Paired DNA Strands with Many Pair Types: Solution¶
Method and contributions¶
Codex (GPT-6) applies equivalence-class counting for the reverse-complement involution and evaluates the numerator modulo \(2P\) , making halving exact for every modulus. See std.cpp. Counting orbits of an involution is a standard method; the contributions here are its application, numerical handling, proof, and implementation.
Let \(q=2K\) be the alphabet size. A structure is represented by a string \(s\) , and exchanging strands gives its reverse complement \(R(s)\) . Since \(R(R(s))=s\) , an equivalence class has one or two representations.
Counting classes¶
There are \(q^N\) strings in total. For odd \(N\) , a fixed string would require its central letter to complement itself, contradicting the pairing definition. For even \(N\) , choosing the first half determines the second, giving exactly \(q^{N/2}\) fixed strings.
Each fixed string forms its own class; all other strings form pairs. The exact integer count is
Exact halving with any modulus¶
\(P\) may be one, even, or composite, so the implementation does not rely on an inverse of two. Let the integer numerator be \(A\) and compute \(r=A\bmod(2P)\) . Both \(A\) and \(2P\) are even, so \(r\) is even. From \(A=2Pt+r\) ,
The final value is already in \([0,P-1]\) and can be XORed into the aggregate.
For odd length, compute \(q^N\bmod(2P)\) . For even length, one exponentiation is enough: with \(h=q^{N/2}\bmod(2P)\) , the numerator residue is \((h^2+h)\bmod(2P)\) . Each query therefore needs one binary exponentiation.
Correctness and complexity¶
Classification into fixed strings and two-element classes proves the counting formula. The equation modulo \(2P\) proves that ordinary integer halving yields the correct residue modulo \(P\) . XORing those individual residues matches the required output.
Each query takes \(O(\log(N+1))\) time and \(O(1)\) auxiliary space, allowing streamed processing. Since \(2P\le2\times10^9\) , the largest multiplication-plus-addition is below \(4\times10^{18}\) and fits signed 64-bit integers. Factoring out \(K\) also avoids inverses, but is an equivalent numerical form of the same counting formula rather than a separate algorithm.