The Most Likely Object to be Seen Through a Window

Abstract

We study data structures to answer window queries using stochastic input sequences. The first problem is the most likely maximal point in a query window: Let [Formula: see text] be constants, with [Formula: see text]. Let [Formula: see text] be a set of [Formula: see text] points in [Formula: see text], for some fixed [Formula: see text]. For [Formula: see text], each point in [Formula: see text] is associated with a probability [Formula: see text] of existence. A point [Formula: see text] in [Formula: see text] is on the maximal layer of [Formula: see text] if there is no other point [Formula: see text] in [Formula: see text] such that [Formula: see text]. Consider a random subset of [Formula: see text] obtained by including, for [Formula: see text], each point of [Formula: see text] independently with probability [Formula: see text]. For a query interval [Formula: see text], with [Formula: see text], we report the point in [Formula: see text] that has the highest probability to be on the maximal layer of [Formula: see text] in [Formula: see text] time using [Formula: see text] space. We solve a special problem as follows. A sequence [Formula: see text] of [Formula: see text] points in [Formula: see text] is given ([Formula: see text]), where each point [Formula: see text] has a probability [Formula: see text] of existence associated with it. Given a query interval [Formula: see text] and an integer [Formula: see text] with [Formula: see text], we report the probability of [Formula: see text] to be on the maximal layer of [Formula: see text] in [Formula: see text] time using [Formula: see text] space. The second problem we consider is the most likely common element problem. Let [Formula: see text] be the universe. Let [Formula: see text] be a sequence of random subsets of [Formula: see text] such that for [Formula: see text] and [Formula: see text], element [Formula: see text] is added to [Formula: see text] with probability [Formula: see text] (independently of other choices). Let [Formula: see text] be a fixed real number with [Formula: see text]. For query indices [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], with [Formula: see text] and [Formula: see text], we decide whether there exists an element [Formula: see text] with [Formula: see text] such that [Formula: see text] in [Formula: see text] time using [Formula: see text] space and report these elements in [Formula: see text] time, where [Formula: see text] is the size of the output.