On the Complexity of
Distributed Network
Decomposition
Alessandro Panconesi
and
Aravind Srinivasan
JOURNAL OF ALGORITHMS 20, 356-374 (1996)
What we saw last week
• Awerbuch, Goldberg, Luby, Plotkin (1989): A network decomposition
can be computed in time:
𝑛𝑂(𝜖 𝑛 )
where 𝜖 𝑛 =
𝑙𝑜𝑔𝑙𝑜𝑔𝑛
𝑙𝑜𝑔𝑛
.
Today
• A network decomposition can be computed in time:
nO 𝜖(𝑛)
where 𝜖 𝑛 =
𝑙𝑜𝑔𝑙𝑜𝑔𝑛
𝑙𝑜𝑔𝑛
=
1
𝑙𝑜𝑔𝑛
• Near-optimal decomposition.
• Completeness result.
.
Setting
• A message-passing distributed network is an undirected graph 𝐺 =
(𝑉, 𝐸) where vertices correspond to processors and edges to
bidirectional communication links.
• The network is synchronous (=rounds).
• We do not charge for local computations.
• Messages are unbounded in size.
Definitions - reminder
• 𝜒: 𝑉 → [𝑘] is a called a 𝑘-coloring of 𝐺 if for every 𝑢, 𝑣 ∈ 𝐸 it holds
that 𝜒 𝑢 ≠ 𝜒(𝑣).
• 𝐼 ⊆ 𝑉 is called an independent set if for every 𝑢, 𝑣 ∈ 𝐼, 𝑢, 𝑣 ≠ 𝐸.
• 𝐼 ⊆ 𝑉 Is called a maximal independent set if it is not a strict subset of
any other independent set.
Definitions - Network Decomposition
• We will assume from now on that 𝑉 = 𝑛.
• Given a graph 𝐺 and a partition 𝑉 =∪𝑖 𝐶𝑖 , the cluster graph 𝐺 is
defined:
𝑉 𝐺 = {𝐶𝑖 }
𝐸(𝐺) = {(𝐶𝑖 , 𝐶𝑗 )|𝑖 ≠ 𝑗, ∃𝑢 ∈ 𝐶𝑖 , ∃𝑣 ∈ 𝐶𝑗 𝑠. 𝑡. 𝑢, 𝑣 ∈ 𝐸)
• The cluster graph is said to be a (𝑑 𝑛 , 𝑐 𝑛 )-decomposition of 𝐺 if:
• Every 𝐺 𝐶𝑖 (the graph induced by 𝐶𝑖 ) is connected and is of 𝑂(𝑑 𝑛 )
diameter;
• The cluster graph 𝐺 is vertex colored with 𝑂(𝑐 𝑛 ) colors.
• Note that a 𝑐-coloring of 𝐺 is a 0, 𝑐 -decomposition.
Why network decomposition?
• Problems like MIS, (Δ + 1)-coloring and others can be solved in
𝑂(𝑑 𝑛 ⋅ 𝑐 𝑛 ) time, given a (𝑑 𝑛 , 𝑐 𝑛 )-decomposition of 𝐺.
• The generic algorithm for such problems, given a cluster
decomposition, will iterate through the color classes, clusters of color
1 being processed first in parallel, clusters of color 2 being processed
next, and so on.
• In each cluster, 𝐶𝑖 , the trivial 𝑂(𝑑 𝑛 ) algorithm (“global
information”) can be used.
Example – Computing MIS
• Suppose we are given an (𝑑, 𝑐)-decomposition, 𝐺 = 𝑉, 𝐸 , of 𝐺.
• For simplicity, assume that 𝑉 = {𝐶𝑖 } are colored differently. In
particular 𝐶𝑖 is of color 𝑖 where 𝑖 = 1, … 𝑐.
• For 𝑖 = 1, compute 𝑀𝐼𝑆 on 𝐶1 and denote this set by 𝐼1 .
• For 𝑖 = 2, … , 𝑐 compute 𝑀𝐼𝑆 on 𝐶𝑖 excluding all vertices from ∪𝑖−1
𝑛=1 𝐼𝑛
and their neighbors.
• The set 𝐼 = ∪ 𝐼𝑖 is a 𝑀𝐼𝑆 of the original graph, 𝐺.
Example – Computing MIS
Example – Computing MIS
compute 𝑀𝐼𝑆 on 𝐶1 and denote this set by 𝐼1
Example – Computing MIS
Example – Computing MIS
compute 𝑀𝐼𝑆 on 𝐶2 , excluding 𝐼1 and their neighbors, and denote this set by 𝐼2
Example – Computing MIS
compute 𝑀𝐼𝑆 on 𝐶3 , excluding 𝐼1 ∪ 𝐼2 and their neighbors, and denote this set by 𝐼3
Example – Computing MIS
compute 𝑀𝐼𝑆 on 𝐶4 , excluding 𝐼1 ∪ 𝐼2 ∪ 𝐼3 and their neighbors, and denote this set by 𝐼4
Example – Computing MIS
The 𝐼 =∪𝑖 𝐼𝑖 is a 𝑀𝐼𝑆 of 𝐺
More definitions
• Given a graph 𝐺 = 𝑉, 𝐸 and a set 𝑆 ⊆ 𝑉, denote by 𝐺[𝑘, 𝑆] the
following graph:
𝑉(𝐺 𝑘, 𝑆 ) = 𝑆
𝐸(𝐺 𝑘, 𝑆 ) = 𝑢, 𝑣 𝑢, 𝑣 ∈ 𝑆, 𝑑𝐺 𝑢, 𝑣 ≤ 𝑘
• Example:
𝐺 = (𝑉, 𝐸)
𝐺[2, 𝑉]
More definitions (2)
• An 𝛼, 𝛽 -ruling set, S, with respect to 𝐺 = 𝑉, 𝐸 and 𝑃 ⊆ 𝑉 is a set
of vertices such that
• 𝑆 ⊆ 𝑃;
• For any two 𝑢 ≠ 𝑣 ∈ 𝑆, 𝑑𝐺 𝑢, 𝑣 ≥ 𝛼;
• For any 𝑢 ∈ 𝑃 there is 𝑣 ∈ 𝑆 such that 𝑑𝐺 𝑢, 𝑣 ≤ 𝛽.
Example 1: any MIS is a (2,1)-ruling set
Example 2: any MIS on G[2,V] is a (3,2)-ruling set
More definitions (3)
• An (𝛼, 𝛽)-ruling forest with respect to 𝐺 = (𝑉, 𝐸) and 𝑃 ⊆ 𝑉 is a
forest of rooted trees 𝐹 = {𝑇𝑖 }, where each tree is a subgraph of 𝐺,
with the following properties:
•
•
•
•
•
For all 𝑖, the root of 𝑇𝑖 , denoted by ri , is in 𝑃;
For every 𝑢 ∈ 𝑃 there is 𝑖 such that 𝑢 ∈ 𝑇𝑖 ;
𝑇𝑖 ∩ 𝑇𝑗 = ∅ for 𝑖 ≠ 𝑗;
For every 𝑖 ≠ 𝑗, 𝑑𝐺 𝑟𝑖 , 𝑟𝑗 ≥ 𝛼;
Tree depth is at most 𝛽.
Ruling forest – black boxes
• Recall from the previous talk that we know how to compute an
(𝛼, 𝛽)-ruling forest given an 𝛼, 𝛽 -ruling set in 𝑂 𝛽 rounds.
• In addition, an 𝑘, 𝑘𝑙𝑜𝑔𝑛 -ruling forest can be computed in 𝑂(𝑙𝑜𝑔𝑛).
Question
• Assume we’re given:
•
•
•
•
A graph 𝐺 such that 𝑉 = 𝐴 ∪ 𝐵;
A 𝑑, 𝑐 -decomposition of 𝐺[𝐴];
We know: maximum degree of 𝐺[𝐵] is at most c−1;
𝑐-coloring of 𝐺[B].
• Is it possible to compute a 𝑑, 𝑐 -decomposition of the whole graph
𝐺?
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
• Input: (𝑑, 𝑐)-decomposition of G[𝐴], 𝑐-coloring of G[𝐵]
• Output: (𝑑, 𝑐)-decomposition of G
1. 𝑉 𝐺 ← 𝑉(𝐺[𝐴]) ∪ {{𝑏}|𝑏 ∈ 𝐵}
2. For i = 1, … , 𝑐 in parallel for each vertex with color 𝑖 choose a color
in [𝑐] not chosen by any of its neighbors.
• Time complexity?
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
c=
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒
Network Decomposition - Strategy
1. Choose a number 𝑝(𝑛) and divide the vertices into high-degree and
low-degree groups (high-degree iff deg 𝐺 𝑣 ≥ 𝑝(𝑛) etc).
2. Collapse high-degree vertices with their neighborhoods, resulting in
a cluster graph 𝐺ℎ𝑖𝑔ℎ . Compute the network decomposition
recursively on 𝐺ℎ𝑖𝑔ℎ .
3. Color the low-degree vertices on the remaining graph 𝐺𝑙𝑜𝑤 .
4. Call 𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒 to get a network decomposition of 𝐺.
Network Decomposition - Strategy (2)
P(n)=6
Choose a number 𝑝(𝑛) and divide the vertices into high-degree and low-degree groups (high-degree iff deg 𝐺 𝑣 ≥ 𝑝(𝑛) etc).
Network Decomposition - Strategy (2)
Collapse high-degree vertices with their neighborhoods, resulting in a cluster graph 𝐺ℎ𝑖𝑔ℎ . Compute the network
decomposition recursively on 𝐺ℎ𝑖𝑔ℎ .
Network Decomposition - Strategy (2)
Color the low-degree vertices on the remaining graph(s).
Network Decomposition - Strategy (2)
Network Decomposition - Strategy (2)
Call 𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒 to get a network decomposition of 𝐺.
Network Decomposition - Strategy (3)
• PROBLEM: It may happen that the neighborhoods of high-degree
vertices intersect! This is a symmetry-breaking problem.
Network Decomposition - Strategy (4)
• Solution: we handle the problem by computing a (3,4)-ruling forest
with respect to the set 𝑃 of high-degree vertices and the graph 𝐺.
• We collapse trees with their root, instead of collapsing
neighborhoods.
• The improvement of this algorithm compared to last week comes
from the computation of a (3,4)-ruling forest instead of a (3,3𝑙𝑜𝑔𝑛)ruling forest. The recursion can terminate faster.
3,4 -ruling forest how?
We will see in a few slides…
Network Decomposition - Algorithm outline
• The algorithm uses two procedures:
• 𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒(𝐺) to compute a (𝑛𝜖 𝑛 , 𝑛𝜖 𝑛 )-decomposition of
some graph 𝐺 with 𝜖 𝑛 = 1/𝑙𝑜𝑔𝑛.
• 𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡(𝑃, 𝐺) to compute a (3,4)-ruling forest with respect to set 𝑃
and graph 𝐺.
• We will show the algorithm achieves 𝑂 𝑛𝜖
• Definition: 𝑂 𝑓 𝑛
=𝑂 𝑓 𝑛
𝑂 1
.
𝑛
running time.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒
• Input: a graph H = (𝑉, 𝐸) with 𝑙 ≤ 𝑛 vertices
• Output: an (8log𝑝 𝑙 , 𝑝)-network decomposition of 𝐻.
1. Define 𝑃 ≔ {𝑣| deg 𝐻 𝑣 ≥ 𝑝} and call 𝐹 = {𝑇𝑖 } ← 𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡(P, H)
2. Construct the graph HF in the following manner:
• 𝑉(HF ) = {𝑇𝑖 }
• E(HF ) = {(Ti , Tj )|i ≠ 𝑗, ∃𝑢 ∈ 𝑇𝑖 , 𝑣 ∈ 𝑇𝑗 : 𝑢, 𝑣 ∈ 𝐸(𝐻)}
3. Call 𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒(HF ).
4. Let 𝑆 ≔ 𝑉\F. Compute a 𝑝-coloring of H 𝑆 .
5. Call 𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑀𝑒𝑟𝑔𝑒 on HF and H[𝑆].
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 – Correctness &
complexity
• Claim 1:
𝑉(𝐻)
𝑉(𝐻)
𝑉(HF ) ≤
≤
𝑝+1
𝑝
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity (2)
• Claim 1:
𝑉(𝐻)
𝑉(𝐻)
𝑉(HF ) ≤
≤
𝑝+1
𝑝
• Claim 2:
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 computes a network decomposition with
clusters of at most 8log𝑝 𝑙 diameter.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity (3)
• Claim 1:
𝑉(𝐻)
𝑉(𝐻)
𝑉(HF ) ≤
≤
𝑝+1
𝑝
• Claim 2:
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 computes a network decomposition with
clusters of at most 8log𝑝 𝑙 diameter.
• Claim 3:
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 computes a network decomposition that uses
at most 𝑝 colors.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity (4)
• What is the time complexity?
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity (5)
• The time complexity 𝑇𝑁𝐷 satisfies the following recursive inequality:
𝑇𝑁𝐷 𝑙 ≤ 𝑇𝑅𝐹 𝑙 + 8 ⋅ 𝑇𝑁𝐷
𝑙
+ 𝑂(𝑝 ⋅ 𝑙𝑜𝑔𝑛)
𝑝
Computing a (3,4)-ruling forest
• In comparison to a (3,3𝑙𝑜𝑔𝑛)-ruling forest, it is not known how to
compute (3, 𝑘)-ruling forest in 𝑂(𝑙𝑜𝑔𝑛) time.
• However, a (3,4)-ruling forest can be computed using network
decomposition, in (hopefully) running time 𝑂 𝑛𝜖(𝑛) .
• We will see that this modification improves the overall time
complexity.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡
• Input: a graph 𝐻 with 𝑙 ≤ 𝑛 vertices and a set 𝑃 ⊆ 𝑉(𝐻).
• Output: a (3,4)-ruling forest with respect to 𝑃 and 𝐻.
1. Compute a (3,3𝑙𝑜𝑔𝑛)-ruling forest 𝐹 = {𝑇𝑘 } with respect to 𝑃 and
𝐻.
2. Within each tree 𝑇𝑘 , the root 𝑟𝑘 will assign numbers 1,2, … , 𝑝
cyclically to the vertices in 𝑇𝑘 ∩ 𝑃. Let 𝑃𝑖 be the set of all vertices in
𝑃 with color 𝑖.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡
3. From each 𝑃𝑖 we obtain a new set 𝑄𝑖 ⊆ 𝑃𝑖 by marking some
elements of 𝑃𝑖 ; 𝑄𝑖 is the set of unmarked vertices in 𝑃𝑖 . Let 𝑄1 = 𝑃1 .
In phases 2, … , 𝑝 repeat the following: in phase 𝑖, any vertex 𝑢 ∈ 𝑃𝑖
such that 𝑑𝐻 𝑢, 𝑣 ≤ 2 for some vertex 𝑣 ∈∪𝑗<𝑖 𝑄𝑗 will be marked.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡
4. Consider 𝐻[2, 𝑄𝑖 ]. In parallel, for all 𝑖, invoke
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒(𝐻 2, 𝑄𝑖 ) to compute a network
decomposition and use it to get an 𝑀𝐼𝑆 𝐼𝑖 in 𝐻[2, 𝑄𝑖 ].
5. 𝐼 =∪𝑖 𝐼𝑖 is a (3,4)-ruling set; use it to compute a 3,4 -ruling forest.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
P(n)=2
Everyone is in 𝑃!
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
Compute a (3,3𝑙𝑜𝑔𝑛)-ruling forest 𝐹 = {𝑇𝑘 } with respect to 𝑃 and 𝐻.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
Within each tree 𝑇𝑘 , the root 𝑟𝑘 will assign numbers 1,2, … , 𝑝 cyclically to the vertices in 𝑇𝑘 ∩ 𝑃.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
𝑄1 = 𝑃1 , 𝑄2 = ∅
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
In parallel, compute 𝑀𝐼𝑆 𝐼𝑖 in 𝐻[2, 𝑄𝑖 ].
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - example
𝐼 =∪𝑖 𝐼𝑖 is a (3,4)-ruling set; use it to compute a 3,4 -ruling forest.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity
• Claim 1:
for all 𝑖 ∈ [𝑝], we have
2𝑙
𝑃𝑖 ≤ .
𝑝
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (2)
• Claim 1:
for all 𝑖 ∈ [𝑝], we have
2𝑙
𝑃𝑖 ≤ .
𝑝
• Claim 2:
The set 𝐼 =∪𝑖 𝐼𝑖 is a (3,4)-ruling set with respect to 𝑃 and 𝐻.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (3)
• What is the time
complexity?
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (4)
• The complexity of 𝑇𝑅𝐹 satisfies the following inequality:
𝑇𝑅𝐹 𝑙 ≤ 𝑂 𝑙𝑜𝑔𝑛 + 𝑂 𝑝 + 2𝑇𝑁𝐷 (2𝑙/𝑝)
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (4)
• The complexity of 𝑇𝑅𝐹 satisfies the following inequality:
𝑇𝑅𝐹 𝑙 ≤ 𝑂 𝑙𝑜𝑔𝑛 + 𝑂 𝑝 + 2𝑇𝑁𝐷 (2𝑙/𝑝)
𝑇𝑁𝐷 𝑙 ≤ 𝑇𝑅𝐹 𝑙 + 8 ⋅ 𝑇𝑁𝐷
𝑙
+ 𝑂(𝑝 ⋅ 𝑙𝑜𝑔𝑛)
𝑝
Time complexities combined
• Combining the inequalities for 𝑇𝑁𝐷 , TRF we get (assuming 𝑝 ≥ 𝑙𝑜𝑔𝑛)
𝑇𝑁𝐷 𝑙 ≤ 𝑂 𝑝𝑙𝑜𝑔𝑛 + 10𝑇𝑁𝐷
2𝑙
𝑝
≤ 𝑂 𝑝𝑙𝑜𝑔𝑛 10log𝑝/2 𝑙
= 𝑓(𝑙, 𝑝)
• The minimum is attained when 𝑝 = 2𝜃(√𝑙𝑜𝑔𝑛) = 𝑂(𝑛𝜖
𝑛
).
THEOREM 1:
Given a graph 𝐺 with 𝑛 vertices, procedure
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 computes an
(𝑛𝜖 𝑛 , 𝑛𝜖 𝑛 )-decomposition of 𝐺 in 𝑂 𝑛𝜖 𝑛
time, where 𝜖 𝑛 = 1/ 𝑙𝑜𝑔𝑛.
Part 2
Near-optimal decomposition
Near-optimal decomposition
• A (𝑙𝑜𝑔𝑛, 𝑙𝑜𝑔𝑛)-decomposition is called a near-optimal
decomposition. We would be interested in computing a near-optimal
decomposition in 𝑂 𝑛𝜖 𝑛 .
• Suppose there is a procedure 𝐴𝐿𝐺 which computes a (𝑑 𝑛 , 𝑐 𝑛 )decomposition in time 𝑂(𝑡 𝑛 ).
• We will prove one can compute a near-optimal decomposition in time
𝑂(𝑡 𝑛 log 𝑛 + 𝑐 𝑛 𝑑 𝑛 log 2 𝑛)
A sequential scheme
• For any index 𝑖 and vertex 𝑣 ∈ 𝑉 define (“cluster” and “boundary”)
𝐶𝑣 ≔ 𝑢 𝑑𝐺 𝑣, 𝑢 ≤ 𝑖
𝐵𝑣 ≔ 𝑢 𝑑𝐺 𝑣, 𝑢 = 𝑖 + 1
• Now, either there exists an index 𝑖,
0 ≤ 𝑖 ≤ log 2 𝑛 − 1
such that 𝐵𝑣 ≤ 𝐶𝑣 or not.
• If there exists no such index, then 𝐺 has 𝑂(𝑙𝑜𝑔𝑛) diameter, so we can
take the whole graph to be one cluster.
• Otherwise let 𝑙𝑣 = 𝑂(𝑙𝑜𝑔𝑛) always be the smallest such index.
A sequential scheme (2)
1. For the procedure, start with any vertex 𝑣.
2. Choose i = 𝑙𝑣 to be the smallest such that 𝐵𝑣 ≤ |𝐶𝑣 |.
3. Proceed by removing the vertices 𝐶𝑣 ∪ 𝐵𝑣 and the edges incident at
them and repeat this process on the remaining graph.
4. Each set 𝐶𝑣 now becomes a cluster and gets color 1.
5. Repeat the above process on the remaining graph (union of 𝐵𝑣 ) to
assign color classes 2,3, ….
A sequential scheme - correctness
• Claim 1:
The diameter of each cluster 𝐶𝑣 is at most 𝑂(𝑙𝑜𝑔𝑛).
• Claim 2:
The coloring of the cluster graph is valid.
• Claim 3:
There are at most 𝑂(𝑙𝑜𝑔𝑛) colors.
A distributed scheme
• Any sequential algorithm can be computed trivially in 𝑂(𝑑) rounds,
where 𝑑 is the diameter of the given graph.
• We seek a faster approach by exploiting the idea of network
decomposition, in a similar manner MIS, for instance, is computed.
• Technical assumption: 𝑑 𝑛 , 𝑐 𝑛 ≥ Ω(logn).
A distributed scheme
1. Use 𝐴𝐿𝐺 to compute a (𝑑(𝑛), 𝑐(𝑛))-decomposition of 𝐺[2𝑙𝑜𝑔𝑛, 𝑉]
2. For 𝑛𝑒𝑤𝑐𝑜𝑙𝑜𝑟 = 1, … , 𝑙𝑜𝑔𝑛 do
• For 𝑐 = 1, … , 𝑂(𝑐(𝑛)), each cluster with color 𝑐 in 𝐺[2𝑙𝑜𝑔𝑛, 𝑉] proceeds as
follows. Each cluster 𝑐 simulates the sequential algorithm to compute the sets
𝐶𝑣 , makes them new clusters with color 𝑛𝑒𝑤𝑐𝑜𝑙𝑜𝑟 (vertices are allowed to
“eat into” old clusters which are not of color 𝑐). Newly colored clusters are
removed from the network.
A distributed scheme - correctness
• Since the initial (𝑑(𝑛), 𝑐(𝑛))-decomposition was computed on
𝐺 2𝑙𝑜𝑔𝑛, 𝑉 , vertices with the same old color that belong to different
clusters are at least 2 𝑙𝑜𝑔 𝑛 + 1 apart.
• In addition, if two clusters receive the same 𝑛𝑒𝑤𝑐𝑜𝑙𝑜𝑟 there are two
possibilities:
• They’re in the same old cluster – by the correctness of the sequential algorithm the
clusters will not have an edge between them.
• They're in different clusters – because of the above bullet and that clusters are at
most 𝑙𝑜𝑔𝑛 in diameter, they don’t interfere with each other.
• Hence the created cluster graph has a valid coloring. There are at most
𝑙𝑜𝑔𝑛 new colors. Additionally, each cluster is at most 𝑙𝑜𝑔 𝑛 in diameter as
in the sequential case.
• We have obtained a (𝑙𝑜𝑔𝑛, 𝑙𝑜𝑔𝑛)-decomposition.
A distributed scheme - time complexity?
1. Use 𝐴𝐿𝐺 to compute a (𝑑(𝑛), 𝑐(𝑛))-decomposition of 𝐺[2𝑙𝑜𝑔𝑛, 𝑉]
2. For 𝑛𝑒𝑤𝑐𝑜𝑙𝑜𝑟 = 1, … , 𝑙𝑜𝑔𝑛 do
• For 𝑐 = 1, … , 𝑂(𝑐(𝑛)), each cluster with color 𝑐 in 𝐺[2𝑙𝑜𝑔𝑛, 𝑉] proceeds as
follows. Each cluster 𝑐 simulates the sequential algorithm to compute the sets
𝐶𝑣 , makes them new clusters with color 𝑛𝑒𝑤𝑐𝑜𝑙𝑜𝑟 (vertices are allowed to
“eat into” old clusters which are not of color 𝑐). Newly colored clusters are
removed from the network.
THEOREM 2:
Suppose there is a procedure 𝐴𝐿𝐺 which
computes a (𝑑 𝑛 , 𝑐 𝑛 )-decomposition in time
𝑂(𝑡 𝑛 ). Then we can compute a near-optimal
decomposition in
2
𝑂(𝑡 𝑛 𝑙𝑜𝑔𝑛 + 𝑐 𝑛 𝑑 𝑛 log 𝑛).
Applications
• Corollary: Given a graph 𝐺 with 𝑛 vertices, the following functions can
be computed in 𝑂(𝑛𝜖 𝑛 ) time in the distributed model of
computation:
•
•
•
•
•
Maximal independent set
(Δ + 1)-vertex coloring
(2Δ − 1)-edge coloring
Maximal matching
(𝑙𝑜𝑔𝑛, 𝑙𝑜𝑔𝑛)-decomposition
Part 3
Completeness
Completeness
• In the previous parts we gave an algorithm for computing a network
decomposition in time 𝑂 𝑛𝜖 𝑛 .
• One can ask a question from a different angle: suppose that we know
how to compute a decomposition in 𝑂(𝑙𝑜𝑔𝑛) but only for some
graphs. Can we use it in conjunction with other methods to compute
a decomposition for all graphs?
• We call this a completeness result: we seek a class of 𝑛-vertex graphs
which is complete for the task of computing a network
decomposition.
Completeness (2)
• We will see a result of the following nature
In order to have a (𝑝𝑜𝑙𝑦𝑙𝑜𝑔𝑛, 𝑝𝑜𝑙𝑦𝑙𝑜𝑔𝑛)-decomposition running in 𝑂(𝑙𝑜𝑔𝑛) time,
we just need to look at graphs of maximum degree Δ ≤ 𝑂(𝑛1/𝑙𝑜𝑔𝑙𝑜𝑔𝑛 ).
Completeness (3)
• In general, let ℎ(𝑛) be any non-decreasing function and let
𝑔 𝑛 = 𝑝 𝑛 1/ℎ(𝑛)
𝑞 𝑛 = 𝑝 𝑛 ℎ(𝑛)
where 𝑝 𝑛 = 𝑂(𝑛1/ 𝑙𝑜𝑔𝑛 ).
• Suppose we have an algorithm, 𝐴𝐿𝐺, that computes a (𝑔 𝑛 , 𝑔 𝑛 )decomposition of graphs with maximum degree Δ ≤ 𝑞(𝑛) in time
𝑂(𝑔 𝑛 ). We wish to compute a decomposition on all graphs.
• For 𝑔 𝑛 = 𝑂(𝑙𝑜𝑔𝑛), we have
ℎ 𝑛 = 𝑙𝑜𝑔𝑛/𝑙𝑜𝑔𝑙𝑜𝑔𝑛
Completeness - strategy
• To compute a network decomposition for all 𝑛-vertex graphs, we will
use a modified version of the mutually recursive procedures
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 and 𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡.
• As one could expect, we will divide the vertices into high-degree and
low-degree ones. This will allow us to use algorithm 𝐴𝐿𝐺 which
operates only on graphs of degree Δ ≤ 𝑞(𝑛).
• In addition, 𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 will compute a (3,6)-ruling forest.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒
• Input: a graph H = (𝑉, 𝐸) with 𝑙 ≤ 𝑛 vertices
• Output: an (𝑔(𝑙), 𝑔 𝑛 log 𝑞 𝑛 𝑙)-network decomposition of 𝐻.
1. Define 𝑃 ≔ {𝑣| deg 𝐻 𝑣 ≥ 𝑞(𝑛)} and call 𝐹 = {𝑇𝑖 }
← 𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡(P, H)
2. Construct the graph 𝐻𝐹 in the following manner:
• 𝑉(𝐻𝐹 ) = {𝑇𝑖 }
• E(HF ) = {(Ti , Tj )|i ≠ 𝑗, ∃𝑢 ∈ 𝑇𝑖 , 𝑣 ∈ 𝑇𝑗 : 𝑢, 𝑣 ∈ 𝐸𝐻 }
3. Call 𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒(𝐻𝐹 ).
4. Let 𝑆 ≔ 𝑉\F. Compute a (𝑔 𝑛 , 𝑔 𝑛 )-decomposition of H 𝑆 using 𝐴𝐿𝐺
with fresh colors.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity
• Claim 1:
𝑉(𝐻)
𝑉(𝐻)
𝑉(𝐻𝐹 ) ≤
≤
𝑞(𝑛) + 1
𝑞(𝑛)
• Claim 2:
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 computes a network decomposition with
clusters of at most 𝑔(𝑙) diameter.
• Claim 3:
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 uses at most 𝑂 𝑔 𝑛 log 𝑙 / log 𝑞 𝑛
colors.
𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒 - Correctness &
complexity (2)
• The complexity of 𝑇𝑁𝐷 satisfies the following inequality:
𝑇𝑁𝐷 𝑙 ≤ 𝑇𝑅𝐹 𝑙 + 12 ⋅ 𝑇𝑁𝐷
𝑙
+ 𝑂(𝑔 𝑛 )
𝑞
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡
• Input: a graph 𝐻 with 𝑙 ≤ 𝑛 vertices and a set 𝑃 ⊆ 𝑉(𝐻).
• Output: a (3,6)-ruling forest with respect to 𝑃 and 𝐻.
1. Partition 𝑃 into disjoint sets 𝑃𝑖 , 𝑖 ∈ 𝑞 , as before, by computing a
(3,3𝑙𝑜𝑔𝑛)-ruling forest and assigning numbers 1,2, . . , 𝑞 cyclically
inside each tree.
2. Let 𝐻𝑖 = 𝐻 4, 𝑃𝑖 . In parallel, for all 𝑖 ∈ [𝑞], compute an MIS 𝐼𝑖 of 𝐻𝑖
by calling 𝑁𝑒𝑡𝑤𝑜𝑟𝑘𝐷𝑒𝑐𝑜𝑚𝑝𝑜𝑠𝑒(𝐻𝑖 ). Let 𝐼 =∪𝑖 𝐼𝑖 .
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡
3. Let 𝐹 = 𝐻[2, 𝐼]. Invoke 𝐴𝐿𝐺 to compute a MIS, 𝐽, of 𝐹. (Δ 𝐹 ≤ 𝑞)
4. The set 𝐽 is a (3,6)-ruling set; use it to compute a (3,6)-ruling
forest.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity
• Claim 1:
for all 𝑖 ∈ [𝑞], we have
2𝑙
𝑃𝑖 ≤ .
𝑞
• Claim 2:
We have
Δ 𝐹 ≤𝑞 𝑛 .
• Claim 3:
The set 𝐽 is a (3,6)-ruling set with respect to 𝑃 and 𝐻.
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (2)
• The complexity of 𝑇𝑅𝐹 satisfies the following inequality:
𝑇𝑅𝐹 𝑙 ≤ 𝑂 𝑙𝑜𝑔𝑛 + 4𝑇𝑁𝐷 (2𝑙/𝑞)+ 𝑂 𝑔(𝑛)
𝑅𝑢𝑙𝑖𝑛𝑔𝐹𝑜𝑟𝑒𝑠𝑡 - Correctness & complexity (3)
• The complexity of 𝑇𝑅𝐹 satisfies the following inequality:
𝑇𝑅𝐹 𝑙 ≤ 𝑂 𝑙𝑜𝑔𝑛 + 4𝑇𝑁𝐷 (2𝑙/𝑞)+ 𝑂 𝑔(𝑛)
𝑇𝑁𝐷 𝑙 ≤ 𝑇𝑅𝐹 𝑙 + 12 ⋅ 𝑇𝑁𝐷
𝑙
+ 𝑂(𝑔 𝑛 )
𝑞
Time complexities combined
• Combining the inequalities for 𝑇𝑁𝐷 , TRF we get (assuming 𝑝 ≥ 𝑙𝑜𝑔𝑛)
𝑇𝑁𝐷
2𝑙
𝑙 ≤ 16𝑇𝑁𝐷
+ 𝑂(𝑔 𝑛 )
𝑞
≤ 𝑂(𝑔 𝑛 )
THEOREM 3:
Let 𝑝 𝑛 = 2 𝑙𝑜𝑔𝑛 . Suppose we want an 𝑂(𝑝 n 1/ℎ 𝑛 )
time algorithm to distributively compute an
(𝑂(𝑝 n 1/ℎ 𝑛 ), 𝑂(𝑝 n 1/ℎ 𝑛 ))-decomposition, for any
non-decreasing function ℎ ⋅ . This is self-reducible to
graphs of maximum degree O(p n ℎ 𝑛 ).
Thank you!
Questions?