Quantum Online Streaming Algorithms with Logarithmic Memory

Abstract

We consider online algorithms with respect to the competitive ratio. Here, we investigate quantum and classical one-way automata with non-constant size of memory (streaming algorithms) as a model for online algorithms. We construct problems that can be solved by quantum online streaming algorithms better than by classical ones in a case of logarithmic or sublogarithmic size of memory.

Classical and Quantum Computations with Restricted Memory

“…Moreover, if a quantum online streaming algorithm gets a single advice bit, then it becomes optimal. Theorem 14 (Khadiev et al [46], Khadiev et al [47]). There is an online minimization problem BHP with the following properties:…”

“…Similar supremacy is shown in the following result. Theorem 13 (Khadiev et al [46]). There is an online minimization problem BHR with the following properties:…”

“…Khadiev, Ziatdinov, Mannapov, Khadieva and Yamilov [46,47] showed that quantum online streaming algorithms can be better than classical ones in the case of sublogarithmic memory (see the following theorem). Theorem 12 (Khadiev et al [46], Khadiev et al [47]). There is an online minimization problem BHP with the following properties:…”

“…Let us consider the case of sublogarithmic memory. Khadiev, Ziatdinov, Mannapov, Khadieva and Yamilov [46,47] showed that quantum online streaming algorithms can be better than classical ones even if the classical algorithms get advice bits. Moreover, if a quantum online streaming algorithm gets a single advice bit, then it becomes optimal.…”

“…Let us again consider the case of polylogarithmic memory. Khadiev, Ziatdinov, Mannapov, Khadieva and Yamilov [46] and same authors with Kravchenko and Rivosh [45] considered this case. They showed that we have a situation similar to the sublogarithmic case.…”