Untangling Planar Curves

Abstract

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Θ(n 3/2 ) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n 2 ), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that Ω(n 3/2 ) facial electrical transformations are required to reduce any plane graph with treewidth Ω( n) to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires Θ(n 3/2 + nk + k 2 ) homotopy moves in the worst case. Finally, we prove that transforming one noncontractible closed curve to another on any orientable surface requires Ω(n 2 ) homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

Tightening Curves on Surfaces via Local Moves

“…Thus, this result extends and generalizes Chang and Erickson's Ω(n 2 ) lower bound for noncontractible curves on the torus [11]. This result is also applied in an upcoming companion paper [10] to derive a quadratic lower bound on the number of facial electrical transformations to reduce a 2-terminal plane graph in the worst case.…”

“…This quadratic bound has been reproved and generalized by several other authors [23,24,30,40,48,49]. Chang and Erickson recently improved the upper bound to O(n 3/2 ) and proved that such bound is optimal in the worst case [11].…”

“…Thus, any such curve can be simplified using O(n 2 ) moves following Steinitz's technique. Chang and Erickson proved a matching Ω(n 2 ) lower bound for non-contractible curves on surfaces with positive genus and no boundary [11]. de Graaf and Schrijver [28] proved that arbitrary curves on the annulus can be tightened using at most O(n 2 ) moves.…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (multi-)curve on any surface can be tightened using at most a quadratic number of moves [11].…”

“…For a more extended survey of the problem and its close relation to electrical transformations, see Chang and Erickson [11].…”