Combinatorial Design Generator
Combinatorial Design Generator generates a Balanced Incomplete Block Design (BIBD) given the number of elements (\\( v \\)), block size (\\( k \\)), and number of blocks (\\( b \\)). It visualizes the incidence structure and displays the blocks.
Methodology Used in Combinatorial Design Generator
A Balanced Incomplete Block Design (BIBD) is defined by parameters \\( (v, b, r, k, \lambda) \\), where \\( v \\) is the number of elements, \\( b \\) is the number of blocks, \\( r \\) is the number of blocks containing each element, \\( k \\) is the block size, and \\( \lambda \\) is the number of blocks containing each pair of elements. The generator ensures the design satisfies:
BIBD Conditions:
\\[ b \cdot k = v \cdot r \\] \\[ r \cdot (k – 1) = \lambda \cdot (v – 1) \\]Algorithm Steps:
1. Validate inputs \\( v \\), \\( k \\), and \\( b \\) against BIBD conditions.
2. Compute \\( r \\) and \\( \lambda \\) using the above equations.
3. Generate blocks using a combinatorial construction (e.g., for specific \\( v \\), \\( k \\), \\( b \\)).
4. Verify each element appears in \\( r \\) blocks and each pair in \\( \lambda \\) blocks.
5. Output the blocks and visualize the incidence structure.
Example Calculation
Sample Input
\\( v = 7 \\), \\( k = 3 \\), \\( b = 7 \\)
Compute \\( r \\) and \\( \lambda \\):
\\[ b \cdot k = v \cdot r \implies 7 \cdot 3 = 7 \cdot r \implies r = 3 \\] \\[ r \cdot (k – 1) = \lambda \cdot (v – 1) \implies 3 \cdot (3 – 1) = \lambda \cdot (7 – 1) \implies 6 = 6\lambda \implies \lambda = 1 \\]Generated Blocks (e.g., for \\( (7, 7, 3, 3, 1) \\)-BIBD):
\\[ \{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,6\}, \{2,5,7\}, \{3,4,7\}, \{3,5,6\} \\]Each element appears in 3 blocks, and each pair appears in 1 block.