Succinct Ordinal Trees Based on Tree Covering

Meng He, J. Ian Munro, University of Waterloo

S. Srinivasa Rao, IT University of Copenhagen

Background: Succinct Data Structures The problem

Modern applications often process huge amounts of data

Examples Web search engines: Google, Altavista,

etc. Bioinformatics application XML databases Spatial databases …

The Solution: Succinct Data Structures

What are succinct data structures Representing data structures using

preferably information-theoretic minimum space

Supporting efficient navigational operations

History of Succinct Data Structures Jacobson 1989

The number of different ordinal trees of n nodes: ( )/(n+1) ≈ 4n/(πn)3/2

Information-theoretic minimum: 2n-O(lg n) bits

Explicit, pointer-based representation: Θ(n lg n) bits

Trees

2nn

Succinct Ordinal Trees Level order unary degree sequence

(LOUDS): Jacobson 1989 Balanced parentheses (BP): Munro &

Raman 1997 Depth first unary degree sequence

(DFUDS): Benoit et al. 1999 Tree covering (TC): Geary et al. 2004

3

Preorder and DFUDS order

1

2

7

5 6

4

8

9 10 11

3

4 5 6

7 8

Navigational Operations Considered Parent Child Level_ancestor Depth Subtree_size LCA …all in O(1) time with 2n+o(n) bits on

the word RAM

Motivations and Objectives Three main representations: BP, DFUDS,

TC

The fact: different representation supports different operations on trees Example: height, node_rankDFUDS, node_rankpost

New problem: a representation supporting all the navigational operations

Motivations and Objectives (Continued)

The assumption: there may be new operations supported by one of these representations

New problem: one representation that can compute an arbitrary word of all the other representations

The Tree Covering Algorithm by Geary et al. The idea

Cover the tree with a set of mini-trees Cover each mini-tree with a set of micro-trees Compute the set of mini-trees (micro-trees) in a

bottom-up, greedy fashion Properties

Any two mini-trees (micro-trees) can only share their root

Size of a mini-tree (micro-tree): M~3M-4 (M’ ~3M’-4) Parameters: M = lg4

n, M’ = lg n / 24

The Tree Covering Algorithm: An Example

M = 8, M’ = 3

Operations supported on TC by Geary et al. The old TC (Geary et al. 2004)

child child_rank depth level_anc nbdesc degree node_rankPRE, node_selectPRE

node_rankPOST, node_selectPOST

New Definitions and Properties: Preorder Changers

Tier-1 preorder changers: Number of Tier-1 preorder changers: at

most twice the number of mini-trees Tier-2 preorder changers are similar

DFUDS Order Changers

Tier-1 DFUDS order changers: Number of Tier-1 DFUDS order changers: at

most four times the number of mini-trees Tier-2 DFUDS order changers are similar

τ*-name of a Node

Preorder numbers: node x τ-names: τ(x)=<τ1(x),τ2(x),τ3(x)>

τ*-names: τ*(x)=<τ1(x),τ2(x),τ3*(x)>

Node 29

τ(29)=<3,

τ*(29)=<3,1,4>

1,5>

Supporting node_selectDFUDS

From the DFUDS number to the τ*-name

From the τ*-name to the τ-name Table lookup

From the τ-name to the preorder number Geary et al. 2004

Computing τ1(x)

1 2 3 4 5 6 7 8 9 …

1

Nodes (DFUDS #)

τ1’s stored …2 1 2

Example: x=8th node in DFUDS

τ1(x)=2

10

Computing τ1(x) (Continued) Dictionary

Universe: n size: O(n / lg4 n) Space cost: o(n) bits (Raman et al,

2002) τ1’s stored

Number of elements stored: O(n / lg4 n) Each element: O(lg n) bits Space cost: o(n) bits

Computing τ2(x) and τ3*(x)

1 2 3 4 5 6 7 8 9 …

1

Nodes

τ2 …1 1 2

τ2(x)=1

10

1 3 1 2

τ3* 1 1 2 1 2 1 3 2 …

τ3*(x)=4

Computing τ2(x) and τ3*(x) (Continued) Dictionary

Universe: n size: O(n / lg n) Space cost: o(n) bits (Raman et al,

2002) τ2’s and τ3

*’s stored Number of elements stored: O(n / lg n) Each element: O(lglg n) bits Space cost: o(n) bits

Other Operations Supported height LCA distance leaf_rank and leaf_select leftmost_leaf and rightmost_leaf leaf_size node_rankDFUDS level_leftmost and level_rightmost level_succ and level_pred

Data Abstraction: Computing a subsequence of BP and DFUDS

The problem: store the tree using TC, and support the computation of a word of its BP or DFUDS sequence

Results: Time: compute a word (Θ(lg n) bits) of

the BP or DFUDS sequence in O(f(n)) time

Space: n/f(n) additional bits

Conclusions A succinct representation of ordinal

trees using 2n+o(n) bits that support all the navigational operations

Our representation also supports level-order traversal, a useful ordering previously supported only with a very limited set of operations

Our encoding schemes supports BP and DFUDS as abstract data types

Open Problems

Support new operations that are not supported by BP, DFUDS or TC

Constant-time computation of a word of BP or DFUDS using o(n) additional bits (or is this possible at all?)

Thank you!