Deterministic Δ + 1 -Coloring in Sublinear (in Δ)
Time in Static, Dynamic and Faulty Networks
Leonid Barenboim
Presented by Idan Hasson
Reminder
• A communication network is represented by an n-vertex graph: 𝐺 = 𝑉, 𝐸
of maximum degree Δ.
•
•
•
•
The vertices = processors, The communication is over the edges.
Each vertex has a unique identity number (ID) consisting of O(log n)
Computation proceeds in discrete synchronous rounds
The running time of an algorithm is the number of rounds it requires
Reminder
• A proper k-coloring 𝜑 of a graph 𝐺 = (𝑉, 𝐸) is a function 𝜑: 𝑉 → 𝑘
(where 𝑘 = {0,1, … , 𝑘 − 1} such that for each 𝑢, 𝑣 𝜖 𝐸 it holds that
𝜑 𝑢 ≠ 𝜑(𝑣).
Last Week..
• We saw an algorithm for distributed (Δ + 1)-Coloring in Linear (in Δ) Time
Our Goal Today
• Overcome the barrier of linear time, and find an algorithm of distributed
(Δ + 1)-Coloring in Sublinear (in Δ) Time
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Some Definitions
• For a vertex v in a graph 𝐺 = 𝑉, 𝐸 , the set of neighbors of v is denoted
Γ 𝑣 .
2
Γ 1 = {2, 3}
1
3
Some Definitions
• An oriented forest is an acyclic subgraph with orientation on the edges with
out degree at most 1.
• Each edge in an oriented forest is directed towards the parent.
2
1
2
3
1
3
Some Definitions
• A k-forest decomposition is a partition of the edge set E of a graph G into
at most k sets, such that each set is an oriented forest.
• Hence, the out degree of each vertex in this case is at most k. (=at most k
parents)
• The set of parents is denoted by Π(𝑣).
Some Definitions
• The arboricity 𝑎 = 𝑎(𝐺) is the minimum number of forests into which the
edge set can be partitioned.
Example
Arboricity meaning
• Arboricity = number of forests
• In a k-forest decomposition there are maximum k parents
• => In a 𝑎(𝐺) − 𝑓𝑜𝑟𝑒𝑠𝑡 𝑑𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 there are maximum 𝑎(𝐺) parents
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Procedure Partition
• Input:
• Graph G
• Positive parameters p, q: p<q
• Goal:
• Compute a vertex disjoint partition 𝐺1 , 𝐺2 , … , 𝐺𝑝
Δ
• Each 𝐺𝑖 has arboricity 𝑂(𝑝) (and some logarithmic factors)
• We use q in the algorithm. It influence the running time.
Black Boxes
Linial algorithm get a graph 𝐺 with n vertices and maximum degree Δ, and
computes a coloring 𝜑 of the graph using 𝑂 Δ2 colors in 𝑂(log ∗ 𝑛) running
time.
Linial
Black Boxes
• Simple Arbdefective algorithm:
• Input:
• A graph 𝐺 with an acyclic orientation of the edges, Integer k>0
• Maximum out-degree is d
• Output:
𝑑
• A partition of G into k subgraphs, each with arboricity ⌊𝑘 ⌋
• Running Time:
• Suppose that the longest consistently oriented path in G has length 𝜆
• The running time is 𝑂(𝜆)
Simple
Arbdefective
Back to Our Algorithm
How do we start?
• First, we use the Linial algorithm for s = 𝑐Δ2 coloring of G. we denote the
resulting coloring by 𝜏0 .
Next step - Grouping
• For a parameter 𝑞 > 𝑝 we group the 𝑂(Δ ) colors into t =
2
Δ2
𝑐Δ2
𝑂( )=
𝑞
𝑞
disjoint color classes.
• 𝑉𝑗 =
𝑣 𝑗 − 1 𝑞 ≤ 𝜏0 (𝑣) ≤ 𝑗 ∗ 𝑞} , 𝑗 ≔ 1,2 … , 𝑡 − 1
𝑣 𝑡 − 1 𝑞 ≤ 𝜏0 𝑣
, 𝑗=𝑡
• For example: q=3, number of colors =11
• Each class with at least q colors and at most 2q
3
3
5
Next step - Orientation
• Let 𝐺1 , 𝐺2 , … , 𝐺𝑡 be the subgraphs induced by vertices of the t colors.
• We orient the edges of G towards end-points of greater color.
Example
<
<
Example
<
<
Next step - Arbdefective
• We invoke Procedure Simple-Arbdefective in parallel on the subgraphs
𝐺1 , 𝐺2 , … , 𝐺𝑡 with the input parameter 𝑘 = 𝑝. (It’s acyclic graph. Why?)
Next step - Arbdefective
𝐺1 , 𝐺2 , … , 𝐺𝑡
𝐾=𝑝
𝐾=𝑝
𝐾=𝑝
𝐺1,1 , 𝐺1,2 , … , 𝐺1,𝑝
𝐺𝑡,1 , 𝐺𝑡,2 , … , 𝐺𝑡,𝑝
𝐺2,1 , 𝐺2,2 , … , 𝐺2,𝑝
Next step - Arbdefective
• As a result we obtain a vertex labeling using p labels in each subgraph.
• The union of all labels results in a p*t labeling: (i, label in 𝐺𝑖 )
Conclusions till Now
• As we saw:
• 𝑘=𝑝
• Max degree 𝑑 = Δ
• Arboricity of each subgraph (=same label) is
𝑑
𝑘
Δ
𝑝
=⌊ ⌋
• Arboricity = maximum number of parents
Δ
=> each vertex has at most ⌊ ⌋ outgoing-edge neighbors in G with the same label as that
𝑝
of v. (because the arboricity is in each subgraph)
Next
• Remove the edges with endpoints of the same label.
• This results in a subgraph 𝐺′ of G that is proper colored using 𝑡 ∗ 𝑝 colors.
• 𝑡=
𝑐Δ2
𝑞
colors by
⇒𝑡∗𝑝=
𝑝
.
𝑞
𝑐Δ2
𝑞
𝑝, and because 𝑝 < 𝑞 we reduced the number of
Next - Induction
• After removing these edges we got G’ which is s = p ∗
colored
• 𝜏1 = the new coloring
• We repeat all stages:
𝑠
𝑐Δ2
𝑞
• 𝐺1 , 𝐺2 , … , 𝐺𝑡 , 𝑡 = 𝑞, each with q colors (last with maximum 2𝑞)
• Orient the graphs
• Simple-Arbdefective, 𝑘 = 𝑝
𝑝
• New coloring with 𝑠 ∗ 𝑞, remove monochromatic edges
Break condition
• We continue while 𝑠 > 𝑝, because the Simple-arbdefective get k=p.
• Total number of rounds: 𝑟 = O(log 𝑞 Δ)
𝑝
• If we set 𝑟 = ⌈log 𝑞 (c ∗ Δ2 )⌉ we get:
𝑝
𝑝
𝑞
𝑟
𝑐Δ2 =
𝑝
𝑞
c∗Δ2
𝑝 −log𝑝
𝑞
𝑞
⌈log𝑞 (c∗Δ2 )⌉
𝑝
𝑐Δ2 ≤
𝑝
𝑞
log𝑞 c∗Δ2
𝑝
1
𝑐Δ2 = 𝑐Δ2 𝑐Δ2 = 1 ≤ 𝑝
• In the last round we have p labels – these are our 𝐺1 , 𝐺2 , … , 𝐺𝑝
𝑐Δ2 =
Total Arboricity
Δ
• As we saw, each vertex has at most ⌊𝑝⌋ outgoing-edge neighbors in G with the same
Δ
label as that of v in each round = we remove maximum ⌊ ⌋ edges of vertex each
𝑝
round.
Δ
• After i rounds we removed maximum i ∗ ⌊𝑝⌋ = maximum number of outgoing-edge
Δ
𝑝
neighbors with the same label in G is i ∗ ⌊ ⌋
Δ
• arboricity is O(log 𝑞 Δ ∗ 𝑝)
𝑝
Total complexity
• Linial – 𝑂 log ∗ 𝑛
• Number of Rounds –O(log 𝑞 Δ)
𝑝
• Each round – 𝑂(𝑞)
• Simple arbdefective is 𝑂(𝜆) and here 𝜆 < 2𝑞 because each subgraph from 𝐺1 , … , 𝐺𝑡
has maximum 2𝑞 colors.
• Total: 𝑂(𝑞 ∗ log 𝑞 Δ + log ∗ 𝑛)
𝑝
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Black Box
Arb-Linial algorithm get a graph 𝐺 with arboricity 𝑎(𝐺), and computes a
coloring 𝜑 of the graph using 𝑂 𝑎2 colors in 𝑂(log ∗ 𝑛) running time.
ArbLinial
Procedure color
• The procedure gets 𝐺, 𝑝, 𝑞 and the vertex v which runs the algorithm
Δ
• First we create a partition 𝐺1 , 𝐺2 , … , 𝐺𝑝 , each with 𝑎 = 𝑂(𝑝)
• How?
Procedure Partition
Procedure color
• Then, each 𝐺𝑖 is properly colored by Arb-Linial algorithm in 𝑐𝑎2 colors.
(The coloring function denoted by 𝜑𝑖 for each 𝐺𝑖 )
Intuition
𝐺1
𝐺2
𝐺
Procedure color
• Goal: Recolor 𝐺𝑖 in such a way that colors of vertices in 𝐺𝑖 do not conflict
with final colors of their neighbors in 𝐺.
How will it work
• We have:
𝜑1
𝜑2
𝜑3
….
𝜑𝑝
How will it work
• We will build a coloring function 𝜓 for entire G.
• For each vertex before the algorithm: 𝜓 𝑣 = 𝑛𝑢𝑙𝑙
• First:
𝜑1
𝜑2
𝜑3
….
𝜑𝑝
𝜓
How will it work
Now we have to find how to recolor 𝐺2 without conflicts with the new
coloring 𝜓 of 𝐺1 .
𝜑1
𝜑2
𝜑3
….
𝜑𝑝
𝜓
How will it work
And so on for p iterations – we will find how to recolor 𝐺𝑖 without conflicts
with the new coloring 𝜓 of 𝐺1 ⋃𝐺2 … ⋃𝐺𝑖−1 .
𝜑1
𝜑2
𝜑3
….
𝜑𝑝
𝜓
What we still don’t know?
• How to recolor?
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Procedure Recolor
• Denote:
• 𝐿 𝑣 = 𝜓 𝑢 𝑢𝜖Γ 𝑣
𝜓 𝑢 ≠ 𝑛𝑢𝑙𝑙} – The list of forbidden colors
• The upper bound of 𝐿 𝑣 is 𝑙 = Δ.
• 𝐿(1) = {𝑔𝑟𝑒𝑒𝑛}
1
𝜓
𝜑
Procedure Recolor
• Denote:
• Γ 𝑣 = 𝑢 𝑢𝜖𝑉 𝐺𝑖
𝑢 ∈ Π(𝑣)} – The parents which are in 𝐺𝑖
• 𝐵 𝑣 = {𝜑(𝑢)|𝑢 ∈ Γ 𝑣 }
• The upper bound of Γ 𝑣 is 𝛽 = 𝑎.
• Γ 1 ={2}
• 𝐵(1) = {𝑏𝑙𝑢𝑒}
1
𝜓
3
𝜑
2
Procedure Recolor
• The procedure accepts as input for each vertex:
• The coloring 𝜑
• 𝐿 𝑣 , 𝑙=Δ
• Γ 𝑣 , 𝐵 𝑣 , 𝛽=𝑎
• The output in iteration i will be 𝜓(𝑣) for each 𝑣 ∈ 𝐺𝑖 .
• Each one of these iterations performed in a separate round.
Steps of algorithm
• For a universal constant c>4 we find 𝜇:
𝑙 +
𝑐 𝛽 ≤ 𝜇 ≤ 2(
• 𝜇 is a prime number. (exists, believe me  )
𝑙 +
𝑐 𝛽)
Steps of algorithm
• Each vertex calculates:
• 𝑎=
𝜑 𝑣
⌈ 𝑐𝛽2 ⌉
, 𝑏 = 𝜑 𝑣 𝑚𝑜𝑑 ⌈
𝑐𝛽 2 ⌉
9 = 1*5 +4
(1,4)
• (𝑎, 𝑏) is a representation of 𝜑(𝑣)
• Each vertex constructs a set of 𝜇 polynomials: 𝑝0𝑣 (𝑥), 𝑝1𝑣 𝑥 , . . , 𝑝𝜇−1𝑣 (𝑥)
• 𝑝𝑖𝑣 𝑥 = 𝑖 + 𝑎 ∗ 𝑥 + 𝑏 ∗ 𝑥 2 (over 𝑍𝜇 )
• Foreach 𝑢, 𝑣 ∈ 𝑉 such that 𝜑 𝑢 ≠ 𝜑 𝑣 the set of polynomials of u and v are
disjoint. (a or b will be different)
Example
• 𝜑 𝑣 = 10, c=4.1, β=4, l=0
•
𝑐𝛽 2 =
•
𝑙 +
• 𝑎=
4.1 ∗ 42 = 8.09 = 9
𝑐 𝛽=
𝜑 𝑣
⌈ 𝑐𝛽2 ⌉
=
• 𝑏 = 𝜑 𝑣 𝑚𝑜𝑑
• 𝜇 = 13
v
4.1 ∗ 4 = 12
10
9
=1
𝑐𝛽 2 = 10 𝑚𝑜𝑑 9 = 1
=10
Example
• 𝑎 = 1, 𝑏 = 1, 𝜇 = 13
• 𝑝𝑖𝑣 𝑥 = 𝑖 + 𝑎 ∗ 𝑥 + 𝑏 ∗ 𝑥 2
• 𝑝0𝑣 𝑥 = 1 ∗ 𝑥 + 1 ∗ 𝑥 2
• 𝑝1𝑣 𝑥 = 1 + 1 ∗ 𝑥 + 1 ∗ 𝑥 2
…..
• 𝑝12𝑣 𝑥 = 12 + 1 ∗ 𝑥 + 1 ∗ 𝑥 2
v
Next Step
• Calculate 𝐿 𝑣 ∩ 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 𝑘 = 0,1, … , 𝜇 − 1} for each 𝑖 ∈ 𝜇
• Choose the index i such that this set has the minimum size. We denote this
set by: 𝐿𝑖 𝑣 .
• In 𝐿(𝑣) there are maximum 𝑙 items
• We intersect 𝐿(𝑣) with 𝜇 disjoint sets (disjoint because 𝑝𝑖𝑣 𝑥 is over 𝑍𝜇 )
• By pigeonhole principle: the size of 𝐿𝑖 𝑣 is at most
𝑙
.
𝜇
Recolor idea
• Denote: 𝐿𝑖 𝑣 = 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 𝑘 = 0,1, … , 𝜇 − 1}\L(𝑣)
• There exists 𝑘 ∈ 0,1, … , 𝜇 − 1 such that: 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 ∈ 𝐿𝑖 𝑣 and
𝑝𝑖𝑣 𝑘 ≠ 𝑝𝑗𝑢 𝑘 for all 𝑢 ∈ Γ(𝑣) (proof immediately(
• 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 ≠ 𝑝𝑗𝑢 𝑘 ′ + 𝑘′𝜇 (𝑝𝑖𝑣 𝑥 is over 𝑍𝜇 )
• 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 not in 𝐿(𝑣), and not in neighbor’s 𝐿𝑗 𝑣
• We can use it!
• Take the smallest k as above: 𝜓 𝑣 = 𝑝𝑖𝑣 𝑘 + 𝑘𝜇
Proof
There exists 𝒌 ∈ 𝟎, 𝟏, … , 𝝁 − 𝟏 such that: 𝒑𝒊𝒗 𝒌 + 𝒌𝝁 ∈ 𝑳𝒊 𝒗 and
𝒑𝒊𝒗 𝒌 ≠ 𝒑𝒋𝒖 𝒌 for all 𝒖 ∈ 𝜞(𝒗)
• Basic idea:
There exists 𝒌 ∈ 𝟎, 𝟏, … , 𝝁 − 𝟏 such that: 𝒑𝒊𝒗 𝒌 + 𝒌𝝁 ∈ 𝑳𝒊 𝒗 and
𝒑𝒊𝒗 𝒌 ≠ 𝒑𝒋𝒖 𝒌 for all 𝒖 ∈ 𝜞(𝒗)
• 𝐿𝑖 𝑣 = 𝑝𝑖𝑣 𝑘 + 𝑘𝜇 𝑘 = 0,1, … , 𝜇 − 1}\L(𝑣)
𝑙
• the size of 𝐿𝑖 𝑣 is at most 𝜇
•
𝐿𝑖 𝑣
𝑙
𝜇
≥ 𝜇 − ≥ 𝑙 + 𝑐𝛽 −
•
𝜇≥
•
𝐶>4
𝑙 +
𝑙
𝑙
= 𝑐𝛽 > 2𝛽
𝑐 𝛽
• There are Γ 𝑣 ≤ 𝛽 parents
• 𝑝𝑖𝑣 𝑥 intersects with 𝑝𝑗𝑢 𝑥 in at most 2 points
• There is at least one number in 𝐿𝑖 𝑣 which is not one of these points
𝑝𝑖𝑣 𝑘 + 𝑘𝜇
𝜇
𝐿𝑖 𝑣
𝑙
𝜇
𝐿(𝑣)
𝑙
Colors
• (Without proof) 𝜓 has at most:
𝑙+𝑂
colors.
𝑙 ∗ 𝛽 + 𝛽2
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Running time
• Procedure Color runtime:
• Partition: 𝑂(𝑞 ∗ log 𝑞 Δ + log ∗ 𝑛)
𝑝
• Parallel execution of Arb-Linial: 𝑂(log ∗ 𝑛)
• p iterations of recolor, each with constant time: 𝑂(𝑝)
• Total: 𝑂 𝑞 ∗ log 𝑞 Δ + log ∗ 𝑛 + 𝑝 = 𝑂(𝑞 ∗ log 𝑞 Δ + log ∗ 𝑛)
𝑝
𝑝
Colors
• 𝜓 has at most:
𝑙+𝑂
𝑙 ∗ 𝛽 + 𝛽2
colors.
• 𝑙=Δ
Δ
• 𝛽 = 𝑎 = O(log 𝑞 Δ ∗ 𝑝)
𝑝
• Total:Δ + 𝑂
Δ
𝑝
Δ
𝑝
Δ ∗ log 𝑞 Δ ∗ + (log 𝑞 Δ ∗ )2
𝑝
𝑝
What if…
3
4
• 𝑝 = Δ , 𝑞 = 2𝑝
• Complexity: 𝑂(𝑞 ∗ log 𝑞 Δ +
log ∗ 𝑛)
3
4
= 𝑂(Δ ∗ 𝑙𝑜𝑔 Δ + log ∗ 𝑛)
𝑝
• Colors: Δ + 1 coloring
• 𝑝=
Δ log 2 Δ , 𝑞 = 2𝑝
• Complexity: 𝑂(𝑞 ∗ log 𝑞 Δ + log ∗ 𝑛) = 𝑂( Δ log 3 Δ + log ∗ 𝑛)
𝑝
• Colors: (1 + o 1 )Δ coloring
Edge Coloring
• An edge coloring of a graph G is a coloring of the edges of G such that
adjacent edges receive different colors.
From 𝑓(Δ) vertex coloring to 𝑓(2Δ − 2) edge
coloring
• Convert each edge to node.
• 2 nodes in the new graph will be connected, if the 2 edges they represent
share a vertex in the original graph.
e1
v2
e2
v2
v4
v1
e3
v3
e4
e2
v4
e4
e1
v1
e3
v3
From 𝑓(Δ) vertex coloring to 𝑓(2Δ − 2) edge
coloring
The Maximum Degree of the New Graph:
e1
Δ−1
Δ−1
e1
2Δ − 2
Conclusion
• From the result of Δ + 1 vertex coloring algorithm, we conclude
2Δ − 2 + 1 = 2Δ − 1
edge coloring algorithm in sublinear time.
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Different Settings of Networks
• Static Graphs – the input graph topology never change.
Begin:
End:
Different Settings of Networks
• Dynamic Graphs – the graph topology changes:
• A subset of vertices 𝑊 ⊆ 𝑉 is removed, and a new set of vertices U is added.
• A subset of edges of E is removed, and a new set of edges is added.
• The number of vertices in the network is upper-bounded by n
Begin:
End:
Different Settings of Networks
• Faulty graphs – the input graph is static, but a subset 𝑈 ⊆ 𝑉 (respectively,
𝐸 ′ ⊆ 𝐸) may lose the computed solution as a result of faults.
Begin:
Middle:
End:
Different Settings of Networks
• The goal of a dynamic distributed algorithm is to compute a solution for the
new graph 𝐺’ given a solution for 𝐺.
• The goal of a distributed algorithm for a faulty network is to compute a new
solution for U such that it is consistent with the solution of V\U.
(respectively for the edges)
Sounds Familiar?(Intuition)
• We saw an algorithm for recoloring the graph, based on partial colored
graph.
• We can use it here!
What are we going to talk about?
• Some definitions in graph theory
• The sublinear algorithm:
Procedure
Partition
Procedure Color
• Analysis
• (Bonus – Optional) Generalization – Intuition only
Procedure
Recolor
Questions?