Concept

Variance — where it appears

How widely a quantity spreads across repeated measurements that ought to agree. For a backtracking search the spread across branch orders can be wider than any difference between the problems, so a single cost is one draw from a distribution rather than a property of the sheet.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

what ten branch orders cost on the same four sheetsnodes of search; the sheets are the same drawings as the sweep abovethe square grid, 4×4, glued69 to 24636, 1 gave upthe square grid, 4×4, cut42 to 55the square grid, 3×3, glued20 to 731the square grid, 3×3, cut25 to 32each bar runs from the cheapest of 8 branch orders to the dearest, on a logarithmic scale; a dot is the middle one

The route, not the sheet

Every cost measured for a glued sheet has been one number from one branch order, and a backtracking search's cost belongs to the pair. Asked under eight orders instead of one, a cut cell's cost barely moves — 42 to 55 nodes — while the torus over the same drawing runs from 69 to 24,636, with one order giving up entirely. The glued sheet's best order costs less than twice the cut sheet's, so most of what a single order charged to the gluing belongs to the route through it.

complexity · Search order
nodes per free letter, cheapest route against the middle onecheapest of eightmiddle of eight× where no route of that kind finished · periods along the bottom0.3110100×2345the square grid, glued×123the triangular grid, glued××1234the honeycomb, glued0.3110100×123the elongated triangular tiling, glued××12the rhombille tiling, glued×123the rhombille tiling, cut

The cheapest route crosses later

A search for a consistent lettering has a threshold: below it the letters propagate and the cost is a third of a node per crease, above it the search backtracks and the cost explodes. The threshold was measured with one branch order. Measured with eight, the cheapest route never starts searching before the typical one, and on most sheets it starts a period or two later — so part of every threshold on the record belongs to the route. And the one cut sheet past its threshold, the rhombille's, spreads across nearly three orders of magnitude of cost, which moves the spread off the gluing and onto the threshold.

complexity · Search order

Named alongside it

The objects these essays reach for when they reach for this one.

BoundaryGluingMeasurementPatchSearchSearch cost

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