An Analysis of the Feasibility of Graph Compression
Techniques for Indexing Regular Path Queries
Frank Tetzel, Hannes Voigt, Marcus Paradies, Wolfgang Lehner
May 19, 2017
1
Regular Path Queries (RPQs)
Matching paths conforming to regular expression
Only distinct (start, end) vertex pairs in result set
v4
v5
final state
start state
s
q1
b
f
Automaton representing (ab)+
b
a
v0
v2
a
a
v3
a
a
a
b
b
v6
a
v1
Search in data graph
start
v2
v3
v6
end
v0 , v2 , v3
v0 , v2 , v3
v0 , v2 , v3
Final result set
2
Processing strategies
Baseline
Guided search with automaton on data graph
Adjacency list on column store
3
Processing strategies
Baseline
Guided search with automaton on data graph
Adjacency list on column store
MR-Index
Store results of RPQs for future use
Treat vertex pairs as edges of a reachability graph
Use graph compression for reachability graph
v0
v2
v6
v3
Reachability graph
3
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1010 1111
1
0 0 1
0
1
1000 1100
Adjacency matrix
4
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1010 1111
1
0
0 0 1
1
1000 1100
Conceptual K2 -tree for k = 2
Adjacency matrix
4
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1010 1111
1
0
0 0 1
1
1000 1100
Conceptual K2 -tree for k = 2
Adjacency matrix
4
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1010 1111
1
0
0 0 1
1
1000 1100
Conceptual K2 -tree for k = 2
Adjacency matrix
4
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1010 1111
1
0
0 0 1
1
1000 1100
Conceptual K2 -tree for k = 2
Adjacency matrix
4
K2 -tree graph compression
Compact representation of a binary relation, e.g., an adjacency matrix
Hierarchical graph compression
0
1
2
3
4
5
6
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
1
0
3
0
0
1
1
0
0
1
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
Adjacency matrix
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0 0 1
1010 1111
1
1000 1100
Conceptual K2 -tree for k = 2
T1 = 1010
T2 = 0011 0011
L = 1010 1111 1000 1100
Bitvector representation
4
Measurements
LDBC social network dataset with scale factor 1 (3 million vertices and 17 million edges)
generate all possible RPQs with path length up to 3 hops, and recursions of them
runtime in ms
200
150
100
base
ADJ
K2
K2C
50
0
−11
−10
−9
−8
−7
−6
−5
−4
x
selectivity in 10
(ADJ - adjacency list, K2 - K2 -tree, K2C - K2 -tree with leaf compression)
5
Space consumption
space consumption in MB
·1010
ADJ
K2
K2C
1.5
1
0.5
0
0
0.5
1
1.5
2
−4
selectivity
·10
Space consumption of queries
6
Measurements
Batch of 300 queries, memory budget of 10 GB
number of uses
time in 103 · s
10
8
6
4
2
base ADJ K2 K2C best
150
100
50
ADJ
K2
K2C
Batch processing with sampled query sets
7
Conclusion and Future Work
Graph compression promising for storing reachability information
K2 -trees not beneficial for all results sets
I
I
Too much overhead for tiny result sets
No good compression for huge result sets
Future Work
Experiment with other K2 -trees to compress uniform 1-regions as well
Improve access time by providing specialized range queries, extracting submatrices
Compare memory consumption and access time with other compact reachability indices
like FERRARI
8
Conclusion and Future Work
Graph compression promising for storing reachability information
K2 -trees not beneficial for all results sets
I
I
Too much overhead for tiny result sets
No good compression for huge result sets
Future Work
Experiment with other K2 -trees to compress uniform 1-regions as well
Improve access time by providing specialized range queries, extracting submatrices
Compare memory consumption and access time with other compact reachability indices
like FERRARI
Thank You
8