We modify the k-d tree on [0, 1] d by always cutting the longest edge instead of rotating through the coordinates. This modification makes the expected time behavior of lowerdimensional partial match queries behave as perfectly balanced complete k-d trees on n nodes. This is in contrast to a result of Flajolet and Puech [J. Assoc. Comput. Mach., 33 (1986), pp. 371-407], who proved that for (standard) random k-d trees with cuts that rotate among the coordinate axes, the expected time behavior is much worse than for balanced complete k-d trees. We also provide results for range searching and nearest neighbor search for our trees.

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Let T be a b-ary tree of height n, which has independent, non-negative, n identically distributed random variables associated with each of its edges, a model previously considered by Karp, Pearl, McDiarmid, and Provan. The value of a node is the sum of all the edge values on its path to the root. Consider the problem of finding the minimum leaf Ä 4 value of T . Assume that the edge random variable X is nondegenerate, has E X -ϱ for n Ä 4 some ) 2, and satisfies bP X s c -1 where c is the leftmost point of the support of X. We analyze the performance of the standard branch-and-bound algorithm for this problem Ž Ž .. n Ž . and prove that the number of nodes visited is in probability ␤ q o 1 , where ␤ g 1, b is a constant depending only on the distribution of the edge random variables. Explicit expres-Ž Ž .. n sions for ␤ are derived. We also show that any search algorithm must visit ␤ q o 1 nodes with probability tending to 1, so branch-and-bound is asymptotically optimal where first-order asymptotics are concerned.

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We consider random multivariate quadtries obtained from n points independently and uniformly distributed on the unit cube of R d. Let N n (y) be the complexity of the standard partial match algorithm for fixed vector y, where y is a vector in R s , 0 < s < d. We study N n = sup y N n (y), the worst-case time for partial match. Among other things, we show that partial match is very stable, in the sense that sup y N n (y)/ inf y N n (y) → 1 in probability.