advance until done

Apply a sequential k-opt move described by chain = [t_1, t_2, ..., t_{2k}]. Edges x_i = (t_{2i-1}, t_{2i}) are removed, edges y_i = (t_{2i}, t_{2i

In-place recursive constructor on path[0..n] (n+1 elements). Keeps path[0] and path[n] fixed.

beta[i, j] = max edge weight on the unique i->j path in the MST spanning the n - 1 non-root nodes. Computed in O(n^2) with a single DFS per so

* Extra edges — for MST, convex hull, candidate forests, etc. * Each edge: [a, b] (0-indexed) or { a, b, color?, alpha?, width?, dash? }.

SOM neurons: draws the closed ring of neuron positions

partial path uses 0-indexed ids (used by branch & bound)

`tour` is a closed Hamiltonian tour: tour[0] == tour[-1], len = n+1.

Outer loop of the faithful LK search with don't-look bits. Each pass scans every node t_1 in turn; whenever an improving move is found from a

Run the faithful sequential k-opt search starting from a single t_1. Returns (best_route, best_gain, best_chain) — the best improving chain ro