Testing Forest-Isomorphismin the Adjacency List Model

Testing Forest-Isomorphism
in the Adjacency List Model
Mitsuru Kusumoto†, Yuichi Yoshida†*
† : Preferred Infrastructure, Inc.
* : National Institute of Informatics.
1

Overview
Given two forests G and H, determine if G ≅ H or G and H are
far from being so by looking at very small parts of G and H.
Outline
 Introduction
Property testing
Problem setting
 Our algorithms
≅
?
2 / 21

Property Testing
We want to solve decision problem as efficiently as possible!!
 Example : Graph connectivity
 Standard setting : BFS is enough. → Θ(n) time.
 Property testing : Check if G is connected or G is far from
being connected. → O(1) time!?
Connected Not connected
4 / 21

Property Testing
Property testing algorithm is a (randomized) algorithm that
checks if input satisfies property P or is far from P with high
probability (e.g., ≥ 2/3) with sublinear query or time complexity.
Main Interest
 What kinds of properties are testable efficiently?
Connected Not connected
We want to
distinguish them
Far from being
connected
Close to being
connected
5 / 21

Graph Property Testing - Review
The efficiency of property testing algorithms depends on the
input models.
Adjacency matrix model
[01010]
[10110]
G = [01001]
[11001]
[00110]
Adjacency list model
v
A
B
C
1
2
3
O(v, 1) = A
O(v, 2) = B
O(v, 3) = C
• Input model for dense graphs. [GGR’98]
• Many properties are testable.
(e.g., connectivity, △-freeness, ... .)
• Necessity & sufficiency for constant-
time testability are known. [Alon+’09]
• Input model for sparse graphs. [GR’02]
[KKR’04]
(e.g., connectivity, H-minor-freeness.)
• But many results assume bounded-
degree condition: degrees of vertices
must be bounded by some constant.
6 / 21

Graph Property Testing - Review
 Only a few efficient algorithms.
 Many hardness results: △-freeness, k-colorability, etc.,
requires Ω(√n) queries. [A+08, B+08, K+04]
Question : Is it possible to obtain efficient algorithms for
fundamental problems without bounded-degree condition?
Adjacency list model
v
A
B
C
1
2
3
O(v, 1) = A
O(v, 2) = B
O(v, 3) = C
• Input model for sparse graphs. [GR’02]
[KKR’04]
(e.g., connectivity, H-minor-freeness.)
• But many results assume bounded-
degree condition: degrees of vertices
must be bounded by some constant.
What happens if we do not assume
the bounded-degree condition?
7 / 21

Forest-Isomorphism
We focus on forest-isomorphism in adjacency list model.
Input : Two forests G and H represented by adjacency lists
and proximity parameter ε > 0.
Query Model : We can access to G and H via following queries:
 deg(v): returns the degree of vertex v.
 adj(v, i): returns a vertex adjacent to v by i-th edge.
 random(): returns a randomly chosen vertex.
≅
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8 / 21

Forest-Isomorphism
Input : Two forests G and H represented by adjacency lists
and proximity parameter ε > 0.
ε-Farness : d(G, H) := # of edge-(additions / deletions) to
transform G to H. (Graph edit distance)
For ε>0, (G, H) are ε-far from being isomorphic ⇔ d(G, H) ≥ εn.
Objective: Determine G≅H or d(G, H) ≥ εn.
≅
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9 / 21

Forest-Isomorphism
Motivation
 Problem is fundamental: Forest is simple structure and
isomorphism is a theoretically important problem.
 Isomorphism was sometimes considered in property testing
literature. [AS’05, AS’08, NS’11]
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10 / 21

Forest-Isomorphism
Related Work
 If there is no restriction on input, graph isomorphism testing
in the adjacency list model requires Ω(√n) queries. [FM’08]
Good motivation for our focus on forests.
 If input is a bounded-degree hyperfinite graph, then graph
isomorphism is constant-time testable. [NS’11]
But if there is no degree bound, testability was unknown.
≅
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11 / 21

Our Contribution
Furthermore, we obtained more general result:
 If the input is a forest, every graph property is testable in
poly(log n) queries in the adjacency list model.
 We use a similar technique with [Newman and Sohler’11].
Query complexity
Upper bound poly(log n)
Lower bound Ω(√log n)
12 / 21

Overview of Our Method
1. Partitioning oracle:
We define a procedure that removes
small fractions of edges to partition
the graph into several parts with
“good” properties.
G
The Partitioning Oracle
H
2. We check if each corresponding
part in G and H is isomorphic or far
from so.
If G, H are far from being isomorphic,
there is at least one corresponding part
in G, H that is also far from being
isomorphic.
14 / 21

Partitioning Oracle
Partitioning Oracle: Given ε>0 and access to G, there exists
integer s=s(ε) and subgraph G’⊆ G s.t.,
 |E(G) – E(G’)| ≤ εn / 3
 Each connected component of G’ is either
s-bounded-degree-tree or s-rooted-tree.
s-rooted tree:
A tree where there exists v ∈ V(T) s.t.
deg(v) ≥ s and (size of each sub-tree) < s.
(We call the vertex v a root.)
s-bounded-degree-tree:
A tree where
(degree of each vertex) < s.
v
15 / 21

Partitioning Oracle
 |E(G) – E(G’)| ≤ εn / 3
We can provide query access to G’.
Alive Edge Query: Check if edge (v, i) still exists in G’.
 The subgraph G’ is chosen deterministically.
 If G ≅ H, then G’ ≅ H’.
v
A
B
C
1
2
3
(v, 1) : not alive
(v, 2) : not alive
(v, 3) : alive
16 / 21

Partitioning Oracle
 |E(G) – E(G’)| ≤ εn / 3
So…
 If d(G, H) = 0 ⇒ d(G’, H’) = 0
G’ and H’ are chosen deterministically.
 If d(G, H) ≥ εn ⇒ d(G’, H’) ≥ εn / 3
We remove at most εn / 3 edges from G and H.
Thus, it is enough to consider the partitioned graphs G’ and H’.
17 / 21

Graph Partition
Suppose that G is obtained through the partitioning oracle.
We split G into the following parts for some constants α,γ>1.
 G[0] := s-bounded degree trees in G
 G[1] := s-rooted trees in G with root degrees in [s, αγ)
 G[2] := s-rooted trees in G with root degrees in [αγ, αγ2)
 G[3] := s-rooted trees in G with root degrees in [αγ2, αγ3)
 ...
O(log n) parts
G[0] G[1]
G[2] ......
18 / 21

Isomorphism between Each Partitions
Graph partition is useful in the following sense.
Lemma. d(G, H) ≤ Σi d(G[i], H[i]).
Proof. Transformation from G[i] to H[i] for each i would transform
G to H. □
Corollary. If d(G, H) ≥ εn, then for βi > 0 with Σ βi = ε,
∃i s.t. d(G[i], H[i]) ≥ βin. □
Thus, it suffices to check the isomorphism between G[i] and H[i]
for each i=0,1,2,….
We set β0=ε/2, β1=β2=…=O(ε / log n).
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Isomorphism between Each Partitions
Testing G[i]≅H[i]
 For i=0 : We can use a tester for the bounded-degree model
[NS’11].
 For i≥1 : We develop a new algorithm.
Sketch : We randomly sample root vertices.
For each root vertex, we randomly sample its subtrees and
create a histogram of subtrees.
After this, we compute the minimum matching between
the histograms in G and H.
This minimum matching turns out to be a good
approximation to d(G, H).
:2
:2
:1
… 20 / 21

Conclusion
 If the input is a forest, every graph property is testable in
poly(log n) queries.
Future Work?
Can we obtain similar results for larger graph class than forests?
Outerplanar graphs, Bounded-tree width graphs,
Scale-free graphs, …
Query complexity
Upper bound poly(log n)
Lower bound Ω(√log n)
Actually O(log^2^poly(1/ε)(n))
21 / 21

Lower bound - Overview
1. We construct two distributions of input, D1, D2.
∀(G, H) ∈ D1, G ≅ H
∀(G, H) ∈ D2, d(G, H) ≥ n/8
2. We reduce the isomorphism testing to checking if two
probabilistic distributions are the same or not. This requires
Ω(√N) queries.
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23 / 21

Lower bound
Let Fk := (n / (2klogn)) copies of a star graph with 2k vertices
(Remark that |Fi| = n / logn)
F3
F2
F1
F0
…
Flogn 24 / 21

Lower bound
Construct two distributions D1, D2 :
D1 : G=H
D2 : randomly assign Fk to
either G or H so that
|V(G)| = |V(H)|.
G = F0 ∪ F1 ∪ … Flogn
H = F0 ∪ F1 ∪ … Flogn
G = ................................
H = ...............................
F0 F1 … Flogn
25 / 21

Lower bound
Because we can perform only “random-sampling” and
(degree/neighbor)-query, checking if G ≅ H is equivalent to
checking two probabilistic distributions are the same.
Lemma. We need Ω(√logn) queries to distinguish D1 and D2.
proba. to observe by
random-sampling
F0 F1 F2 Flogn
G
H
G=H
26 / 21

Lower bound
Lemma. ∀(G, H) ∈ D2, d(G, H) ≥ n/8
Proof.
Let Φ:V(G)→V(H) be a bijection achieves minimum graph edit
distance. It holds that
d(G, H) ≥ Σv∈V(G) |deg(v) – deg(Φ(v))| / 2.
If we restrict v in the sum to the root of stars, we obtain
d(G, H) ≥ Σk=2,3,4,... (n / (2k logn)) ∙ 2k-1/2 ≥ n/8. □
Thus, Ω(logn) lower bound holds.
Φ
27 / 21

Testing Forest-Isomorphismin the Adjacency List Model

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