Gnome Sort: coordinate compression and a Fenwick tree¶
Problem setter's solution: vector ranks, Fenwick inversion counting and prefix-minimum records. Codex (GPT-6) supplies the proof and complete C++17 program. This matches merge counting asymptotically but uses a different data structure. Both are parallel options; merge counting remains the reference.
Ranks and an input-order scan¶
Sort the two-dimensional vectors lexicographically to assign unique ranks \(r_i\) , mapped back to their original positions. Equal first coordinates require comparison of the second; coordinate sums cannot replace lexicographic order. Distinct vectors ensure unique ranks.
Scan in original input order. A Fenwick tree stores frequencies of processed ranks. Before processing vector \(i\) , there are \(i-1\) processed vectors, so
counts earlier larger ranks, exactly the inversions whose right endpoint is this position. Accumulate it, then insert the current rank, obtaining \(I=\sum_i d_i\) .
A separate scan counts strict prefix-minimum records \(R\) : count the first vector, and thereafter count a vector only if it is smaller than the current minimum.
From inversions to comparisons¶
Use the movement proof in the main editorial: each adjacent swap removes one inversion, so there are \(I\) successful checks and \(I+n-R\) failed checks. Their sum is
The Fenwick tree counts each inversion once by its right endpoint, and the prefix scan gives \(R\) exactly, proving the answer. Use 64-bit integers for inversion counts and the final result.
Complexity and tradeoffs¶
Sorting and Fenwick operations take \(O(n\log n)\) time; vectors, ranks and the tree use \(O(n)\) space. These bounds match merge counting. This version turns counting into prefix-frequency queries during a scan, which also provides a natural basis for rank statistics with insertions. It stores ranks and extra arrays and requires sorting for compression first. Equal asymptotic bounds do not imply equal actual speed; measurements give full-range time and memory behavior.
Same-input measurements¶
Both programs agree on all 10 formal inputs. Five alternating timing runs and input, source and binary hashes are in the full record; see the reference environment. Largest per-input median times are 0.142160 seconds for the reference and 0.148444 seconds for this option. Peak working sets are 12.79 and 18.53 MiB respectively, including process/runtime overhead. This implementation is below the 0.5-second indicative limit on every measured formal input. This conclusion concerns the supplied implementation, these inputs and the reference environment. Finite testing does not replace the proof or guarantee that every implementation with the same bound passes.