Quorum Systems Quorum minimum number of people who

Quorum Systems

Quorum minimum number of people who must be present at a meeting (of a committee, etc) before it can proceed and its decisions, etc can be considered valid -- Oxford Dictionary l a subset of an universal set U l n E. G. q 1={1, 2} and q 2= {2, 3} are quorums under U={1, 2, 3} 2

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Quorum systems l a collection C of subsets (called quorums) of U satisfying n l Intersection Property: q 1, q 2 : q 1, q 2 C: q 1 q 2 ; E. G. {{1, 2}, {2, 3}, {1, 3}} is a quorum system under U={1, 2, 3} 4

Coterie a small group of people with shared activities, interests, tastes, etc, esp one that tends to be exclusive -- Oxford Dictionary l A quorum system satisfying the minimality property: l n Minimality Property: q 1, q 2 : q 1, q 2 C: q 1 q 2. 5

Construction of Quorums l Majority Quorums: n Any set containing (n+1)/2 nodes is a quourm of a coterie n n=5, (n+1)/2 =3: {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5} and {3, 4, 5} are quorums. 6

Construction of Quorums l Tree Quorums n Structure: 1 2 4 3 5 6 7 7

Construction of Quorums l Tree Quorums of a coterie (root to leaf path): {1, 2 , 4}, {1, 2, 5}, {1, 3, 6}, {1, 3, 7}, {2, 4, 3, 6}, {2, 4, 3, 7}, {2, 5, 3, 6}, {2, 5, 3, 7}, {2, 4, 6, 7}, {2, 5, 6, 7}, {4, 5, 3, 6}, {4, 5, 3, 7}, {4, 5, 6, 7} 8

Construction of Quorums l Hierarchical Quorums: n Structure 1 2 3 4 5 6 7 8 9 9

Construction of Quorums l Hierarchical Quorums: {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3 , 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}, {4, 5, 7, 8}, {4, 5, 7, 9}, {4, 5, 8, 9}, {4, 6, 7, 8}, {4, 6, 7, 9}, {4, 6, 8, 9}, {5, 6, 7, 8}, {5, 6, 7, 9}, {5, 6, 8, 9}. {1, 2, 7, 8}, {1, 2, 7, 9}, {1, 2, 8, 9}, {1, 3, 7, 8}, {1, 3, 7, 9}, {1, 3, 8, 9}, {2, 3, 7, 8}, {2, 3, 7, 9}, {2, 3, 8, 9} 10

Construction of Quorums n Cohorts Quorums (level, crumblingwall, wheel, triangle, Lovas, …) n Structure one primary cohort n with supporting cohort at rear n e. g. quorums: {1, 2, 4, 5} {3, 4, 6} {5, 6} n 1 3 5 2 4 6 11

Construction of Quorums l Grid-based Quorums: n Structure: w columns by h rows grid 12

Construction of Quorums Mercedes structure: for BS channel allocation 13

Construction of Quorums Blossom structure: for BS channel allocation 14

Construction of Quorums l Geometric-Based Qorums: n Triangle (coterie: find a path from leftmost column to the bottommost row) 15

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