Super-fast 3-ruling sets
Kishore Kothapalli and Sriram Pemmaraju
PRESENTED BY MICHAL BEN HAIM
29/10/2016
3-ruling sets - definition
A 3-ruling set of a graph G = (V,E) is a vertex-subset
S⊆ 𝑉 that is independent and satisfies the property
that every vertex 𝑣 ∈ 𝑉 is at a distance of at most 3
from some vertex in S.
Example of 3-ruling sets
1
2
3
𝑆 = {1}
5
4
6
7
Example of 3-ruling sets
This is not 3- ruling set!
1
2
3
𝑆 = {1,2}
5
4
6
7
Example of 3-ruling sets
1
2
3
5
4
𝑆 = {1,3,4,6,7}
6
7
Example of 3-ruling sets
This is not 3- ruling set!
1
3
2
4
𝑆 = {1}
3
4
2
5
1
5
Algorithms goal

Algorithm RulingSet-HG computes a 3-ruling set
for a tree G.

RulingSet-HG terminates in
𝑂 𝑙𝑜𝑔𝑙𝑜𝑔𝑛 2 𝑙𝑜𝑔𝑙𝑜𝑔𝑙𝑜𝑔𝑛 rounds with high
probability
The algorithm

Let G = (V,E) be a graph with n vertices,
maximum degree ∆.

Let i* be the smallest positive integer such that
1𝑖∗
2
∆ ≤ 6 ∙ log 𝑛 .

Conclusion: i* = O(log log ∆).
The algorithm
For 1 ≤ 𝑖 ≤ 𝑖 ∗ :

1𝑖
2
M1 – vertices with degree > ∆ join to M1 with probability
6∙log 𝑛
1𝑖−1
∆2

1𝑖
2
M2 – vertices with degree ≤ ∆ join to M2 with probability
6∙log 𝑛
1𝑖
∆2

W - vertices 𝑣 ∈ 𝑉 \(𝑀1 ∪ 𝑀2 ) such that dist(v, 𝑀1 ∪ 𝑀2 ) ≤2.
The algorithm

n=16

∆= 5

Deg
13
Deg
81
Deg
12
3
Deg
13
1
1
2
5 ≈ 2.23
75
Deg
61
Deg
42
≤ 6𝑙𝑜𝑔16)
Deg
Deg 5
Deg
52
Deg
23
Deg
31

i*=1 (5
1𝑖∗
2
Deg
92
Deg
10
1
Deg
11
1

Deg
15
2
Deg
14
1
Deg
16
1
1𝑖
2
d𝑒𝑔(𝑣) > ∆ join to M1 with
6∙log 𝑛
probability
1𝑖−1
∆2

1𝑖
2
deg(v)≤ ∆ join to M2 with
6∙log 𝑛
probability
1𝑖
∆2

𝑣 ∈ 𝑉 \(𝑀1 ∪ 𝑀2 ) and
dist(v, 𝑀1 ∪ 𝑀2 ) ≤2 join to W.
Who can join M1?

n=16

∆= 5

Deg
13
Deg
61
Deg
42
Deg
81
Deg
14
3
Deg
16
1
≤ 6𝑙𝑜𝑔16)
1
2
5 ≈ 2.23
Deg
75
Deg
52
Deg
23
Deg
31

i*=1 (5
1𝑖∗
2
Deg
12
2
Deg
15
1
Deg
13
1
Deg
92
Deg
10
1
Deg
11
1

d𝑒𝑔 𝑣 > 2.23 join to M1 with
6∙log 𝑛
probability
1𝑖−1
∆2
Who can join M2?

n=16

∆= 5

Deg
13
Deg
52
Deg
23
Deg
31

≤ 6𝑙𝑜𝑔16)
1
2
5 ≈ 2.23
Deg
75
Deg
61
Deg
42
Deg
81
Deg
14
3
Deg
16
1
i*=1 (5
1𝑖∗
2
Deg
12
2
Deg
15
1
Deg
13
1
Deg
92
Deg
10
1
Deg
11
1

1𝑖
2
deg(v)≤ ∆ join to M2 with
6∙log 𝑛
probability
1𝑖
∆2
Who join W?

n=16

∆= 5

Deg
13
Deg
52
Deg
23
Deg
3
1

Deg
81
Deg
14
3
Deg
16
1
Deg
12
2
Deg
15
1
≤ 6𝑙𝑜𝑔16)
1
2
5 ≈ 2.23
Deg
75
Deg
61
Deg
42
i*=1 (5
1𝑖∗
2
Deg
13
1
Deg
92
Deg
10
1
Deg
11
1

𝑣 ∈ 𝑉 \(𝑀1 ∪ 𝑀2 ) and
dist(v, 𝑀1 ∪ 𝑀2 ) ≤2 join to W
M1
M2
W
Who stay in V?

n=16

∆= 5

Deg
13
Deg
52
Deg
23
Deg
3
1

Deg
81
Deg
14
3
Deg
16
1
Deg
12
2
Deg
15
1
≤ 6𝑙𝑜𝑔16)
1
2
5 ≈ 2.23
Deg
75
Deg
61
Deg
42
i*=1 (5
1𝑖∗
2
Deg
13
1
Deg
92
Deg
10
1
Deg
11
1

𝑣 ∈ 𝑉 \(𝑀1 ∪ 𝑀2 ) and
dist(v, 𝑀1 ∪ 𝑀2 ) ≤2 join to W
M1
M2
W
The algorithm

In each iteration the algorithm add to subset I the
result of MIS on G[M1UM2] .

After all i* iterations the algorithm compute MIS on
𝑉\(𝑀1 ∪ 𝑀2 ∪ 𝑊) and add the result to subset I.

The 3-ruling set on the graph is subset I.
Lemma 6

For 1 ≤ 𝑖 ≤ 𝑖 , with probability at least 1 −
∗
1𝑖
2
1
𝑛2
, all
vertices still in V have degree at most ∆ at the end
of iteration i. (similar to lemma 1)
Corollary 7

1
,
2
𝑛
With probability at least 1 −
after all 𝑖 ∗ iterations
of the for-loop in Algorithm RulingSet-HG, the graph G
has maximum degree at most 6 log n.
Lemma 8

Consider an arbitrary iteration 1 ≤ 𝑖 ≤ 𝑖 ∗ with
2
probability at least 1 − , the maximum degree of a
𝑛
vertex in G[Mj ], j = 1, 2 is at most 12 · log n.
Lemma 8
2
The lemma: Consider an arbitrary iteration 1 ≤ 𝑖 ≤ 𝑖 ∗ with probability at least 1 − 𝑛 ,
the maximum degree of a vertex in G[Mj ], j = 1, 2 is at most 12 · log n.


Proof :
With probability 1 −
1
𝑛2
for all 𝑣 ∈ 𝑉 at the beginning of
1𝑖−1
2
an iteration i 𝑑𝑒𝑔 𝑣 ≤ ∆ . (lemma 6)
 deg 𝑀j (𝑣) - the degree of vertex 𝑣 ∈ Mj in G[Mj] for
j ∈ {1,2} .

𝐸 deg 𝑀1 (𝑣) ≤ ∆
1𝑖−1
2
∙
6 log 𝑛
1𝑖−1
∆2
= 6 log 𝑛 .
Lemma 8
2
The lemma: Consider an arbitrary iteration 1 ≤ 𝑖 ≤ 𝑖 ∗ with probability at least 1 − 𝑛 ,
the maximum degree of a vertex in G[Mj ], j = 1, 2 is at most 12 · log n.


Proof :
1𝑖
2
M2 – vertices with degree ≤ ∆ join to M2 with
6∙log 𝑛
probability
1𝑖
∆2

1𝑖
2
𝐸 deg 𝑀2 (𝑣) ≤ ∆ ∙
6 log 𝑛
1𝑖
∆2
= 6 log 𝑛 .
Lemma 8
2
The lemma: Consider an arbitrary iteration 1 ≤ 𝑖 ≤ 𝑖 ∗ with probability at least 1 − 𝑛 ,
the maximum degree of a vertex in G[Mj ], j = 1, 2 is at most 12 · log n.

Proof :

Using Chernoff bounds we conclude that
Pr[deg 𝑀𝑗 𝑣 ≥ 12 ∙ 𝑙𝑜𝑔𝑛] ≤ 𝑒

−2𝑙𝑜𝑔𝑛
1
𝑛
=
1
𝑛2
𝑓𝑜𝑟 𝑗 ∈ {1,2}
with probability at least 1 − the maximum degree
of 𝐺[𝑀1 ∪ 𝑀2] is at most 12 log n under the
assumption 𝑑𝑒𝑔 𝑣 ≤ ∆
1𝑖−1
2
.
Lemma 8
2
The lemma: Consider an arbitrary iteration 1 ≤ 𝑖 ≤ 𝑖 ∗ with probability at least 1 − 𝑛 ,
the maximum degree of a vertex in G[Mj ], j = 1, 2 is at most 12 · log n.

Proof :

Without the assumption and with union bound : with
2
probability at least 1 − the maximum degree
𝑛
of 𝐺[𝑀𝑗] for j = 1, 2 is at most 12 log n
Theorem 9

Algorithm RulingSet-HG computes a 3-ruling set of a
tree G.

RulingSet-HG terminates in
𝑂 𝑙𝑜𝑔𝑙𝑜𝑔𝑛 2 ∙ 𝑙𝑜𝑔𝑙𝑜𝑔𝑙𝑜𝑔𝑛 rounds with high
probability.
Theorem 9

‫חישוב‬
MIS
Example:
M1
M2
W
W
Theorem 9

Example:
2
1
M1
3
I
W
W
Theorem 9 - Total running time

i* times (the worst case) MIS subroutine for 𝑀1 ∪ 𝑀2
subset.

MIS subroutine for V subset in the end of the
algorithm.

MIS runs in 𝑂 𝑙𝑜𝑔𝑙𝑜𝑔𝑛 ∙ 𝑙𝑜𝑔𝑙𝑜𝑔𝑙𝑜𝑔𝑛 rounds by
Barenboim’s algorithm.

RulingSet-HG runs in 𝑂 𝑙𝑜𝑔𝑙𝑜𝑔∆ ∙ 𝑙𝑜𝑔𝑙𝑜𝑔𝑛 ∙ 𝑙𝑜𝑔𝑙𝑜𝑔𝑙𝑜𝑔𝑛
= 𝑂 𝑙𝑜𝑔𝑙𝑜𝑔𝑛
2
∙ 𝑙𝑜𝑔𝑙𝑜𝑔𝑙𝑜𝑔𝑛
Super-fast t-ruling sets
Tushar Bisht, Kishore Kothapalli, Sriram V. Pemmaraju
PRESENTED BY MICHAL BEN HAIM
29/10/2016
t-ruling sets - definition
A t-ruling set of a graph G = (V,E) is a vertex-subset
S⊆ 𝑉 that is independent and satisfies the property
that every vertex 𝑣 ∈ 𝑉 is at a distance of at most t
from some vertex in S.
Rapid sparsification- reminder:
1
2
In the last
iteration
3
Mi
Wi
5
4
S= {1,3,4,5}
6
7
Theorem 1

Let G be an arbitrary n-vertex graph with maximum
degree ∆.

With high probability, The rapid sparsification
algorithm with input G and f runs in 𝑂(log 𝑓 ∆) rounds
and produces a vertex-subset S⊆ 𝑉(G) such that
∆(𝐺[𝑆]) ∈ 𝑂(𝑓𝑙𝑜𝑔𝑛), and every vertex in V is either in
S or has a neighbor in S.
The algorithm t-RulingSet-GG
𝑡−1−𝑖
(log 𝑛) 𝑡−1
 𝑓𝑖−1
=2
 𝑓𝑡−1
= log 𝑛
 𝑆0


i=1,2,…t-2
=𝑉
For i=1 to t-1 the algorithm call the rapid
sparsification algorithm with 𝐺 𝑆𝑖−1 𝑎𝑛𝑑 𝑓𝑖−1 and save
the result is 𝑆𝑖 .
return the result of MIS algorithm on 𝐺 𝑆𝑡−1 .
Lemma 1

For each i, 0 ≤ 𝑖 ≤ 𝑡, with high probability every
vertex in V is at most i hops from some vertex in 𝑆𝑖 .
Thus, with high probability, 𝑆𝑡 is a t-ruling set.
Theorem 2

With high probability, Algorithm t-RulingSet-GG runs in
time 𝑂 𝑡 ∙ log 𝑛
1
𝑡−1
+ exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛)).
Proof :
 ∆𝑖 - max degree of vertex in 𝐺[𝑆𝑖 ] .
for 1 ≤ 𝑖 ≤ 𝑡 − 1:
 𝑆𝑖 = Rapid sparsification algorithm on 𝐺[𝑆𝑖−1 ] 𝑎𝑛𝑑 𝑓𝑖−1 ∶

 ∆𝑖 ∈
 𝑆𝑖
𝑂(log 𝑛 ∙ 𝑓𝑖−1 )
calculate in O log 𝑓𝑖−1 ∆𝑖−1 = 𝑂( log 𝑛
1
𝑡−1
+ 𝑙𝑜𝑔𝑙𝑜𝑔 𝑛) rounds
Theorem 2
With high probability, Algorithm t-RulingSet-GG runs in time
𝑂 𝑡 ∙ log 𝑛

Proof :

∆𝑡−1 ∈ 𝑂 log 𝑛 ∙ 𝑓𝑡−1 = 𝑂 (log 𝑛 )2
 𝑆𝑡

1
𝑡−1
+ exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛)).
=MIS(𝐺[𝑆𝑖−1 ] ) runs in exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛)).
Total of :
𝑂((𝑡 − 1)( log 𝑛
𝑂 𝑡 log 𝑛
1
𝑡−1
1
𝑡−1
+𝑙𝑜𝑔𝑙𝑜𝑔 𝑛)) + exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛))=
+ exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛)).
Corollary 1

For 𝑡 ≤
𝑙𝑜𝑔𝑙𝑜𝑔𝑛 we can compute a t-ruling set in
𝑂 𝑡 ∙ log 𝑛
1
𝑡−1
round and if 𝑡 > 𝑙𝑜𝑔𝑙𝑜𝑔𝑛 we
can compute a t-ruling set in exp(𝑂( 𝑙𝑜𝑔𝑙𝑜𝑔𝑛)).