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Improved Bounds for Minimum
Fault-Tolerant Gossip Graphs
Toru Hasunuma1 and Hiroshi Nagamochi2
1
Institute of Socio-Arts and Sciences,
The University of Tokushima
2 Department of Applied Mathematics and Physics,
Kyoto University
Contents
• Introduction
– Fault-Tolerant Gossiping Problem
– Results
• Construction of Fault-Tolerant Gossip Graphs
• Fault-Tolerant Gossip Graphs Based on
Hypercubes
• Fault-Tolerant Gossip Graphs Based on Circulant
Graphs
• A Lower Bound
• Conclusion
Gossiping Problem
• There are n persons such that each person has a unique
message.
• All the n persons want to know all the n messages by
telephone.
• In each telephone call, the two persons exchange every
message which they have at the time of the call.
What is the minimum number of calls?
The minimum number of calls was determined to be 2𝑛 − 4 for
𝑛 > 2.
• Tijdeman [1971]
• Baker and Shostak [1972]
Fault-Tolerant Gossiping Problem
• There are n persons such that each person has a unique
message.
• All the n persons want to know all the n message by
telephone.
• In each telephone call, the two persons exchange every
message which they have at the time of the call.
• At most k telephone calls fail.
– The messages in a failed call are not exchanged.
What is the minimum number of calls?
• Let 𝝉(𝒏, 𝒌) be the minimum number of calls for the faulttolerant gossiping problem on 𝑛 persons with at most 𝑘
failed calls.
𝑘-Fault-Tolerant Gossip Graphs
• The 𝑘-fault tolerant gossiping problem can be modeled by a
(multiple) graph 𝐺 = (𝑉, 𝐸) with edge-ordering 𝜌: 𝐸 →
{1,2, ⋯ , 𝐸 } .
• Define a 𝑘-fault-tolerant gossip graph as an ordered graph in
which for any ordered pair of distinct vertices 𝑢 and 𝑣, there are
𝑘 + 1 ascending paths from 𝑢 to 𝑣.
• The size of a 𝑘-fault-tolerant gossip graph is an upper bound on
𝜏 𝑛, 𝑘 .
32
An example of a 2-fault-tolerant
gossip graph
36
1
23
18
6
18
24
2
19
3
13
12 4
14
15
22
28
7
17
31
21
5
30
29
27
20
33
26
8
16
11
10
34
25
9
35
Comparison of Upper Bound Results
Previous Results
Berman
and
Hawrylycz
[1986]
Haddad
et al.
[1987]
Haddad
et al.
[1987]
When 𝑛 is a
power of two
Hou and
Shigeno
[2009]
𝑘+
𝑘
+ 2𝑝
2
min
Our Results
improv
ement
3
(𝑛 − 1)
2
𝑛−1
(𝑛 − 1) + 𝑝
+ 2𝑝
2 −1
𝑘+1
𝑛 log 2 𝑛
+1
,
log 2 𝑛
2
𝑘+1
𝑛 log 2 𝑛
+1
log 2 𝑛
2
𝑛 𝑛−1
𝑛𝑘
+
2
2
𝑘
+𝑝
2
𝑛−1
(𝑛 − 1) + 𝑝
+ 2𝑝
2 −1
𝑛 log 2 𝑛 𝑛𝑘
+
2
2
2𝑛 log 2 𝑛 + 𝑛
𝑘−1
2
𝑛 > 12
Comparison of Lower Bound Results
Previous Results
Berman
and Paul
[1986]
Berman
and Paul
[2002]
Hou and
Shigeno
[2009]
Our Result
Improv
ement
𝑘+4
(𝑛 − 1) − 2 𝑛 + 1
2
𝑖𝑓 𝑘 ≤ 𝑛 − 2,
𝑘+3
(𝑛 − 1) − 2 𝑛
2
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑘 𝑛−1
2𝑛 − 2 +
2
𝑛+
𝑛𝑘
2
− log 2 𝑛
3𝑛 − 5
1
𝑛+1
+
𝑛𝑘 +
− log 2 𝑛
2
2
2
𝑛
𝑘>
2
𝑛>4
Construction of Fault-Tolerant Gossip
Graphs
• The general method by Haddad et al.
– Cumulate the edge set of a copy of an ordered
graph (𝐺, 𝜌) iteratively.
𝐸(𝐺ℎ−1 )
ℎ ∙ (𝐺, 𝜌)
𝜌′ 𝑒 = 𝜌 𝑒 + ℎ − 1 |𝐸 𝐺 |
⋮
⋮
𝐸(𝐺3 )
𝜌′ 𝑒 = 𝜌 𝑒 + 3|𝐸 𝐺 |
𝐸(𝐺2 )
𝜌′ 𝑒 = 𝜌 𝑒 + 2|𝐸 𝐺 |
𝐸(𝐺1 )
𝜌′ 𝑒 = 𝜌 𝑒 + |𝐸 𝐺 |
𝐺 = 𝐺0
𝜌′ 𝑒 = 𝜌 𝑒
Folded Number
• A path from 𝑢 to 𝑣 in an ordered graph 𝐺, 𝜌 is called
𝑠-folded if it is a series of 𝑠 (maximal) ascending paths.
• The folded number of an 𝑠-folded ascending path is
defined to be 𝑠.
𝑢
1-2-5-6
3-4-7-9
8-10-13
11-12-14
15-16-17
𝑃 ⋯ 4-folded ascending path
𝑣
Edge-Disjoint Ascending Paths in ℎ ⋅
(𝐺, 𝜌)
If there is an 𝑠-folded ascending path from 𝑢 to 𝑣, then
there are (ℎ − 𝑠) edge-disjoint ascending paths from 𝑢 to 𝑣 in ℎ ∙
(𝐺, 𝜌) for any integer ℎ ≥ 𝑠.
3 edge-disjoint ascending path from 𝑢 to 𝑣 in 7 ∙ (𝐺, 𝜌)
P6
P5
P4
P3
P2
P1
P = P0
𝑢
7-4 = 3
4
𝑣
Lemma 1:
If there are 𝒑 edge-disjoint 𝑠-folded ascending paths from 𝑢 to 𝑣,
then there are 𝒑(ℎ − 𝑠) edge-disjoint ascending paths from 𝑢 to 𝑣
in ℎ ∙ (𝐺, 𝜌) for any integer ℎ ≥ 𝑠.
𝑃
𝐸 𝑃 ∩ 𝐸 𝑃′ = ∅
𝑃′
Lemma 1:
If there are 𝑝 edge-disjoint 𝑠-folded ascending paths from 𝑢 to 𝑣, then
there are 𝑝(ℎ − 𝑠) edge-disjoint ascending paths from 𝑢 to 𝑣 in ℎ ∙
(𝐺, 𝜌) for any integer ℎ ≥ 𝑠.
Corollary 1:
Let (𝐺, 𝜌) be an ordered graph with 𝑛 vertices and 𝑚 edges. If for any
ordered pair of vertices 𝑢 and 𝑣, there are 𝑝 edge-disjoint paths from
𝑢 to 𝑣 in (𝐺, 𝜌) such that their folded numbers are at most 𝑠, then
𝑘+1
𝑠+
⋅ (𝐺, 𝜌) is a 𝑘-fault-tolerant gossip graph, thus
𝑝
𝑘+1
𝜏 𝑛, 𝑘 ≤ 𝑠 +
𝑝
𝑚
⋮
When k =7, i.e.,
we need 8
edge-disjoint
ascending
paths ,
Corollary 2
consider 6 ⋅
𝐺, 𝜌 .
However, it is
sufficient to
consider 4 ⋅
(𝐺, 𝜌) .
⋮
𝐺5
5 edge-disjoint ascending paths
𝐺4
5 edge-disjoint ascending paths
𝐺3
4 edge-disjoint ascending paths
𝐺2
3 edge-disjoint ascending paths
𝐺1
2 edge-disjoint ascending paths
𝐺 = 𝐺0
1 ascending path
5 edge-disjoint
𝑃𝑒
paths from u to v 𝑃
𝑑
Newly created edgedisjoint ascending paths
𝑃𝑐
𝑢
𝑃𝑏
𝑃𝑎
𝑣
⋮
⋮
𝐺5
add
add
add
add
add
5 edge-disjoint ascending paths
𝐺4
add
add
add
add
add
5 edge-disjoint ascending paths
𝐺3
add
add
add
add
add
4 edge-disjoint ascending paths
𝐺2
add
add
add
add
add
3 edge-disjoint ascending paths
𝐺1
add
add
add
add
add
add
add
add
add
add
2 edge-disjoint ascending paths
1 ascending path
𝐹0
𝐹1
𝐺=
𝐹2 𝐺0
𝐹3
𝐹4
𝐺0
𝐸 𝐺
⋮
⋮
⋮
Newly created edgedisjoint ascending paths
for any two edges 𝑒 ∈ 𝐹𝑖 and 𝑒 ′ ∈ 𝐹𝑗 , 𝜌 𝑒 < 𝜌(𝑒 ′ ) if 𝑖 < 𝑗
5 edge-disjoint 𝑃𝑒
paths from u
𝑃𝑑
to v
𝑢 𝑃𝑐
𝑃𝑏
𝑃𝑎
𝑣
The sum of the
folded numbers
of edge-disjoint
paths = 10
𝐸 𝐺
𝑢
⋮
⋮
⋮
⋮
⋮
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
add
𝐹0
𝐹1
𝐹2
𝐹3
𝐹4
When 𝑘 =7
8
It is sufficient to add
the subsets k+q+1
times, where q is the
sum of folded numbers
of the edge-disjoint
paths from u to v.
𝑣
Theorem 1:
Let (𝐺, 𝜌) be an ordered graph with 𝑛 vertices. Suppose that
– 𝐸(𝐺) can be decomposed into 𝑙 subsets 𝐹0 , 𝐹1 , ⋯ , 𝐹𝑙−1 such that
for any two edges 𝑒 ∈ 𝐹𝑖 and 𝑒 ′ ∈ 𝐹𝑗 , 𝜌 𝑒 < 𝜌(𝑒 ′ ) if 𝑖 < 𝑗 ;
– for any two vertices 𝑢 and 𝑣, there are 𝑝 edge-disjoint paths from
𝑢 to 𝑣 such that the sum of their folded numbers is 𝑞, and the last
edges of 𝑟𝑖 paths are in 𝐹𝑖 for 0 ≤ 𝑖 < 𝑙.
Then,
𝜏 𝑛, 𝑘 ≤
0≤𝑖≤𝑤 |𝐹𝑖 𝑚𝑜𝑑 𝑙
where 𝑤 is an integer satisfying
0≤𝑖≤𝑤
𝑟𝑖 𝑚𝑜𝑑 𝑙 ≥ 𝑘 + 𝑞 + 1.
|,
Fault-Tolerant Gossip Graphs
Based on Hypercubes
𝑄𝑝 : 𝑝-dimensional hypercube
• 𝑉 𝑄𝑝 =
𝑢1 , 𝑢2 , ⋯ , 𝑢𝑝
𝑢𝑖 = 0 𝑜𝑟 1, 1 ≤ 𝑖 ≤ 𝑝 }
• 𝐸 𝑄𝑝 =
The dimension of an edge
is defined to be j.
𝑢1 , 𝑢2 , ⋯ , 𝑢𝑝 𝑣1 , 𝑣2 , ⋯ , 𝑣𝑝
𝑢𝑗 ≠ 𝑣𝑗 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑗 𝑎𝑛𝑑
𝑢𝑖 = 𝑣𝑖 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑖 ≠ 𝑗}
• The edge-ordering 𝜌 is defined so that for any two edge 𝑒 and 𝑒′, if the
dimension of 𝑒 is smaller than that of 𝑒′, then 𝜌 𝑒 < 𝜌(𝑒 ′ ).
1
Q1 :
Q2 :
0
1
1
3
Q3 :
00
1
3
01
2
10
000
4
9
11
001
5
010
10
6
011
2
100
7
11
110
12
8
101
4
111
𝑄4 :
9
𝑢 = 0000,
𝑣 = 1001
Haddad et al. showed that
in 𝑄𝑝 for any ordered pair of
vertices u and v, there are 𝑝
edge-disjoint paths from u to v
such that their folded
numbers are at most one.
11
0000
17
0010
0100
25
3
26
27
0001
0011
1010
12
1100
29
30
31
22
1001
15
23
1110
8
4
We can easily check that the
sum of their folded numbers is
𝑝 − 1.
0111
0101
13
21
7
20
10
1000
0110
28
5
18
1
2
19
6
32
24
1011
14
1111
1101
16
Applying Theorem 1 to Hypercube
• Let 𝐹𝑖 be the set of edges with dimension 𝑖, i.e., 𝐹𝑖 =
𝑛
𝑝−1
2
= .
2
• There exists exactly one path whose last edge is in 𝐹𝑖 , i.e.,
𝑟𝑖 = 1 for 1 ≤ 𝑖 ≤ 𝑛.
• The sum of the folded numbers is 𝑝 − 1.
𝑛
2
• 𝜏 𝑛, 𝑘 ≤ 0≤𝑖≤𝑤 |𝐹𝑖 𝑚𝑜𝑑 ℓ | = 𝑤 + 1 .
• 𝑤 is an integer satisfying 0≤𝑖≤𝑤 𝑟𝑖 𝑚𝑜𝑑 ℓ ≥ 𝑘 + 𝑞 + 1.
•
0≤𝑖≤𝑤 1
= 𝑤 + 1 ≥ 𝑘 + 𝑝 − 1 + 1 = 𝑘 + 𝑝 = 𝑘 + log 2 𝑛 .
Therefore, 𝜏 𝑛, 𝑘 ≤
𝑛
2
𝑤+1 =
𝑛(𝑘+log2 𝑛)
.
2
General Case
• The ℎ, 𝑝 -hypercube 𝑄ℎ,𝑝 is defined as the graph obtained ℎ copies
of 𝑄𝑝 by selecting one vertex from each 𝑄𝑝 and identifying such ℎ
vertices as a single vertex 𝑥 called the center vertex.
𝑛−1
• By letting ℎ = 𝑝 , 𝑛′ = 𝑉 𝑄ℎ,𝑝 ≥ 𝑛.
2 −1
• Since 𝜏 𝑛, 𝑘 ≤ 𝜏 𝑛′ , 𝑘 , we consider edge-disjoint paths in 𝑄
𝑄𝑝
𝑄4,𝑝 :
𝑄𝑝
𝑥
𝑄𝑝
𝑄𝑝
𝑛−1
2𝑝 −1
,𝑝 .
Edge-Disjoint Paths in 𝑄
𝑛−1
2𝑝 −1
,𝑝
• Let 𝑢, 𝑣 ∈ 𝑉(𝑄 𝑛−1 ,𝑝 ) such that 𝑢 and 𝑣 are in distinct copies of
2𝑝 −1
𝑄𝑝 .
• Construct edge-disjoint paths from 𝑢 to 𝑣 by concatenating the
edge-disjoint paths from 𝑢 to the center vertex 𝑥 in the copy of
𝑄𝑝 and the edge-disjoint paths from 𝑥 to 𝑣 in the copy of 𝑄𝑝 .
• Let 𝑃𝑖 (resp., 𝑃′𝑖 ) be the path from 𝑢 to 𝑣 whose last edge (resp.,
the path from 𝑥 to 𝑣 whose first edge) has the dimension 𝑖.
• Define
–
–
–
–
𝑅1 = 𝑃1 ⊙ 𝑃′2
𝑅2 = 𝑃2 ⊙ 𝑃′3
⋮
𝑅𝑝−1 = 𝑃𝑝−1 ⊙ 𝑃′𝑝
– 𝑅𝑝 = 𝑃𝑝 ⊙ 𝑃′1
𝑄𝑝
𝑢
𝑄𝑝
𝑥
𝑣
Applying Theorem 1 to 𝑄
𝑛−1
2𝑝 −1
,𝑝
𝑛−1
• Let 𝐹𝑖 be the set of edges with dimension 𝑖, i.e., 𝐹𝑖 = 𝑝
∙ 2𝑝−1 .
2 −1
• There exists exactly one path whose last edge is in 𝐹𝑖 , i.e., 𝑟𝑖 = 1 for 1 ≤ 𝑖 ≤
𝑝.
• The sum of the folded numbers is at most 2𝑝 − 1.
𝑛−1
• 𝜏 𝑛′, 𝑘 ≤ 0≤𝑖≤𝑤 |𝐹𝑖 𝑚𝑜𝑑 ℓ | = 𝑝
∙ 2𝑝−1 𝑤 + 1 .
2 −1
• 𝑤 is an integer satisfying 0≤𝑖≤𝑤 𝑟𝑖 𝑚𝑜𝑑 ℓ ≥ 𝑘 + 𝑞 + 1.
• 𝑤 + 1 = 0≤𝑖≤𝑤 1 ≥ 𝑘 + 2𝑝 − 1 + 1 = 𝑘 + 2𝑝.
Therefore,
𝜏 𝑛′, 𝑘 ≤
𝑛−1
𝑘
𝑝−1
∙
2
𝑘
+
2𝑝
≤
+𝑝
𝑝
2 −1
2
𝑛−1 +
𝑛−1
+ 2𝑝 .
𝑝
2 −1
Fault-Tolerant Gossip Graphs Based on
Circulant Graphs
• Let 𝑛 be an integer which is not a power of two.
The 2 log 2 𝑛 -regular graph 𝑅 𝑛 is defined as follows:
• 𝑉 𝑅 𝑛
= 0,1, ⋯ , 𝑛 − 1
• 𝐸 𝑅 𝑛
=
𝑢, 𝑣
𝑣 ≡ 𝑢 + 2𝑖 𝑚𝑜𝑑 𝑛 , 0 ≤ 𝑖 ≤ log 2 𝑛 − 1}.
Fault-Tolerant Gossip Graphs Based on
Circulant Graphs
• The span of an edge 𝑢, 𝑣 , where 𝑣 ≡ 𝑢 + 2𝑖 𝑚𝑜𝑑 𝑛 , is defined
to be 𝑖.
• The edge-ordering 𝜌 is defined so that for any two edge 𝑒 and 𝑒′,
if the span of 𝑒 is greater than that of 𝑒′, then 𝜌 𝑒 < 𝜌(𝑒′).
The edges with span 0
The edges with span 1
The edges with span 2
The edges with span 3
Edge-Disjoint Paths in 𝑅(𝑛)
Case
The set of 𝟐 log 𝟐 𝒏 edge-disjoint paths
𝑙 ≥ 2 and 𝑡1 = log 2 𝑛 − 1
𝑆 𝑃∗ ∪ 𝑂𝑃∗ ∪ {𝑀𝑃∗ }
𝑙 ≥ 2, 𝑡1 ≤ log 2 𝑛 − 2, and 𝑣 ≠
2𝑡𝑙 (2𝑡1 −𝑡𝑙 +1 − 1)
𝑆 𝑃∗ ∪ 𝐽𝑃∗ ∪ {𝑀𝑃∗ }
𝑙 ≥ 2, 𝑡1 = log 2 𝑛 − 2, and 𝑣 =
2𝑡𝑙 (2𝑡1 −𝑡𝑙 +1 − 1)
(𝑆 𝑃′ − {𝐿𝑡𝑙−1 𝑃′ } ∪ 𝑂𝑃∗ ∪ {𝐷𝑃∗ }
∪ {𝐿𝑡𝑙−1 𝐶𝑃∗ }
𝑙 ≥ 2, 𝑡1 ≤ log 2 𝑛 − 3, and 𝑣 =
2𝑡𝑙 (2𝑡1 −𝑡𝑙 +1 − 1)
(𝑆 𝑃′ − {𝐿𝑡𝑙−1 𝑃′ } ∪ 𝐽𝑃∗ ∪ {𝐷𝑃∗ }
∪ {𝐿𝑡𝑙−1 𝐶𝑃∗ }
𝑙 = 1 and 𝑡1 = log 2 𝑛 − 1
𝑆 𝑃∗ ∪ 𝑂𝑃∗
𝑙 = 1 and 𝑡1 = log 2 𝑛 − 2
𝑆 𝑃∗ ∪ 𝑂𝑃∗∗
𝑙 = 1 and 𝑡1 ≤ log 2 𝑛 − 3
𝑆 𝑃∗ ∪ 𝑊𝑃∗
Applying Theorem 1 to 𝑅(𝑛)
• Let 𝐹𝑖 be the set of edges with span log 2 𝑛 − 𝑖 − 1, i.e., 𝐹𝑖 = 𝑛.
• There are exactly two paths whose last edge is in 𝐹𝑖 , i.e., 𝑟𝑖 = 2 for 0 ≤
𝑖 ≤ log 2 𝑛 − 1.
• The sum of the folded numbers is at most 2 2 log 2 𝑛 − 1 .
• 𝜏 𝑛, 𝑘 ≤ 0≤𝑖≤𝑤 |𝐹𝑖 𝑚𝑜𝑑 ℓ | = 𝑛 𝑤 + 1 .
• 𝑤 is an integer satisfying 0≤𝑖≤𝑤 𝑟𝑖 𝑚𝑜𝑑 ℓ ≥ 𝑘 + 𝑞 + 1.
• 2 𝑤 + 1 = 0≤𝑖≤𝑤 2 ≥ 𝑘 + 4 log 2 𝑛 − 1.
Therefore,
𝑘−1
𝜏 𝑛, 𝑘 ≤ 𝑛 𝑤 + 1 = 2𝑛 log 2 𝑛 + 𝑛
.
2
• When 𝑘 is even, we can slightly improve this bound since by adding at
2𝑛
most edges in 𝐹(𝑘 2−1) 𝑚𝑜𝑑 log2 𝑛 , we can construct k+1 edge-disjoint
3
ascending paths.
𝑘-Fault Tolerant Broadcast Graph
• A 𝑘-fault-tolerant broadcast graph is an ordered graph in
which there are 𝑘 + 1 edge-disjoint ascending paths from one
vertex to any other vertex.
• Let 𝜇(𝑛, 𝑘) be the minimum number of edges in a 𝑘-faulttolerant broadcast graph.
Theorem 2 (Berman and Hawrylycz)
𝜇 𝑛, 𝑘 =
𝑘+2
𝑛−1
2
𝑘+1
𝑛
2
𝑖𝑓 𝑘 ≤ 𝑛 − 2
𝑖𝑓 𝑘 > 𝑛 − 2
Broadcast Number
• The 𝑣-broadcast number of (𝐺, 𝜌) is the number of vertices to
which there is an ascending path from 𝑣.
• The broadcast number of (𝐺, 𝜌) is the minimum 𝑣-broadcast
number over all vertices of (𝐺, 𝜌) .
Theorem 3 (Berman and Paul)
The broadcast number of any ordered tree (𝑇, 𝜌) is at most log 2 𝑛 .
𝑥
∃𝑥, the 𝑥-broadcast number is at most log 2 𝑛
Defect Number
• For an ordered graph 𝐻, 𝜌𝐻 , let 𝜂(𝐻,𝜌𝐻 ) (𝑢, 𝑣) be the number of
ascending paths from 𝑢 to 𝑣.
• Define the defect number 𝜓 𝐻,𝜌 𝑣; 𝑘 of 𝑣 with respect to 𝑘 as
𝜓 𝐻,𝜌𝐻 𝑣; 𝑘 = max { 0 , 𝑘 + 1 − min𝑢∈𝑉 𝐻 − 𝑣 𝜂 𝐻,𝜌𝐻 𝑢, 𝑣 }
• Suppose that 𝐺, 𝜌 is a k-fault-tolerant gossip graph with 𝑛 vertices,
where 𝑛 ≥ 3.
• Let 𝐺, 𝜌 𝑖 be the spanning subgraph of 𝐺, 𝜌 having all the edges
of order at most 𝑖.
• By Berman and Hawrylycz’s Theorem, we can see that
for any vertex 𝑣 in 𝐺, 𝜌 3(𝑛−1) −1 , there exists a vertex 𝑢 such that
2
there is at most one ascending path from 𝑢 to 𝑣, i.e.,
𝜂 𝐺,𝜌 3
(𝑢, 𝑣) ≤ 1 , thus, 𝜓 𝐺,𝜌 3
𝑣; 𝑘 ≥ 𝑘.
2(𝑛−1) −1
• Therefore, 𝜓
2(𝑛−1) −1
𝐺,𝜌 3
2(𝑛−1) −1
𝑣; 𝑘 = 𝑘 𝑜𝑟 𝑘 + 1.
• Let 𝑥 be a vertex whose broadcast number is equal to the
broadcast number of 𝐺, 𝜌 𝑛−1 .
• Let 𝐵𝑖 (𝑥) be the set of vertices to which there is an
ascending path from 𝑥 in 𝐺, 𝜌 𝑖 .
• By Berman and Paul’s Theorem, |𝐵𝑛−1 𝑥 | ≤ log 2 𝑛 .
• Adding the edge with order 𝑖 + 1 to 𝐺, 𝜌 𝑖 , at most one
vertex newly receive the message originated from 𝑥, thus,
|𝐵𝑖+1 𝑥 | ≤ |𝐵𝑖 𝑥 | + 1.
𝐵𝑖 (𝑥)
𝑖
• Therefore, |𝐵 3(𝑛−1) −1 𝑥 | ≤ log 2 𝑛 +
2
3
(𝑛
2
− 1) − 𝑛.
• 𝜓
•
𝐺,𝜌 3
2(𝑛−1) −1
𝑣; 𝑘 = 𝑘 𝑜𝑟 𝑘 + 1.
|𝐵 3(𝑛−1) −1 𝑥 | ≤ log 2 𝑛 +
2
3
(𝑛
2
− 1) − 𝑛
• Let 𝑛𝑘 be the number of vertices with defect number 𝑘 in
𝐺, 𝜌 3(𝑛−1) −1.
2
• Since any vertex with defect number 𝑘 is in 𝐵 3(𝑛−1) −1 𝑥 ∪ 𝑥 ,
2
3
2
𝑛𝑘 ≤ log 2 𝑛 +
𝑛 − 1 − 𝑛 + 1.
• For any vertex 𝑤 in 𝐺, 𝜌 3(𝑛−1) −1, if the defect number of w is 𝑘
2
(resp., 𝑘 + 1), then there must exist at least 𝑘 (resp. 𝑘 + 1) edges
3
with order > (𝑛 − 1) − 1 which are incident to 𝑤 in (𝐺, 𝜌).
2
• Therefore, 𝐸 𝐺
≥
3
2
𝑛−1
−1+
1
(𝑘𝑛𝑘
2
+ (𝑘 +
Conclusion
• We have refined a general method by Haddad et al. for constructing
fault-tolerant gossip graphs.
• Applying the result to edge-disjoint paths in ((ℎ, 𝑝)-)hypercubes, we
have improved the upper bounds by Haddad et al on 𝜏(𝑛, 𝑘).
• We have presented edge-disjoint paths whose folded numbers are
at most two in circulant graphs, from which we obtain an upper
𝑛𝑘
bound of + 𝑂(𝑛 log 𝑛) on 𝜏(𝑛, 𝑘).
2
• We have shown a new lower bound on 𝜏 𝑛, 𝑘 , which improves the
𝑛
previously known lower bounds when 𝑘 > and 𝑛 > 4.
2
•
𝑛𝑘
2
+ 𝑂 𝑛 ≤ 𝜏 𝑛, 𝑘 ≤
• Problem: 𝜏 𝑛, 𝑘 =
𝑛𝑘
2
𝑛𝑘
2
+ 𝑂(𝑛 log 𝑛)
+Θ 𝑛 ?