An Exponential Separation
Between Randomized
and Deterministic Complexity
in the LOCAL Model
Yi-Jum Chang
Tsvi Kopelowitz
Seth Pettie
University of Michigan University of Michigan University of Michigan
1
Preliminaries
The LOCAL Model
DetLOCAL
RandLOCAL
deg(1)=2
deg=2
1
deg(2)=3
deg(5)=2
5
deg=2
deg=3
2
deg(4)=2
4
deg(3)=3
3
deg=2
deg=3
Graph Notation:
“
◦ The distance between v and u in G dist G (v, u).
◦ The neighborhood of v N v = {u| v, u ∈ E}
◦ The r neighborhood of v –
N r v = {u|dist G v, u ≤ r}
Locally Checkable Labeling (LCL):
“
graph problems whose solution can be
verified in O(1) rounds, given a suitable
labeling of the graph.
𝜟-Sinkless Coloring:
“
𝚫 – Sinkless Orientation
“
Linial’s coloring
In the DetLOCAL model the initial
Θ(log n) – bit IDs can be viewed as an
nO 1 coloring of the graph.
“
1
4
3
2
“
Theorem 1
Let G be a graph which has been kcolored. Then it is possible to
determinstically re-color G using 5Δ2 Log k
colors in one round.
(N. Linial. Locality in distributed graph algorithms. SIAM J. Comput., 1992)
Theorem 2
“
There exists a universal constant β > 0 such that
there is a DetLOCAL algorithm that computes a 𝛽 ⋅ Δ2 −coloring of a graph in
O(log ∗ n − log ∗ Δ + 1) time.
(N. Linial. Locality in distributed graph algorithms. SIAM J. Comput., 1992)
We will see that
Randomized algorithm
Deterministic algorithm
𝛀(𝐥𝐨𝐠 𝐓) 𝛀(𝐓)
Each of the results has a very compelling take-away massage
Fast Δ −coloring of trees requires
ramdom bits.
Randomized lower bounds imply
higher deterministic lower bounds.
Deterministic lower bounds imply
randomized lower bounds.
2
Lower bounds for Δ-coloring Δ-regular
Trees
Prove that on degree-Δ graphs with girth Ω(log Δ 𝑛),
Δ −coloring takes Ω(log Δ log n) time in RandLOCAL and
Ω(log Δ 𝑛) in DetLOCAL.
“
Theorem 4
Any RandLOCAL algorithm for Δcoloring a graph with degree at most Δ
and error probabilty p takes at least
1
t = min ϵ log 3 Δ+1 ln
, ϵ log Δ n ≥ 1 rounds
p
for a sufficient small ϵ > 0.
Proof
◦Let Δ ≥ 3.
◦There exist a bipartite Δ- regular graphs with
girth Ω(log Δ n).
Δ
Δ
Δ
Δ
Proof
◦This graph is trivially Δ-edge colorable.
Δ
Δ
Δ
Δ
Proof
◦Any Δ- coloring of such a graph is also a valid
Δ-sinkless coloring.
Δ
Δ
Δ
Δ
Proof
◦Any t-round Δ-sinkless coloring algorithm with
error probability 𝑝 can be transformed into a
(t-1)-round algorithm with error probability
𝟒 𝟐𝚫
𝟏
𝟏
𝚫+𝟏 𝐩𝟑 𝚫+𝟏
<
𝟏
𝟕𝐩𝟑 𝚫+𝟏
Proof
◦Iterating this process t times, it follows that there
exists a 0-round Δ-sinkless coloring algorithm with
error probability 𝑂 𝐩
𝐩
𝟏
𝟑 𝚫+𝟏
𝐭
≤𝐩
𝟏
𝟑 𝚫+𝟏
𝐭
, Note that:
𝟏
𝟑 𝚫+𝟏
1
ϵ log3 Δ+1 ln
p
Proof
◦We obtain that the error probability is 𝑂 𝐩
Note that:
𝐩
𝟏
𝟑 𝚫+𝟏
=𝐩
1
ln
𝑝
𝐭
≤𝐩
𝟏
𝟑 𝚫+𝟏
−𝜖
=𝑒
𝟏
𝟑 𝚫+𝟏
1
ϵ log3 Δ+1 ln
p
1 1−𝜖
−ln
𝑝
𝐭
,
Proof
The graph is 𝚫 −regular and
the vertices undifferentiated
by IDs.
Any 0-round RandLOCAL
algorithm colors vertex
independently according to the
same distribution.
The probability that any vertex
involved in a forbidden
𝟏
configuration is at least 𝟐 .
𝚫
Proof
◦We obtain that the error probability is Ω
◦We have ϵ log 3
Δ+1
ln
1
p
≥1⇒Δ<
1
Δ2
1
ln .
𝑝
◦We get:
1
≥
2
Δ
1
−2ln ln
p
𝑒
≫𝑒
1 1−𝜖
− ln
𝑝
.
Proof
◦In conclusion:
◦We obtain that the error probability is
𝟏
𝟑 𝚫+𝟏
𝐭
−
1 1−𝜖
ln𝑝
𝑂 𝐩
≤𝑂 𝑒
◦We obtain that the error probability
is Ω 𝑒
−
1 1−𝜖
ln
𝑝
≪Ω
1
Δ2
.
CONTRADICTION:
The lower bound is bigger than the upper bound
“
Corollary 2
Any RandLOCAL algorithm for Δcoloring a graph with global error
1
probabilty
takes Ω log Δ log n
time.
poly n
“
Theorem 5
Any DetLOCAL algorithm that Δ-colors
degree-Δ graphs with girth Ω log Δ n or
degree-Δ trees requires Ω(log Δ n) time.
Proof
◦Let 𝒜𝐷𝑒𝑡 be a DetLOCAL algorithm that
Δ −colors a graph in 𝑡 = 𝑡(𝑛, Δ) rounds.
◦ Let G be the input graph.
The idea:
We will construct a RandLOCAL algorithm -𝓐𝑹𝒂𝒏𝒅
that taking 𝑶(𝒕) rounds and simulate 𝓐𝑫𝒆𝒕 . And we
will use the lower bound from Theorem 4.
Proof
We construct 𝒜Rand as follow:
G:
Proof
We construct 𝒜Rand as follow:
Before the first round each vertex locally
generates a random 𝑛-bit ID.
1
78
8
18
G:
52
3
99
23
12
9
Proof
We construct 𝒜Rand as follow:
Assume for the time being that these IDs
are unique, and therefore constitute a
2𝑛 -coloring of G.
1
78
8
18
G:
52
3
99
23
12
9
Proof
We construct 𝒜Rand as follow:
Assume for the time being that these IDs
are unique, and therefore constitute a
2𝑛 -coloring of G.
1
78
8
18
G:
52
3
99
23
12
9
Proof
We construct 𝒜Rand as follow:
Let 𝐺 ′ = (𝑉, {𝑢, 𝑣}|𝑑𝑖𝑠𝑡𝐺 𝑢, 𝑣 ≤ 2𝑡 + 1}
G:
G’:
Proof
We construct 𝒜Rand as follow:
We apply one step of Linial’s recoloring
algorithm.
G’:
Proof
We construct 𝒜Rand as follow:
We apply one step of Linial’s recoloring
algorithm.
Linial’s WHAT?!
G’:
Let 𝐺 be a graph which has been 𝑘colored. Then it is possible to
determinstically re-color 𝐺 using 5Δ2 𝐿𝑜𝑔 𝑘
colors in one round.
Proof
We construct 𝒜Rand as follow:
We obtain a coloring with palette size
𝑂 Δ′2 log 2𝑛 = 𝑂(𝑛3 ).
G’:
Proof
We construct 𝒜Rand as follow:
This step of Linial’s algorithm in G’ is
simulated in G using 𝑂(𝑡) time.
G’:
Proof
We construct 𝒜Rand as follow:
This step of Linial’s algorithm in G’ is
simulated in G using 𝑂(𝑡) time.
G:
Proof
We construct 𝒜Rand as follow:
Now we will use these colors as
3 log 𝑛 + 𝑂 1 −bit IDs.
1
G:
3
4
2
5
5
6
3
2
1
Proof
We construct 𝒜Rand as follow:
We simulate 𝒜𝐷𝑒𝑡 on G for 𝑡 steps.
1
G:
3
4
2
5
5
6
3
2
1
Proof
We construct 𝒜Rand as follow:
We simulate 𝒜𝐷𝑒𝑡 on G for 𝑡 steps.
1
G:
3
4
2
5
5
6
3
2
1
Proof
We construct 𝒜Rand as follow:
We simulate 𝒜𝐷𝑒𝑡 on G for 𝑡 steps.
1
G:
3
4
2
5
5
6
3
2
1
Proof
We construct 𝒜Rand as follow:
We simulate 𝒜𝐷𝑒𝑡 on G for 𝑡 steps.
And we have 𝚫 −coloring in G.
1
G:
3
4
2
5
5
6
3
2
1
Proof
𝒜𝐷𝑒𝑡 is deterministic, so the only way 𝒜𝑅𝑎𝑛𝑑
can fail is if the initial 𝑛-bit IDs fail to be unique.
This occurs with probability
𝒑<
𝒏𝟐
𝟐𝒏
𝓐𝑹𝒂𝒏𝒅 is RandLOCAL algorithm⇒By
Theorem 4 𝓐𝑹𝒂𝒏𝒅 takes
𝟏
𝛀(𝐦𝐢𝐧 𝐥𝐨𝐠 𝚫 𝐥𝐨𝐠
, 𝐥𝐨𝐠 𝚫 𝐧 = 𝛀(𝐥𝐨𝐠 𝚫 𝐧)
𝐩
𝓐𝑫𝒆𝒕 takes t=𝛀(𝐥𝐨𝐠 𝚫 𝐧) steps.
3
Gaps in Deterministic Time Complexity
Definition: Hereditary graph class
A closed under removing vertices and
edges graph class.
Examples:
forests, triangle-free graphs etc…
Recalls that:
Theorem 2
There exists a universal constant β > 0 such that
there is a DetLOCAL algorithm that computes a 𝛽 ⋅ Δ2 −coloring of a graph in
O(log ∗ n − log ∗ Δ + 1) time.
(N. Linial. Locality in distributed graph algorithms. SIAM J. Comput., 1992)
Locally Checkable Labeling (LCL):
graph problems whose solution can be
verified in O(1) rounds, given a suitable
labeling of the graph.
Theorem 6
“
Let 𝒫 be an LCL graph problam with parameters r, Σ
and C, and let 𝒜 be a DetLOCAL algorithm for solving
𝒫. Let β be the universal constant from Theorem 2.
Suppose that the cost of 𝒜 on instances of 𝒫 with n
vertices, where the instance is taken from a hereditary
graph class, is at most f Δ + ϵ log Δ n time, where
1
f Δ ≥ 0 and ϵ =
is a constant. Then there
4+4logβ+4r
exists a DetLOCAL algorithm 𝒜 that solves 𝒫 on the
same instances in O((1 + f Δ ) log ∗ n − log ∗ Δ + 1 ) time.
Proof
◦Notice that for any instance of 𝒫 with n vertices
and ID length ℒ, it must be that ℒ ≥ log 𝑛 and so
the running time of 𝒜 om such instances is
𝜖ℒ
bounded by 𝑇 Δ, ℒ ≤ 𝑓 Δ +
.
log Δ
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
1. Let 𝜏 = 1 + log 𝛽 .
1
78
8
18
G:
52
3
99
23
12
9
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
2. We use Linial’s coloring technique to produce short
IDs of length ℒ′ that are distinct wihtin distance
4𝑓 Δ + 2𝜏 + 2𝑟.
1
78
8
18
G:
52
3
99
23
12
9
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
The IDs are unique and therefore constitute a
2log 𝑛 -coloring.
1
78
8
18
G:
52
3
99
23
12
9
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
Let 𝐺 ′ = (𝑉, {𝑢, 𝑣}|𝑑𝑖𝑠𝑡𝐺 𝑢, 𝑣 ≤ 4𝑓(Δ) + 2𝜏 + 2𝑟}
1
78
8
18
G:
52
3
99
23
12
9
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
Let 𝐺 ′ = (𝑉, {𝑢, 𝑣}|𝑑𝑖𝑠𝑡𝐺 𝑢, 𝑣 ≤ 4𝑓(Δ) + 2𝜏 + 2𝑟}
1
78
8
18
G’:
52
3
99
23
12
9
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
We apply one step of Linial’s recoloring
algorithm.
G’:
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
We apply one step of Linial’s recoloring
algorithm.
G’:
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
Notice that Δ′ ≤ Δ4𝑓
G’:
Δ +2𝜏+2𝑟
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
We simulate the algorithm of Theorem 2 on G’
and produces a 𝛽 ⋅ Δ8𝑓 Δ +4𝜏+4𝑟 -coloring.
G’:
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
This step of Linial’s algorithm in G’ is simulated
in G using 𝑂(𝑓 Δ + 𝜏 + 𝑟) time.
G’:
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
Now we will use these colors as
ℒ ′ = ( 8𝑓 Δ + 4𝜏 + 4𝑟 𝑙𝑜𝑔Δ + 𝑙𝑜𝑔𝛽) −bit IDs.
1
G:
G’:
3
4
2
5
5
6
3
2
1
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
Now we will use these colors as
ℒ ′ = ( 8𝑓 Δ + 4𝜏 + 4𝑟 𝑙𝑜𝑔Δ + 𝑙𝑜𝑔𝛽) −bit IDs.
1
G:
G’:
3
4
2
5
5
6
3
2
1
Proof
◦Let 𝐺 = 𝑉, 𝐸 be an instance of 𝒫. The
algorithm 𝒜′ in 𝐺 works as follows:
3. We apply 𝒜 on G while implicitly assuming that
′
the graph size is 2 ℒ and using the shorter IDs.
1
G:
3
4
2
5
5
6
3
2
1
Proof
The run time of this execution of 𝒜 is:
𝛜𝓛′
𝜖( 8𝑓 Δ + 4𝜏 + 4𝑟 𝑙𝑜𝑔Δ + 𝑙𝑜𝑔𝛽)
𝐟 𝚫 +
=𝑓 Δ +
log Δ
𝐥𝐨𝐠 𝚫
= 1 + 8𝜖 𝑓 Δ + 1 +
𝜖log 𝛽
log Δ
≤
Proof
The run time of this execution of 𝒜 is:
𝛜𝓛′
𝜖( 8𝑓 Δ + 4𝜏 + 4𝑟 𝑙𝑜𝑔Δ + 𝑙𝑜𝑔𝛽)
𝐟 𝚫 +
=𝑓 Δ +
log Δ
𝐥𝐨𝐠 𝚫
= 1 + 8𝜖 𝑓 Δ + 1 +
𝜖log 𝛽
log Δ
≤
1 + 8𝜖 𝑓 Δ + 𝜏
Proof
The run time of this execution of 𝒜 is:
𝛜𝓛′
𝜖( 8𝑓 Δ + 4𝜏 + 4𝑟 𝑙𝑜𝑔Δ + 𝑙𝑜𝑔𝛽)
𝐟 𝚫 +
=𝑓 Δ +
log Δ
𝐥𝐨𝐠 𝚫
= 1 + 8𝜖 𝑓 Δ + 1 +
𝜖log 𝛽
log Δ
≤
≤ 𝟐𝐟 𝚫 + 𝝉
1 + 8𝜖 𝑓 Δ + 𝜏
Proof
Time complexity of step 2 is:
4𝑓 Δ + 2𝜏 + 2𝑟 ⋅ 𝑂(log ∗ 𝑛 − log ∗ Δ + 1)
Proof
Therefore the total time complexity is
4𝑓 Δ + 2𝜏 + 2𝑟 ⋅ 𝑂 log ∗ 𝑛 − log ∗ Δ + 1 + 2𝑓 Δ + 𝜏
=O((1 + f(Δ)) ⋅ log ∗ 𝑛 − log ∗ Δ + 1 )
Proof
Why labeling of 𝑢 ∈ 𝑉 is legal?
Due to the hereditary property of the
graph.
Consequence
Deterministic time complexity of a problem
can either be solved very efficiently (i.e. in
O((1 + f Δ ) log ∗ n − log ∗ Δ + 1 ) of requires
Ω(𝑓 Δ + log Δ 𝑛).
“
Corollary 3
The time complexity of any LCL
problem on any hereditary graph class
that has constant Δ in the DelLOCAL model
is either Ω(log n) or O(log ∗ n).
Thanks!
ANY QUESTIONS?