In this work, we study the notion of representation obliviousness in the context of pseudodeterministic streaming algorithms. A (randomized) streaming algorithm A is pseudodeterministic, if for every stream D, there is a ''canonical value'' g(D) so that with probability at least 2/3, A on input stream D outputs g(D). Intuitively, a randomized algorithm is representation oblivious if the output distribution of the algorithm does not depend on the representation of the input. We investigate this notion in the context of streaming algorithms, more specifically, distinct elements estimation (F 0 estimation) in data streams. In this context, representation obliviousness captures the idea that the output distribution of an algorithm for estimating F 0 should only depend on the set of distinct elements of the stream. This is a natural notion, as we note that standard streaming algorithms are representation-oblivious in this sense. We prove that any representation oblivious pseudodeterministic streaming algorithm for estimating F 0 must use Ω(n) space, where [n] is the universe. More generally, we prove that any representation oblivious pseudodeterministic t(n)-pass streaming algorithm requires Ω(n/t(n)) space. This lower bound matches the space requirement of the straightforward multi-pass deterministic algorithm that exactly computes F 0 .

Given a non-deterministic finite automaton (NFA) A with m states, and a natural number n (presented in unary), the #NFA problem asks to determine the size of the set L(A,n) of words of length n accepted by A. While the corresponding decision problem of checking the emptiness of L(A,n) is solvable in polynomial time, the #NFA problem is known to be #P-hard. Recently, the long-standing open question --- whether there is an FPRAS (fully polynomial time randomized approximation scheme) for #NFA --- was resolved by Arenas, Croquevielle, Jayaram, and Riveros in [ACJR19]. The authors demonstrated the existence of a fully polynomial randomized approximation scheme with a time complexity of ~O(m 17 n 17 • 1/ε 14 • log (1/δ)), for a given tolerance ε and confidence parameter δ. Given the prohibitively high time complexity in terms of each of the input parameters, and considering the widespread application of approximate counting (and sampling) in various tasks in Computer Science, a natural question arises: is there a faster FPRAS for #NFA that can pave the way for the practical implementation of approximate #NFA tools? In this work, we answer this question in the positive. We demonstrate that significant improvements in time complexity are achievable, and propose an FPRAS for #NFA that is more efficient in terms of both time and sample complexity. A key ingredient in the FPRAS due to Arenas, Croquevielle, Jayaram, and Riveros [ACJR19] is inter-reducibility of sampling and counting, which necessitates a closer look at the more informative measure --- the number of samples maintained for each pair of state q and length i <= n. In particular, the scheme of [ACJR19] maintains O(m 7 /n 7 ε 7 ) samples per pair of state and length. In the FPRAS we propose, we systematically reduce the number of samples required for each state to be only poly-logarithmically dependent on m, with significantly less dependence on n and ε, maintaining only ~O(n 4 /ε 2 ) samples per state. Consequently, our FPRAS runs in time ~O((m 2 n 10 + m 3 n 6 ) • 1/ε 4 • log 2 (1/δ)). The FPRAS and its analysis use several novel insights. First, our FPRAS maintains a weaker invariant about the quality of the estimate of the number of samples for each state q and length i <= n. Second, our FPRAS only requires that the distribution of the samples maintained is close to uniform distribution only in total variation distance (instead of maximum norm). We believe our insights may lead to further reductions in time complexity and thus open up a promising avenue for future work towards the practical implementation of tools for approximate #NFA.

In today's digital age, it is becoming increasingly prevalent to retain digital footprints in the cloud indefinitely. Nonetheless, there is a valid argument that entities should have the authority to decide whether their personal data remains within a specific database or is expunged. Indeed, nations across the globe are increasingly enacting legislation to uphold the "Right To Be Forgotten" for individuals. Investigating computational challenges, including the formalization and implementation of this notion, is crucial due to its relevance in the domains of data privacy and management. This work introduces a new streaming model: the 'Right to be Forgotten Data Streaming Model' (RFDS model). The main feature of this model is that any element in the stream has the right to have its history removed from the stream. Formally, the input is a stream of updates of the form (a, Δ) where Δ ∈ {+, ⊥} and a is an element from a universe U. When the update Δ=+ occurs, the frequency of a, denoted as f a , is incremented to f a +1. When the update Δ=⊥, occurs, f a is set to 0. This feature, which represents the forget request, distinguishes the present model from existing data streaming models. This work systematically investigates computational challenges that arise while incorporating the notion of the right to be forgotten. Our initial considerations reveal that even estimating F 1 (sum of the frequencies of elements) of the stream is a non-trivial problem in this model. Based on the initial investigations, we focus on a modified model which we call α-RFDS where we limit the number of forget operations to be at most α fraction. In this modified model, we focus on estimating F 0 (number of distinct elements) and F 1 . We present algorithms and establish almost-matching lower bounds on the space complexity for these computational tasks.

Given a family of sets (S1, S2,... SM) over a universe Ω, estimating the size of their union in the data streaming model is a fundamental computational problem with a wide variety of applications. The holy grail in the field of streaming is to seek design of algorithms that achieve (ε, δ)-approximation with poly(log |Ω|, ε-1, log δ-1) space and update time complexity.

In this paper we study the problem of estimating the size of the union of sets $S_1, \dots, S_M$ where each set $S_i \subseteq Ømega$ (for some discrete universe $Ømega$) is implicitly presented and comes in a streaming fashion. We define the notion of Delphic sets to capture class of streaming problems where membership, sampling, and counting calls to the sets are efficient. In particular, we show our notion of Delphic sets capture three well known problems: Klee's measure problem (discrete version), test coverage estimation, and model counting of DNF formulas. The Klee's measure problem corresponds to computation of volume of multi-dimension axis aligned rectangles, i.e., every d-dimension axis-aligned rectangle can be defined as $[a_1,b_1] \times [a_2,b_2] \times łdots \times [a_d, b_d]$. The problem of test coverage estimation focuses on the computation of coverage measure for a given testing array in the context of combinatorial testing, which is a fundamental technique in the context of hardware and software testing. Finally, given a DNF formula $\varphi = T_1 \vee T_2 \vee łdots \vee T_M$, the problem of model counting seeks to compute the number of satisfying assignments of $\varphi$. The primary contribution of our work is a simple and efficient sampling-based algorithm, called \hybrid, for estimating the of union of sets in streaming setting. Our algorithm has the space complexity of $O(Rłog |Ømega|)$ and update time is $O(Rłog R \cdot łog(M/δ) \cdot łog|Ømega|)$ where, $R = Ołeft(łog (M/δ)\cdot \varepsilon^2 \right).$ Consequently, our algorithm provides the first algorithm with linear dependence on d for Klee's measure problem in streaming setting for $d>1$, thereby settling the open problem of Tirthpura and Woodruff (PODS-12). Furthermore, a straightforward application of our algorithm lends to an efficient algorithm for coverage estimation problem in streaming setting. We then investigate whether the space complexity for coverage estimation can be further improved, and in this context, we present another streaming algorithm that uses near-optimal $O(tłog n/\varepsilon^2)$ space complexity but uses an update algorithm that is in $\rm P ^\rm NP $, thereby showcasing an interesting time vs space trade-off in the streaming setting. Finally, we demonstrate the generality of our Delphic sets by obtaining a streaming algorithm for model counting of DNF formulas. It is worth remarking that we view a key strength of our work is the simplicity of both the algorithm and its theoretical analysis, which makes it amenable to practical implementation and easy adoption.