lumpn / pairwise-voting

Pair generator for sparse pairwise voting

Pairwise voting

Unique pair generator for sparse pairwise voting.

Algorithm

The basic idea is to generate edge cycles over a fully connected graph of N nodes by generating edges by stepping through the nodes like a linear congruential generator using coprimes of N.

int previousIndex = 0;
for (int i = 1; i <= numNodes; i++)
{
    var index = (previousIndex + coprime) % numNodes;
    print("{0}, {1}", previousIndex, index);
    previousIndex = index;
}

Example

For a graph of 15 nodes, find coprimes 1, 2, 4, 7 and generate cycles:

coprime	n0	n1	n2	n3	n4	n5	n6	n7	n8	n9	n10	n11	n12	n13	n14	n15
1	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	0
2	0	2	4	6	8	10	12	14	1	3	5	7	9	11	13	0
4	0	4	8	12	1	5	9	13	2	6	10	14	13	7	11	0
7	0	7	14	6	13	5	12	4	11	3	10	2	9	1	8	0

All edges (n_i, n_i+1) are guaranteed to be unique across all generated cycles.

Comprime threshold

The reason we stop at coprime 7 in the example above is because while coprimes larger than N / 2 generate additional unique directed edges they don't generate unique undirected edges. E.g. coprime 8 would generate the edge (0, 8) in its first step but we already have the edge (8, 0) from the last step of comprime 7's cycle.

Graph radius

If our goal is to generate pairs that minimize the radius of the sparse graph of one cycle, then we should pick the largest coprime smaller than N / 2 first, because that connects the farthest nodes directly, halving the radius. Following a binary split pattern through the coprimes, we can cut the radius of the graph in half in each step.

Completeness

If N is prime then this algorithm finds all edges of the fully connected graph.

Proof: All numbers 1, 2, 3, ..., (N-1)/2 are coprimes of N and they are all smaller than N / 2. Therefore the algorithm generates cycles for each one of them, effectively generating (N-1)/2 cycles of length N, resulting in a total of N * (N-1)/2 unique edges. But that's exactly the number of edges in a fully connected graph. Therefore the algorithm finds all of them.

Finding an algorithm that generates all edges if N is not prime is left as an exercise to the reader. 😘