 log n -scheme by Kannan, Naor, and Rudich use n 2 intervals. We aim at using n log c n intervals We use intervals of the following form, for k = 1 , . . . , log n : x a x (a+b) k k I (k,a,b) x k 2x k 1 N level k: x k 12 / 31

Spine decomposition v 1 v i F 1 F i v s F s Nodes classified as either heavy or apex . 13 / 31

Trees with bounded spine decomposition depth d = 0 ( 1 ) 14 / 31

Trees with bounded spine decomposition depth d = 0 ( 1 ) F ( n , d ) = forests with ≤ n nodes, v 1 and spine-decomposition depth ≤ d . We aim at using nd 2 intervals for v i F 1 F ∈ F ( n , d ) Induction of k = log n F i Difficult case: F containing a tree T v s of size larger than 2 k , i.e., 2 k < | T | ≤ 2 k + 1 . F s 14 / 31

General idea I(v ) I(v ) I(v ) 1 2 s bin J level k+1 J 1 J 2 J s x k+1 level k x k I(F ) I(F ) I(F ) 1 2 s c |F | c |F | c |F | k 1 k 2 k s s I( U F ) i=1 i 15 / 31

Tuning of the parameters (1/3) I(v ) I(v ) I(v ) 1 2 s bin J level k+1 J 1 J 2 J s x k+1 level k x k I(F ) I(F ) I(F ) 1 2 s c |F | c |F | c |F | k 1 k 2 k s s I( U F ) i=1 i For 1 ≤ i < s , the length of I ( v i ) must satisfy s � | I ( v i ) | ≈ c k | F i | + x k + 1 + | I ( v i + 1 ) | ≈ c k ( | F i | ) + i · x k + 1 . j = i Bin J to be of length | J | ≈ c k · 2 k + 1 + ( s + 1 ) · x k + 1 suffices. 16 / 31

Tuning of the parameters (2/3) Since s ≤ d , we must have | J | be approximately c k + 1 2 k + 1 ≈ c k 2 k + 1 + d · x k + 1 Choose the values of c k so that: d · x k + 1 c k + 1 − c k ≈ 2 k + 1 We set k 2 k 1 � c k ≈ j 1 + ǫ , and thus x k ≈ d · k 1 + ǫ j = 1 17 / 31

Tuning of the parameters (3/3) Let A k ≈ N / x k and B k ≈ c k 2 k / x k . x a x (a+b) k k I (k,a,b) 1 x k 2x k N level k: x k where 1 ≤ a ≤ A k and 1 ≤ b ≤ B k . Thus, N ≈ c log n · n = O ( n ) . The number of level- k intervals is O ( A k · B k ) = O ( nd 2 k 2 ( 1 + ǫ ) / 2 k ) , yielding a total of O ( nd 2 ) intervals, as desired. 18 / 31

The general case: uses the folding-decomposition v 1 v 1 * = v 1 v 2 u 1 u 2 v i v s v 2 * u 2 v 2 = u 1 v i v j F 1 v j v s F i F 1 F i * F s F 2 = F s * F 1 (a) (b) 19 / 31

Ancestry preservation DFS traversal in T that visits apex children first. For any node u , let DFS ( u ) be the DFS number of u . v 1 v 1 * = v 1 v 2 u 1 u 2 v i v s v 2 * u 1 u 2 v 2 v i v j = F 1 v j F i v s F 1 F i * F s F 2 = F s * F 1 (a) (b) Lemma Node v is an ancestor of u in T if and only if at least one of the following two conditions hold ◮ C1: v is an ancestor of u in T ∗ ; ◮ C2: APEX ( v ) is ancestor of u in T ∗ and DFS ( v ) < DFS ( u ) . 20 / 31

Ordering the intervals Lemma Node v is an ancestor of u in T if and only if at least one of the following two conditions hold ◮ C1: v is an ancestor of u in T ∗ ; ◮ C2’: APEX ( v ) is ancestor of u in T ∗ and I ( v ) ≺ I ( u ) . v 1 I(v ) I(v ) 1 2 k+1 v 2 k I(F ) I(F ) 1 2 2 F 1 1 F 2 (a) (b) label ( u ) = ( I ( u ) , I ( APEX ( u ))) 21 / 31

Compact encoding of I ( APEX ( v )) It is sufficient to encode: ◮ its level k ′ ◮ two shifts b ′ left and b ′ right in [ 1 , B k ′ ] x a' x (a'+b') k' k' I k',a',b' x a" N k' level k': I(v) level k: 22 / 31

Outline Informative Labeling Scheme Why should we fight for constants? Optimal ancestry-labeling scheme Small universal posets Conclusion 23 / 31

Graph arboricity The arboricity of a graph is the minimum number of forests into which its edges can be partitioned. Corollary ( Kannan, Naor, Rudich [STOC ’88] ) There exists an adjacency labeling scheme for the family of graphs with arboricity at most k with labels of at most ( k + 1 ) log n bits. High level correspondence between: adjacency/arboricity for graphs and ancestry/tree-dimension for posets 24 / 31

Partially ordered sets Poset ( X , ≤ ) ◮ reflexivity: x ≤ x ◮ antisymmetry: ( x ≤ y and y ≤ x ) ⇒ x = y ◮ transitivity: ( x ≤ y and y ≤ z ) ⇒ x ≤ z ( X , ≤ ′ ) is an extension of ( X , ≤ ) if: ∀ x , y ∈ X , x ≤ y ⇒ x ≤ ′ y The dimension of a poset ( X , ≤ ) is the smallest number of linear (i.e., total order) extensions of ( X , ≤ ) the intersection of which gives rise to ( X , ≤ ) . 25 / 31

Universal posets A poset ( X , ≤ X ) contains a poset ( Y , ≤ Y ) as an induced suborder if there exists an injective mapping φ : Y → X such that for any two elements a , b ∈ Y : a ≤ Y b ⇐ ⇒ φ ( a ) ≤ X φ ( b ) . Definition A poset ( U , ≤ ) is called universal for a family of posets F if ( U , ≤ ) contains every poset in F as an induced suborder. 26 / 31