Improved Bounds for Minimum Fault-Tolerant Gossip Graphs

1. 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

2. 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

3. 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 for . • Tijdeman [1971] • Baker and Shostak [1972]

4. 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 be the minimum number of calls for the fault-tolerant gossiping problem on persons with at most failed calls.

5. -Fault-Tolerant Gossip Graphs • The -fault tolerant gossiping problem can be modeled by a (multiple) graph with edge-ordering . • Define a -fault-tolerant gossip graph as an ordered graph in which for any ordered pair of distinct vertices and , there are ascending paths from to . • The size of a -fault-tolerant gossip graph is an upper bound on 32 19 3 31 13 36 20 2 4 12 1 14 21 An example of a 2-fault-tolerant gossip graph 23 5 30 15 22 18 10 28 34 29 6 33 11 27 25 18 26 9 7 8 24 35 17 16

6. Comparison of Upper Bound Results

7. Comparison of Lower Bound Results

8. Construction of Fault-Tolerant Gossip Graphs • The general method by Haddad et al. • Cumulate the edge set of a copy of an ordered graph iteratively.

9. Folded Number • A path from to in an ordered graph is called -folded if it is a series of (maximal) ascending paths. • The folded number of an -folded ascending path is defined to be . -folded ascending path

10. Edge-Disjoint Ascending Paths in If there is an -folded ascending path from to , then there are edge-disjoint ascending paths from to in for any integer . 3 edge-disjoint ascending path from to in P6 P5 7-4 = 3 P4 P3 P2 4 P1 P = P0

11. Lemma 1: If there are edge-disjoint -folded ascending paths from to , then there are edge-disjoint ascending paths from to in for any integer .

12. Lemma 1: If there are edge-disjoint -folded ascending paths from to , then there are edge-disjoint ascending paths from to in for any integer . Corollary 1: Let be an ordered graph with vertices and edges. If for any ordered pair of vertices and , there are edge-disjoint paths from to in such that their folded numbers are at most , then is a -fault-tolerant gossip graph, thus

13. 5 edge-disjoint ascending paths When k =7, i.e., we need 8 edge-disjoint ascending paths , Corollary 2 consider However, it is sufficient to consider . 5 edge-disjoint ascending paths 4 edge-disjoint ascending paths 3 edge-disjoint ascending paths 2 edge-disjoint ascending paths 1 ascending path 5 edge-disjoint paths from u to v Newly created edge- disjoint ascending paths

14. add add add add add 5 edge-disjoint ascending paths add add add add add 5 edge-disjoint ascending paths add add add add add 4 edge-disjoint ascending paths 3edge-disjoint ascending paths add add add add add add add add add add 2 edge-disjoint ascending paths add add add add add 1 ascending path Newly created edge- disjoint ascending paths for any two edges and , if 5 edge-disjoint paths from u to v

15. add add add add add The sum of the folded numbers of edge-disjoint paths = 10 add add add add add When 7 add add add add add add add add add add add add add add add add add add add add 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.

16. Theorem 1: Let be an ordered graph with vertices. Suppose that • can be decomposed into subsets such that for any two edges and , if ; • for any two vertices and , there are edge-disjoint paths from to such that the sum of their folded numbers is , and the last edges of paths are in for . Then, , where is an integer satisfying

17. Fault-Tolerant Gossip Graphs Based on Hypercubes : -dimensional hypercube • The edge-ordering is defined so that for any two edge and , if the dimension of is smaller than that of , then . The dimension of an edge is defined to be j. 1 3 Q1: Q2: Q3: 1 110 010 100 5 000 0 7 00 10 1 10 11 9 3 4 12 6 2 1 8 01 11 111 001 011 101 2 4

18. 9 11 17 19 28 3 27 5 25 26 7 Haddad et al. showed that in for any ordered pair of vertices u and v, there are edge-disjoint paths from u to v such that their folded numbers are at most one. We can easily check that the sum of their folded numbers is . 18 20 1 10 12 15 13 21 23 2 8 4 1110 0110 1010 0010 6 0100 1100 1000 0000 29 30 31 32 22 24 14 16 1111 0111 0001 1001 1011 0011 0101 1101

19. Applying Theorem 1to Hypercube • Let be the set of edges with dimension , i.e., • There exists exactly one path whose last edge is in , i.e., for . • The sum of the folded numbers is . • is an integer satisfying Therefore,

20. General Case • The -hypercube is defined as the graph obtained copies of by selecting one vertex from each and identifying such vertices as a single vertex called the center vertex. • By letting , . • Since , we consider edge-disjoint paths in .

21. Edge-Disjoint Paths in • Let such that and are in distinct copies of . • Construct edge-disjoint paths from to by concatenating the edge-disjoint paths from to the center vertex in the copy of and the edge-disjoint paths from to in the copy of . • Let (resp., ) be the path from to whose last edge (resp., the path from to whose first edge) has the dimension . • Define

22. Applying Theorem 1to • Let be the set of edges with dimension , i.e., • There exists exactly one path whose last edge is in , i.e., for . • The sum of the folded numbers is at most . • is an integer satisfying Therefore,

23. Fault-Tolerant Gossip Graphs Based on Circulant Graphs • Let be an integer which is not a power of two. The-regular graph is defined as follows:

24. Fault-Tolerant Gossip Graphs Based on Circulant Graphs • The span of an edge where , is defined to be . • The edge-ordering is defined so that for any two edge and , if the span of is greater than that of , then . The edges with span 0 The edges with span 1 The edges with span 2 The edges with span 3

25. Edge-Disjoint Paths in

26. Applying Theorem 1to • Let be the set of edges with span , i.e., • There are exactly two paths whose last edge is in , i.e., for . • The sum of the folded numbers is at most . • is an integer satisfying Therefore, • When is even, we can slightly improve this bound since by adding at most edges in , we can construct k+1 edge-disjoint ascending paths.

27. -Fault Tolerant Broadcast Graph • A -fault-tolerant broadcast graph is an ordered graph in which there are edge-disjoint ascending paths from one vertex to any other vertex. • Let be the minimum number of edges in a -fault-tolerant broadcast graph. Theorem 2 (Berman and Hawrylycz)

28. Broadcast Number • The -broadcast number of is the number of vertices to which there is an ascending path from . • The broadcast number of is the minimum -broadcast number over all vertices of . Theorem 3 (Berman and Paul) The broadcast number of any ordered tree is at most . the -broadcast number is at most

29. Defect Number • For an ordered graph let be the number of ascending paths from to . • Define the defect number of with respect to as • Suppose that is a k-fault-tolerant gossip graph with vertices, where • Let be the spanning subgraph of having all the edges of order at most . • By Berman and Hawrylycz’s Theorem, we can see that for any vertex in , there exists a vertex such that there is at most one ascending path from to , i.e., , thus, . • Therefore, .

30. Let be a vertex whose broadcast number is equal to the broadcast number of . • Let be the set of vertices to which there is an ascending path from in . • By Berman and Paul’s Theorem, . • Adding the edge with order to , at most one vertex newly receive the message originated from , thus, • Therefore, .

31. . • Let be the number of vertices with defect number in . • Since any vertex with defect number is in , . • For any vertex in , if the defect number of w is (resp., ), then there must exist at least (resp. ) edges with order > which are incident to in . • Therefore, . • Substituting the upper bound on , we obtain .

32. 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 (-)hypercubes, we have improved the upper bounds by Haddad et al on . • We have presented edge-disjoint paths whose folded numbers are at most two in circulant graphs, from which we obtain an upper bound of on . • We have shown a new lower bound on , which improves the previously known lower bounds when and . • Problem: