Lower Bounds for Induced Cycle Detection in Distributed Computing

The distributed subgraph detection asks, for a fixed graph H, whether the n-node input graph contains H as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently. In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the n-node input graph contains H as an induced subgraph or not. We first show a ??(n) lower bound for detecting the existence of an induced k-cycle for any k ? 4 in the CONGEST model. This lower bound is tight for k = 4, and shows that the induced variant of k-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for 5 ? k ? 7, this ??(n) lower bound cannot be improved via the two-party communication framework. We then show how to prove stronger lower bounds for larger values of k. More precisely, we show that detecting an induced k-cycle for any k ? 8 requires ??(n^{2-?{(1/k)}}) rounds in the CONGEST model, nearly matching the known upper bound O?(n^{2-?{(1/k)}}) of the general k-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman [DISC 2019]. Finally, we investigate the case where H is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing

Distributed Quantum Interactive Proofs

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs. In this paper, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The first result of this paper shows that by using distributed quantum interactive proofs, the number of interactions can be significantly reduced. More precisely, our result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to three. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well. As a corollary of our results, we show that there exist 5-turn/3-turn distributed quantum interactive protocols with small certificate size for problems that have been considered in prior works on distributed interactive proofs such as [Kol, Oshman, and Saxena PODC 2018, Naor, Parter, and Yogev SODA 2020]. We then utilize the framework of the distributed quantum interactive proofs to test closeness of two quantum states each of which is distributed over the entire network

Brief Announcement: Distributed Quantum Interactive Proofs

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an n-node network G can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including G itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs. In this brief announcement, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The main result of this paper shows that by using quantum distributed interactive proofs, the number of interactions can be significantly reduced. More precisely, our main result shows that for any constant k, the class of languages that can be decided by a k-turn classical (i.e., non-quantum) distributed interactive protocol with f(n)-bit certificate size is contained in the class of languages that can be decided by a 5-turn distributed quantum interactive protocol with O(f(n))-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to 3-turn. Since no similar turn-reduction classical technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well

Idiopathic REM Sleep Behavior Disorder: Implications for the Pathogenesis of Lewy Body Diseases

Objectives. Both results of the odor identification and cardiac 123I-metaiodobenzylguanidine accumulation have been investigated for their potential to enhance the detection of pathogenesis resembling that of Lewy body-related α-synucleinopathies in patients clinically diagnosed as having idiopathic REM sleep behavior disorder. Methods. We performed both the Odor Stick Identification Test for Japanese and 123I-metaiodobenzylguanidine scintigraphy in 30 patients with idiopathic REM sleep behavior disorder, 38 patients with Parkinson's disease, and 20 control subjects. Results. In idiopathic REM sleep behavior disorder, reduced odor identification score and an early or delayed heart to mediastinum ratio on 123I-metaiodobenzylguanidine were almost as severe as in Parkinson's disease patients. Delayed cardiac 123I-metaiodobenzylguanidine uptake was even more severe in the idiopathic REM sleep behavior disorder group than in the Parkinson's disease group. Conclusions. Reduced cardiac 123I-metaiodobenzylguanidine uptake, which is independent of parkinsonism, may be more closely associated with idiopathic REM sleep behavior disorder than olfactory impairment

Transient Automatic Writing Behavior following a Left Inferior Capsular Genu Infarction

A 79-year-old, right-handed woman was admitted to the hospital with decreased spontaneity. Brain magnetic resonance imaging showed a left inferior capsular genu infarction. 99m TC-ECD single-photon emission computed tomography revealed a left-dominant diffuse hypoperfusionin the basal ganglia and frontal lobe. The patient showed abulia and increased writing activity without motor or sensory deficit. The writing was mainly perseverative, and words written along lines were legible and without spatial distortions. This augmented writing behavior disappeared on day 21. The writing characteristic was more similar to automatic writing behavior than hypergraphia. Dissociation between speech and writing behavior was present in our patient. We suggest that a disconnection within the frontal-subcortical circuit contributed to the development of motor perseveration in writing