On Approximate Colored Path Counting.

Given an ordinal tree T on n nodes in which each node is assigned a color from {0,1,… ,C-1} , an approximate colored path counting query asks for an approximation of the number, , of distinct colors assigned to nodes in a query path. We first present data structures that can compute a 2-approximate answer, i.e., a number in [, 2] , and achieve three different time/space trade-offs: i) an O(n)-word structure with O( ^λ n) query time for any constant 0<λ <1 , ii) an O(n n) -word structure with O( n) query time and iii) an O(n ^λ n) -word structure with O(1) query time. The first trade-off beats the O( n/ n) query time of the linear-word 2-approximate structure in previous work. We then design an O(n)-word structure which can compute in O(ϵ ^-2 n) time a (1±ϵ ) -approximate answer, i.e, a number in [(1-ϵ ), (1+ϵ ) ] , for any ϵ∈ (0,1) . Previously, when the space cost is O(n) words, the only known solution computes a (1±ϵ ) -approximate answer in O(ϵ ^-4 ^2 n) time with success probability no less than 1-δ , where δ is an arbitrary constant in (0, 1); our solution not only has faster query time but also always returns a (1±ϵ ) -approximation. When designing (1±ϵ ) -approximate solutions, our techniques also yield an O(n)-word structure that can answer a colored type-2 path counting query in O() time; this query reports the number of occurrences of each distinct color in a query path. This result improves the best previous linear-word solution in which the query time is O( n) .