1. Succinct Representation of Labeled Graphs
Jérémy Barbay, Luca Castelli Aleardi, Meng He, J. Ian Munro

2. Background: Succinct Data Structures
What are succinct data structures
Representing data structures using ideally information-theoretic minimum space
Supporting efficient navigational operations
Jacobson 1989
Why succinct data structures
Large data sets in modern applications: textual, genomic, spatial or geometric
An implementation: Delpratt et al. 2006

3. Motivations and Objectives
Initial Problem: representing unlabeled graphs succinctly
Connectivity information: degree, neighbors, adjacency
Jacobson 89, Munro and Raman 97, Chuang et al. 98, Chiang et al. 01, Castelli Aleardi et al. 05 & 06
New Problem: representing labeled graphs succinctly
Properties of an object is often modeled as labels on vertices or edges
Label-based queries

4. Planar Triangulations
Planar Triangulations: A planar graph whose faces are all triangles
Applications: computer graphics, computational geometry

5. An Example: Terrain and Triangulations
Geometric Modeling and Computer Graphics Research Groups,
University of Genova

6. Planar Triangulations and Realizers
Planar Triangulation T
Number of vertices: n; number of edges: m
External face: (v0, v1, vn-1)
Realizer (Schnyder 1990)
A partition of the set of internal edges into three set of directed edges T0, T1, T2
the edges incident to any internal vertex v in ccw order are: one outgoing edge in T0, zero or more incoming edges in T2, one outgoing edges in T1, zero or more incoming edges in T0, one outgoing edge in T2 and zero or more incoming edges in T1

7. Planar Triangulations and Realizers (Continued)
Property
T0 is a spanning tree of T / {v1, vn-1}
T1 is a spanning tree of T / {v0, vn-1}
T2 is a spanning tree of T / {v0, v1}
Canonical spanning tree T0
T0 with edges (v0, v1) and (v0, vn-1)
Canonical Ordering

8. Realizer: An Example T2 T0 T1

9. Three Traversal Orders on a Planar Triangulationsπ2 T2 1 2 3 4 5 6 7 8 9 10 11 4 3 2 6 5 10 9 8 7 1 7 9 1 8 2 6 5 4 3 π0 π1 T0 T1

10. Operations on Unlabeled Planar Triangulations
Old operations
adjacency
Degree
New operations
select_neighbor_ccw(x, y, r): the rth neighbor of x staring from y in ccw order if x and y are adjacent
rank_neighbor_ccw(x, y, z): the number of neighbors of vertex x between y and z in ccw order if y and z are neighbors of x
Conversions between the numbers of vertices under π0, π1 and π2

11. Succinct Representation of Unlabeled Planar Triangulations
Observation
For any node x, its children in T1 (or T2), listed in ccw (or cw) order, have consecutive numbers in π1 or π2
Approach
Represent T0, T1 and T2 using different types of parentheses in orders π0, π1 and π2, respectively
Combine the three sequences into a multiple parenthesis sequence
Results
Space: 2m log26 + o(m) bits
Time: O(1)

12. Succinct Unlabeled Planar Triangulations: An Example1 2 3 4 5 6 7 8 9 10 11
adjacency(4,6)
=true
degree(6) =8
rank_neighbor_ccw(4, 5, 1) =3 π0 π1 T0 T1
( ( [ [ [ [ ) ( ] ( ] ( ] { [ [ ) { ) ( } ] { ) ( } ] { [ [ [ [ ) … …

13. Vertex Labeled Planar Triangulations
Notion
Number of labels: σ
Number of vertex-label pairs: t
Operations
lab_degree(α,x): number of neighbors of vertex x associated with label α
lab_select_ccw(α,x,y,r): rth vertex labeled α among neighbors of vertex x after vertex y in ccw order if y is a neighbor of x
lab_rank_ccw(α,x,y,z) : number of neighbors of vertex x labeled α between vertices y and z in ccw order if y and z are neighbors of x

14. Vertex Labeled Planar Triangulations: An Example
{b,c} 1 2 3 4 5 6 7 8 9 10 11
lab_degree(a,5) =3
lab_select_ccw(c,6,4,2) =5
lab_rank_ccw(b,6,10,5) =4
{a,b}
{a,b}
{b}
{b}
{a}
{c}
{b}
{a,c}
{a,b}
{a,b,c}
{c}

15. Succinct Index for Vertex Labeled Triangulations
ADT
node_label: f(n, σ, t) time
Succinct Index
Space: t∙o(lg σ) bits
Time: O((lg lg lg σ)2(f(n, σ, t)+lglg σ) time for lab_degree, lab_select_ccw and lab_rank_ccw

16. Succinct Representation of vertex labeled planar triangulations
Space: t lg σ+t∙o(lg σ) bits
Time:
node_label : O(1)
lab_degree, lab_select_ccw and lab_rank_ccw : O((lg lg lg σ)2lglg σ)

17. Edge Labeled k-Page Graphs
Notion
Number of labels: σ
Number of edge-label pairs: t
Operations
lab_adjacency(α,x,y): whether there is an edge labeled α between vertices x and y
lab_degree_edge(α,x): the number of edges of vertex x that are labeled α
lab_edges(α,x): the edges of vertex x that are labeled α

18. Edge Labeled k-Page Graphs
Results
Space: kn + t(lg σ + o(lg σ)) bits
Time:
lab_adjacency and lab_degree_edge: O(k lglg σ(lg lg lg σ)2)
lab_edges: O(d lglg σ lg lg lg σ+k) , where d is the number of such edges

19. More Results
Succinct representation of unlabeled k-page graphs for large k
Succinct representation of edge Labeled k-page graphs for large k
The results on edge labeled k-page graphs also apply to edge labeled planar graphs (k = 4)

20. Conclusions Summary Subsequent Work and Open Problems
First succinct representations of labeled graphs
Two preliminary results on succinct representations of unlabeled graphs
Subsequent Work and Open Problems
Edge labeled planar triangulations (done)
Vertex labeled k-page graphs

21. Thank you!