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For n points spread randomly over a square of side L, the optimal tour length approaches 0.7124·√(nL²). The minimum spanning tree gives a hard lower bound, and doubling it gives a constructive upper bound via the double-tree heuristic.
BHH asymptotic
L_opt ≈ 0.7124 · √(n · A) for n points in an area A
Tour count
(n − 1)! / 2 distinct tours on n cities
The estimate is 0.7124 × √100000 ≈ 225.3 units, and there are 181440 distinct tours.
Walking each tree edge twice visits every city and returns home, and shortcutting repeated visits can only shorten it.