1. Qingxia Liu qxliu.nju@gmail.com
Optimal SenTree: representing RDF sentence as a tree with minimal reversed triples
Qingxia Liu

2. Outline Introduction Preliminaries The algorithm Analysis Related work
Conclusion
Reference

3. Introduction RDF Sentence Existence of Blank Node
A set of b-connected triples
Existence of Blank Node
Knowledge Representing necessity
Representing Predicates that have more than two arguments
Population (America, 312.8million,2012)
Population (America,bn_1)
Value(bn_1,312.9million)
Year(bn_1, 2012)

4. 438 (55.9%) published blank nodes
Introduction
Existence of Blank Node (con.)
Mallea et al. (ISWC 2011)[1]
783 domains crawled
438 (55.9%) published blank nodes
All domains
Avg %bnode = 7.5%
Domains in LOD
Avg %bnode = 6.1%
By domain
Average percentage in terms of each domain

5. Introduction Existence of Blank Node (con.) Falconet_v05 By document
total: ,885,647,
has bnode: 8,692,959, (39.72%)
by sentence
total: ,643,593,653
In bnDoc: 855,234,540
has bnode: 208,591,775, (12.69%--in total, %--in BnDoc)
complex: , (2.29% -- in bnSentence)
After remove literal, parallel edges, self-loop, m!=n-1
bnDoc: RDF document that contains blank node;
bnSentence: RDF sentence that contains blank node

6. The problem How to represent an RDF sentence
A tree-oriented browser will feel natural [3]
To maintain the semantic
Each triple represents once and only once
Minimal number of inversed triples in the representing

7. Our Method Overview Entity relation direction info. partial order
Skeleton Graph
RDF Sentence
Cost Graph
Minimum Spanning Tree
Edmonds Algorithm
Optimum SentTree
simplification
post processing
assign cost

8. Preliminaries SenTree An out tree
For a directed labeled multigraph G, an SenTree of G is an out tree ST=(V,E,f,g,h,r) that
f: V->V(G) , is a surjection. if v∈V and u∈V(G), f(v)=u
g:V(G)->V, for every vertex in G assign a vertex in ST that can act as parent
h: E->E(G) a bijection that if eST∈E and eG∈E(G), h(eST)=eG, then eST and eG satisfy one of the relations bellow:
Case 1(parallel): f(tail(eST))=tail(eG),f(head(eST))=head(eG),label(eST)=label(eG);
Case 2(inverse): f(tail(eST))=head(eG),f(head(eST))=tail(eG),label(eST)=label(eG);

9. Preliminaries SenTree Unsmoothness Optimum SenTree
For edge e in a SenTree
Case 1 (parallel), unsmth(e) = 0;
Case 2 (inveres), usmth(e) = 1;
For a SenTree ST
usmth(𝑆𝑇)= 𝑒 𝑖 ∈𝐸(𝑆𝑇 usmth(𝑒 𝑖 )
Optimum SenTree
The SenTree of graph G that has the minimum unsmoothness among all SentTrees of G
Goal: minST 𝑒 𝑖 ∈𝐸(𝑆𝑇 usmth(𝑒 𝑖 )

10. Preliminaries Skeleton Graph
A simple digraph describes the relationship of resource terms (uris and blank nodes) in an RDF graph
the Skeleton Graph of an RDF graph G is a simple digraph Gs=(Vs,Es) that
Vs=V(G)-Literal(G)
Es: any two vertexes u,v∈Vs, if exist a statement (u, p, v) in G, then ordered pair <u,v> ∈Es
Simplification Operations
Delete Literals
Delete parallel edges
Delete loops

11. Preliminaries Cost Graph
A weighted symmetric graph (every edge has its inverse edge)
A cost graph of a directed graph G satisfies that:
For every pair of adjacent vertexes u,v in G, there are only two directed edges e1=(u’,v’) and e2=(v’,u’) in GC
If edges between u,v in G in both direction, then cost(e1)=cost(e2)=0;
Else if all in one direction, for example u→v, then cost(e1)=0, cost(e2)=1;

12. The algorithm Input Output
a directed labeled graph G of triples in an RDF sentence
Output
Optimal SenTree STOPT of G

13. The algorithm Simplification Delete Literals Delete parallel edges
res1
lit
Simplification
Delete Literals
Reduce the problem size
Do not affect the b-connectivity of RDF sentence
Never has out degree
If act as inner vertex, at least one edge inversed
All hanging as leaves, never gain unsmoothness
Delete parallel edges
Will be hanging as leaves, never gain unsmoothness
Delete loops
res1
lit

14. The algorithm Assign cost Init Gc as empty set
For each edge e=<u,v> in Gs
if <u,v>∈Gc, set cost(<u,v>) = 1;
Else put <u,v>,<v,u> into Gc, set cost(<u,v>)=0, cost(<v,u>)=1

15. The algorithm
Edmonds Algorithm
(Gc , r) → MST
O(m+nlogn) [4]

16. The algorithm Post processing Init ST = {r} DFS traverse the MST
When processing vertex u
For each edge <u,v> in MST (u∈ST)
If cost(<u,v>)=0, add all edge (u,p,v)∈G as children of u into ST, one of v as main vertex to expand in the next step
If cost(<u,v>)=1, select one edge (v,p,u)∈G, put the inversed as children of u into ST, v as the main vertex to expand in the next step
Other not visited edges (u,p,x)∈G, hanging as leaves of u into ST
x: literals, resources that expand at other place

17. Analysis SenTree Optimal SenTree vertexes Edges
V(G)-Literal(G)=V(Gs)=V(Gc)=V(MST)
A literal connected to at lease one vertex in V(G)-Literal(G)
Literals processed as x in post processing
Edges
Processed once and only once
Optimal SenTree
Simplification : never gain the unsmoothness
Cost of Gc: max num of unsmoothness will gain when visiting along an directed edge
MST
Edge inversed only when cost(<u,v>)=1 and inversed only one edge

18. References
Mallea, Alejandro, Marcelo Arenas, Aidan Hogan, and Axel Polleres. "On blank nodes." In The Semantic Web–ISWC 2011, pp Springer Berlin Heidelberg,
Baase, Sara. Computer Algorithms: Introduction To Design & Analysis, 3/E. Pearson Education India, 2000.
Berners-Lee, Tim, et al. "Tabulator: Exploring and analyzing linked data on the semantic web." Proceedings of the 3rd International Semantic Web User Interaction Workshop. Vol
H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,” Combinatorica 6 (1986),

19. Thank you!
Any Questions?