The document discusses cyclic quorums, which are a type of quorum design where groups (quorums) are assigned to people such that certain intersection and membership properties are satisfied with minimal size k. A cyclic quorum imposes the additional restriction that the quorums have a cyclic property where members of each quorum are spaced cyclically. The document provides an example of an optimal cyclic quorum design and discusses techniques for performing exhaustive searches to find optimal cyclic quorum designs such as pruning and isomorphic rejection.

1. Cyclic Quorum
Wai-Shing Luk
luk@fudan.edu.cn

2. Introduction
• Originally proposed by [Luk and Wong 97]
• Now utilized in different areas:
– Power-saving protocols/wakeup scheduling
– Control channel establishment in cognitive radio
network
– Anti-jamming communication
– Suffix array construction

3. Quorum Design Problem
• N people are assigned to k groups (quorum)
such that:
1. For each pair of quorums, there is at least one
person who is a member of both groups (Non-
empty intersection property)
2. Each quorum contain exactly k people
3. Each person joins exactly k quorums
• Question:
– How the groups are designed with minimal k?

4. Example: N=8, k=4
• B0 = { 0, 1, 2, 4 }
• B1 = { 1, 2, 3, 5 }
• B2 = { 2, 3, 4, 6 } <--
• B3 = { 3, 4, 5, 7 }
• B4 = { 4, 5, 6, 0 }
• B5 = { 5, 6, 7, 1 } <--
• B6 = { 6, 7, 0, 2 }
• B7 = { 7, 0, 1, 3 }

5. Some Facts
• For a fixed k, the maximal value of N is k(k-1) +
1. Hence, the theoretical lower bound of k for
a given N is ~sqrt(N).
• If N = k(k-1)+1 and n=k-1 is prime power, the
solution can be constructed via a finite
projective plane of order n.
• In general, optimal solution is very hard to
find.

6. Cyclic Quorum
• Impose one more restriction: a set of quorums
has a cyclic property.
• In the previous example,
Bi = { 0+i, 1+i, 2+i, 4+i } modulo 8.
• Advantage: the whole solution can be
specified by just one single quorum. Hence,
easy to hard coded in hardware.
• Exhaustive search method is used for finding
the optimal solution.

7. Techniques for Exhaustive Search
• Pruning:
– Detect partial solutions which cannot generate
optimal solution.
• Isomorphic rejection:
– E.g. Let B1 = {a0, a1, …, an}. We can always assume
that a0 = 1 and a1 = 2.
– Also we can search only with ak floor(N/2)

8. Table of Optimal Cyclic Quorums