Algorithm 1. FindPUGS algorithm: finding (nearly-optimal) prototypes using greedy search.

Result: Two ordered lists, N and R, of the minimum number of prototypes required for each circle and their rotations relative to the previous circle.

T ← the number of circles

c ← the length by which radii should grow

Algorithm FindPUGS (T, c)

N ← [1]

R ← [0]

for t = 1,2,…,T – 1 do

m ← N[−1];

n ← m + 1;

p ← 0;

while True do

$d_a\leftarrow \sqrt{2t^2{c}^2-2t^2{c}^2\cos (\frac{\mathrm{\pi }}{m})}$;

$d_b\leftarrow \sqrt{2(t+1{)}^2{c}^2-2(t+1{)}^2{c}^2\cos (\frac{\mathrm{\pi }}{n})}$;

for i = 0,1,…,4mn do

$\mathrm{\theta }\leftarrow \frac{i\mathrm{\pi }}{m\ast n\ast 16}$

if d1(t,c,m,n,θ) > db and d2(t,c,m,n,θ) > da then

$p\leftarrow n$

break

end

end

if p>0 then

N.append(p);

R.append(θ);

break;

end

$n\leftarrow n+1$

end

end

return N, R;

Procedure d1 (t,c,m,n,θ)

dists ← [];

for i = 0,…,m – 1 do

for j = 0,…,n − 1 do

dist $\leftarrow \sqrt{t^2{c}^2+(t+1{)}^2{c}^2-2t(t+1)c^2\cos (\frac{2i\mathrm{\pi }}{m}-\frac{2j\mathrm{\pi }}{n}-\frac{\mathrm{\pi }}{n}-\mathrm{\theta })}$;

dists.append(dist);

end

end

return min(dists);

Procedure d2 (t,c,m,n,θ)

dists ← [];

for i = 0,…,m − 1 do

for j = 0,…,n − 1 do

dist $\leftarrow \sqrt{t^2{c}^2+(t+1{)}^2{c}^2-2t(t+1)c^2\cos (\frac{2i\mathrm{\pi }}{m}-\frac{2j\mathrm{\pi }}{n}+\frac{\mathrm{\pi }}{m}-\mathrm{\theta })}$;

dists.append(dist);

end

end

return min(dists);