On a conjecture by Plaisted and Hong

JOURNAL

OF ALGOR~I’HMS

9,597-598 (1988)

NOTE On a Conjecture

by Plaisted and Hong

S. OLARIU, S. TOIDA, AND M. ZUBAIR Department of Computer Science, Old Dominion University, Norfolk, Virginia 23.508 Received December 7,1987; accepted March 7,1988

Recently, P&ted and Hong [l] proposed a heuristic algorithm to triangulate a set of n points in the Euclidian plane. Their main result states that the heuristic produces a triangulation within a ratio of O(log n) to the cost of an optimal triangulation. At the same time Plaisted and Hong [l] conjectured that their heuristic produces a triangulation which is, in fact, a constant ratio to the optimal triangulation. (For relevant definitions and terminology, the reader is referred to [l].) The purpose of this note is to disprove this conjecture: we note that if a set of points forms a convex polygon, then Plaisted and Hong’s algorithm simply performs the ring heuristic to obtain a triangulation. Moreover, for suitably chosen convex polygons, the ring heuristic turns out to be G(log n) to an optimal triangulation. To make our exposition self-contained we shall briefly sketch how to obtain for every n (n 2 4), a convex polygon which achieves this bound: consider a rectangle R.,,,,, with edges ab, bc, cd, da such that ab = cd = 2/97 and ad = bc = n. Next, with ab (cd) as a diameter, draw the circle C,, (Ccd), and consider the convex set defined by RabedW C,, U Cc,. Finally, place [n/2] points on the arc ab and the remaining [n/2] points on the arc cd. For convenience, we shall label the vertices clockwise as ~*=a

,...,

x,,,,~,=~,x,~,~,+~=c

,.,.,

x,=d.

By construction, we have xixi 2 n, whenever 1 I i I [n/2], in/21 + 1 I j _< n. Now a straightforward argument shows that the cost of the optimal triangulation satisfies EL (OPT(P)) (To see this, take arbitrary

triangulations

< 4n.

(1)

T,, Tz of the sets of points

597 Ol%-6774/88

$3.00

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598

OLARIU,

TOIDA,

AND ZUBAIR

P, = {x1,. . . , x,~,~]} and P2 = {~,~,~,+r,. . . , xn}, together with one of the edges ac or bd.) Similarly, it is easy to see that the cost of the triangulation obtained by the ring heuristic satisfies EL(RH(P))

2 n[log nj,

(4

since there are at least llog n] edges in the triangulation of length at least n. Now (1) and (2) combined show that the SI(log n) bound is tight. An interesting open question related to Plaisted and Hong’s paper can be stated as follows: is it the case that if in Plaisted and Hong’s algorithm an optimal triangulation is found for each convex polygon obtained in Step 1, then the global triangulation is within a constant factor from the optimal? ACKNOWLEDGMENT We thank the referee for many constructive comments which simplified our exposition.

REFERENCE 1. D. A. PLAISTED AND J. HONG, (1987), 405-437.

A heuristic triangulation algorithm, J. Algorithms

8, No. 3