We consider the Unit Interval Selection problem in the one-pass random order streaming model. In this setting, an algorithm is presented with a sequence of n unit-length intervals on the line that arrive in uniform random order, one at a time, and the objective is to output (an approximation of) a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, within these space constraints, a (2/3)-approximation can be achieved in such streams, and this is best possible, in that going beyond such an approximation factor requires space Ω(n) [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, where the expectation is taken over the stream order. More specifically, we give a one-pass streaming algorithm with expected approximation factor 0.7401 that uses space O(|OPT|), where OPT denotes an optimal solution. We also show that random order algorithms with expected approximation factor above 8/9 require space Ω(n), and algorithms that compute a better than 2/3-approximation with probability above 2/3 also require Ω(n) space. On a technical level, we design an algorithm for the restricted domain [0, Δ), for some constant Δ, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain [0, Δ), we run O(Δ) recursive instances of our algorithm, with each instance targeting the situation where a specific interval of an optimal solution arrives first. We establish the interesting property of our algorithm that it performs worst when the input stream consists solely of a set of independent intervals. It then remains to analyse the algorithm on these simple instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds established by [Chakrabarti et al., Theory Comput. 2016].

We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the L most recent intervals, for some integer L. We give the following results: 1) In the unit-length intervals case, we give a 2-approximation sliding window algorithm with space Õ(|OPT|), and we show that any sliding window algorithm that computes a (2-ε)-approximation requires space Ω(L), for any ε > 0. 2) In the arbitrary-length case, we give a (11/3+ε)-approximation sliding window algorithm with space Õ(|OPT|), for any constant ε > 0, which constitutes our main result. We also show that space Ω(L) is needed for algorithms that compute a (2.5-ε)-approximation, for any ε > 0. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass 2-approximation streaming algorithm by Cabello and Pérez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a (4+ε)-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

We study the Set Cover problem in the one-pass edge-arrival streaming model. In this model, the input stream consists of a sequence of tuples (S, u), indicating that element u is contained in set S. This setting captures the streaming Dominating Set problem and is more general and harder to solve than the Set Cover set-arrival setting, where entire sets with all their elements arrive in the stream one-by-one. We prove the following results (n is the size of the universe, m is the number of sets):