The Quantum Complexity of Set Membership

We study the quantum complexity of the static set membership problem: given a subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of bits so that queries of the form `Is x \in S?' can be answered. The goal is to use a small table and yet answer queries using few bitprobes. This problem was considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where lower and upper bounds were shown for this problem in the classical deterministic and randomized models. In this paper, we formulate this problem in the "quantum bitprobe model" and show tradeoff results between space and time.In this model, the storage scheme is classical but the query scheme is quantum.We show, roughly speaking, that similar lower bounds hold in the quantum model as in the classical model, which imply that the classical upper bounds are more or less tight even in the quantum case. Our lower bounds are proved using linear algebraic techniques.Comment: 19 pages, a preliminary version appeared in FOCS 2000. This is the journal version, which will appear in Algorithmica (Special issue on Quantum Computation and Quantum Cryptography). This version corrects some bugs in the parameters of some theorem

Explicit deterministic constructions for membership in the bitprobe model

We look at time-space tradeoffs for the static membership problem in the bit-probe model. The problem is to represent a set of size up to n from a universe of size m using a small number of bits so that given an element of the universe, its membership in the set can be determined with as few bit probes to the representation as possible. We show several deterministic upper bounds for the case when the number of bit probes, is small, by explicit constructions, culminating in one that uses o(m) bits of space where membership can be determined with [lg lgn] + 2 adaptive bit probes. We also show two tight lower bounds on space for a restricted two probe adaptive scheme

Validity of diffusion-weighted magnetic resonance imaging in the evaluation of acute pyelonephritis in comparison with contrast-enhanced computed tomography

Purpose: Applications of diffusion-weighted magnetic resonance imaging outside the brain have gained increasing importance in recent years, and recent studies have shown the usage of diffusion-weighted (DW) imaging in diagnosing pyelonephritis based on renal cortical and medullary apparent diffusion coefficient (ADC) values. The aim of this study was to assess the validity of DW magnetic resonance (MR) imaging in comparison with contrast-enhanced computed tomography (CECT) in diagnosing pyelonephritis. Material and methods: A cross-sectional observational study was conducted for a period of six months in a tertiary hospital in Coimbatore. All patients with clinical and laboratory diagnosis of acute pyelonephritis, who were referred for radiological imaging (CECT), were taken into the study. Out of 112 patients with a clinical and laboratorial diagnosis of acute pyelonephritis (APN), who underwent both DW MR and CECT, diagnosis of APN was made in 100 patients based on CECT, while in 12 cases the investigation (CECT) was negative. Finally, these 100 patients were included in the study. The validity of DW MR imaging in diagnosing APN was assessed by deriving sensitivity, specificity, and positive and negative predictive value in comparison with CECT findings. Results: The validity report of DW MR imaging in the detection of APN showed a very high sensitivity (96-100%) and specificity (86-90%) and very low false positives (6-10%) and negatives (< 5%), and it also showed that in the areas of affected renal parenchyma ADC values were consistently lower compared to unaffected renal parenchyma. Conclusion: Based on the generated hypothesis, DW MR imaging of the kidneys seems to be highly sensitive and specific for the detection of focal or diffuse infections within the kidney in comparison with CECT