Restoring Minds after Swaps — Alternative Solution¶
This applies the combined-cycle construction of Evans, Huang, and Nguyen. Codex (GPT-6) adapted its cycle orientation and verified alternative.cpp.
Construction¶
Use the standard solution's body-to-mind cycles \(C_1,\ldots,C_c\) . For cycle \(a_1,\ldots,a_t\) , its rotated order is \(a_2,\ldots,a_t,a_1\) . Output zero if there are no nontrivial cycles. Otherwise:
- Swap \(y\) with the first vertex of \(C_c,\ldots,C_2\) .
- For \(C_1\) , swap \((x,a_1)\) , swap \(y\) along its rotated order, then swap \((x,a_2)\) .
- Swap \(x\) along each rotated order of \(C_2,\ldots,C_c\) .
- Swap \((x,y)\) .
Invariant and Bound¶
Step 1 chains the displaced minds. Step 2 restores the first cycle, leaves \(y\) holding \(x\)'s mind, and gives \(x\) the next cycle's second mind. Each traversal restores that cycle and collects the next needed mind. Finally \(x\) holds \(y\)'s mind; the last swap restores both. Pairs never repeat.
For \(s\) affected bodies, use \(s+c+2\le2N+1\) swaps, \(O(N+M)\) time, and \(O(N)\) space. This saves swaps when \(c\ge3\) . No global optimality for the particular history is claimed.