ON THE COMMUNICATION COMPLEXITY
OF PROPERTY TESTING IN GRAPHS
Joint work with Orr Fischer and
Shay Gershtein
THE “CONGEST” NETWORK MODEL
Network graph: 𝐺 = 𝑉, 𝐸
Input to node 𝑣 ∈ 𝑉: its neighbors in 𝐺
Computation in synchronous rounds
𝑂 log 𝑛 bits per edge per round
Goal: does 𝐺 satisfy property 𝑃?
Yes: all nodes accept
No: some node rejects
EXAMPLES OF HARD PROBLEMS
Θ 𝑛 rounds:
Distinguishing diameter 4 from 5 [Holzer and Wattenhofer ‘12]
 3 2 − 𝜖 -approx of the diameter [Abboud, Censor-Hillel, Khoury ’16]
𝐶𝑘 -freeness for odd 𝑘 [Drucker, Kuhn, O. ’14]
Θ
𝑛 rounds:
MST, … [Das Sarma et. al. ’12]
3/2-approx of the diameter [Holzer and Wattenhofer ‘12], [Frishknecht
et al. ‘12]
𝐶4 -freeness [Drucker, Kuhn, O. ’14]
OPEN PROBLEM
Is there a “natural” graph property that requires 𝜔 𝑛
rounds?
WHAT ABOUT PROPERTY-TESTING?
Does 𝐺 satisfy 𝑃, or is 𝐺 𝜖-far from satisfying 𝑃?
Need to add/remove 𝜖 𝐸 edges
to get a graph satisfying 𝑃
TESTING TRIANGLE-FREENESS, 𝐸 = Θ 𝑛
Observation:
If 𝐺 is 𝜖-far from triangle-free, then
𝐺 contains Θ 𝜖𝑛2 edge-disjoint triangles
Algorithm:
Sample each edge w.p. 𝑂 1/ 𝜖𝑛2
Announce sampled edges to neighbors
Locally check for triangles
2
TESTING SUBGRAPH-FREENESS
[Censor-Hillel, Fischer, Schwartzman,Vasudev’16]: 𝑂 1 𝜖 2
for triangle-freeness (general model)
[Fraigniaud,Rapaport,Salo,Todinca’16]:
𝑂 1 𝜖 2 for all connected 4-node graphs
What about 𝐶5 ? 𝐾5 ? …
LOWER BOUNDS FROM COMMUNICATION COMPLEXITY
Alice
Bob
COMMUNICATION COMPLEXITY OF
GRAPH PROPERTY TESTING
THE MODEL
𝑘 players with private inputs 𝐸1 , … , 𝐸𝑘 ⊆ 𝑛 2
Does 𝐺 = 𝑛 , 𝑘𝑖=1 𝐸𝑖 satisfy 𝑃, or is it 𝜖-far?
Communication: shared blackboard
“General model”
Upper bounds: edges can be duplicated
Lower bounds: no edge duplication
“BUILDING BLOCKS”
Can efficiently implement:
Sampling a random subgraph
Estimating the average degree
Random walk
…
TESTING TRIANGLE-FREENESS
Upper bound: 𝑂 𝑘 𝑛𝑑
‘06])
1 4
+ 𝑘 2 (adapting [Alon et. al.
Also: simultaneous protocol with CC of 𝑂 𝑘 𝑛 when 𝑑 = 𝑂
and 𝑂 𝑘 𝑛𝑑 1 3 when 𝑑 = Ω 𝑛
𝑛
TESTING TRIANGLE-FREENESS: LOWER BOUNDS
𝑑 = Θ 1 , two players, one-way: Ω
𝑛
By reduction from Boolean Hidden Matching
“Evidence of hardness”: finding a triangle edge when 𝐺 is
𝜖-far from ∆-free
𝑑 = Θ
𝑛 , three players, one-way*: Ω 𝑛1
4
 Alice and Bob talk back-and-forth
 Charlie observes, then outputs the answer
𝑑 = Θ
𝑛 , three players, simultaneous: Ω
𝑛
THE HARD DISTRIBUTION [INSPIRED BY ALON ET. AL. ‘06]
Charlie
Tripartite graph 𝐺 = 𝑈 ∪ 𝑉1 ∪ 𝑉2 , 𝐸
The edges are iid 𝐵𝛾
𝑉1
𝑛
𝑈
E #triangles = Θ 𝑛3/2
W.h.p., 𝐺 is Θ 1 -far from triangle-free
𝑉2
Let 𝑝 be the probability that 𝑒 ∈ Δ
THE HARD DISTRIBUTION [INSPIRED BY ALON ET. AL. ‘06]
Tripartite graph 𝐺 = 𝑈 ∪ 𝑉1 ∪ 𝑉2 , 𝐸
The edges are iid 𝐵𝛾
𝑛
Θ 𝑛1
4
Θ 𝑛1
4
𝑉1
𝑈
𝑢
Upper bound:
Fix 𝑢 ∈ 𝑈
Alice and Bob send Θ 𝑛1
4
edges from 𝑢
𝑉2
THE PROTOCOL’S GOAL
Charlie should output {𝑖, 𝑗} ∈ 𝑋𝐶 such that:
𝑖, 𝑘 ∈ 𝑋𝐴
Pr ∃𝑘:
𝑀𝐴 , 𝑀𝐵 ≥ 1 − 𝛿
𝑗, 𝑘 ∈ 𝑋𝐵
𝑖, 𝑗 is “covered”
Alice and Bob should provide Charlie
with Ω
𝑛 edges that “look covered”.
𝑖
𝑘
𝑗
THE PROTOCOL’S GOAL
Charlie should output {𝑖, 𝑗} ∈ 𝑋𝐶 such that:
𝑖, 𝑘 ∈ 𝑋𝐴
Pr ∃𝑘:
𝑀𝐴 , 𝑀𝐵 ≥ 1 − 𝛿
𝑗, 𝑘 ∈ 𝑋𝐵
𝑀𝐴 , 𝑀𝐵 are “good” if:
𝑘
(Pr 𝑒 covered 𝑀𝐴 , 𝑀𝐵 − 𝑝) ≥ Ω
𝑒∈𝑉1 ×𝑉2
𝑖
𝑛
Prior probability of being
In a triangle
𝑗
BOUNDING THE SUM OF COVER-PROBABILITIES
Pr 𝑖, 𝑗 covered 𝑀𝐴 , 𝑀𝐵
≤
𝑘 Pr
𝑖, 𝑘 ∈ 𝑋𝐴 ∧ 𝑗, 𝑘 ∈ 𝑋𝐵 𝑀𝐴 , 𝑀𝐵
=
𝑘 Pr
𝑖, 𝑘 ∈ 𝑋𝐴 |𝑀𝐴 ⋅ Pr 𝑗, 𝑘 ∈ 𝑋𝐵 |𝑀𝐵
𝑒∈𝑉1 ×𝑉2 (Pr
≤
𝑒 covered 𝑀𝐴 , 𝑀𝐵 − 𝑝)
𝛾
Sum of
𝐿1 distance
between
Pr
𝑒 ∈ 𝑋𝐴 |𝑀
𝐴 − 𝑛
𝑒∈𝑈×𝑉
1
posterior and prior on Alice’s input
𝛾
Sum of
𝐿1 distance
between
Pr
𝑒 ∈ 𝑋𝐵 |𝑀
𝐵 − 𝑛
𝑒∈𝑈×𝑉
2
posterior and prior on Bob’s input
BOUNDING THE SUM OF COVER-PROBABILITIES
Sum of 𝐿1 distance between
posterior and prior on Alice’s input
≤
𝑒∈𝑈×𝑉1 𝐷𝐾𝐿 (
𝑋𝑒 |𝑀𝐴 ∥ 𝑋𝑒 )
(“Linear Pinsker”)
≤ 𝐷𝐾𝐿 ( 𝑋|𝑀𝐴 ∥ 𝑋 )
With 𝑛1
4
communication bits: typically ≤ 𝑛1
4
BOUNDING THE SUM OF COVER-PROBABILITIES
𝑒∈𝑉1 ×𝑉2 (Pr
𝑒 covered 𝑀𝐴 , 𝑀𝐵 − 𝑝)
𝛾
Sum
of
𝐿
distance
between
≤ 𝑒∈𝑈×𝑉1 Pr1 𝑒 ∈ 𝑋𝐴 |𝑀𝐴 −
posterior and prior on Alice’s 𝑛input
“Usually” O 𝑛1
≤𝑂
𝑛
4
𝛾
Sum
of
𝐿
distance
between
𝑒∈𝑈×𝑉2 Pr 𝑒1∈ 𝑋𝐵 |𝑀𝐵 − 𝑛
posterior and prior on Bob’s input
“Usually” O 𝑛1
4
SIMULTANEOUS PROTOCOLS?
Tripartite graph 𝐺 = 𝑈 ∪ 𝑉1 ∪ 𝑉2 , 𝐸
The edges are iid 𝐵𝛾
𝑛
Θ 𝑛1
4
Θ 𝑛1
4
𝑉1
𝑈
𝑢
Upper bound:
Fix 𝑢 ∈ 𝑈
Alice and Bob send Θ 𝑛1
4
edges from 𝑢
𝑉2
OPEN PROBLEMS: SUBGRAPH-FREENESS
“Real” lower bounds for testing triangle-freeness
Multi-round
Testing triangle-freeness, not “finding a triangle”
Going to larger subgraphs
Embedding back in the CONGEST model