Lower bounds for testing Euclidean Minimum Spanning Trees

Abstract

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G = ( P , E ) on a set of points in the two-dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj et al. [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] gave a 1-sided-error non-adaptive property-tester for this task of query complexity O ˜ ( n ) . We show that every non-adaptive (not necessarily 1-sided-error) property-tester for this task has a query complexity of Ω ( n ) , implying that the test in [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] is of asymptotically optimal complexity. We further prove that every adaptive property-tester has query complexity of Ω ( n 1 / 3 ) . Those lower bounds hold even when the input graph is promised to be a bounded degree tree.