# Path and Ancestor Queries on Trees with Multidimensional Weight Vectors

We consider an ordinal tree T on n nodes, with each node assigned a d-dimensional weight vector w∈{1,2,...,n}^d, where d ∈N is a constant. We study path queries as generalizations of well-known orthogonal range queries, with one of the dimensions being tree topology rather than a linear order. Since in our definitions d only represents the number of dimensions of the weight vector without taking the tree topology into account, a path query in a tree with d-dimensional weight vectors generalize the corresponding (d+1)-dimensional orthogonal range query. We solve ancestor dominance reporting problem as a direct generalization of dominance reporting problem, n)/( n)^d-2+k)in timeØ(^d-1n+k)n)^d-1/( n)^d-2) words, and space of Ø(n^d-2n) words, where k is the size of the output, for d ≥ 2. We also achieve a tradeoff of Ø(n^d-2+n) words of space, with query time of Ø((^d-1 n)/( n)^d-2+k), for the same problem, when d ≥ 3. We solve path successor problem in Ø(n^d-1n) words of space and time Ø(^d-1+n) for d ≥ 1 and an arbitrary constant > 0. We propose a solution to path counting problem, with Ø(n(n/n)^d-1) words of space and Ø((n/n)^d) query time, for d ≥ 1. Finally, we solve path reporting problem in Ø(n^d-1+n) words of space and Ø((^d-1n)/(n)^d-2+k) query time, for d ≥ 2. These results match or nearly match the best tradeoffs of the respective range queries. We are also the first to solve path successor even for d = 1.

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