Abstract. This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n 3/2 ), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 3 , 2 3 .) For sparse tables, the construction cost has expectation O(n), standard deviation O( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.
We present an analysis of the effect of the last-come-first-served heuristic on a linear probing hash table. We study the behavior of successful searches, assuming searches for all elements of the table are equally likely. It is known that the Robin Hood heuristic achieves minimum variance over all linear probing algorithms. We show that the last-comefirst-served heuristic achieves this minimum up to lower-order terms. An accurate analysis of this algorithm is made by introducing a new transform which we call the Diagonal Poisson Transform as it resembles the Poisson Transform. We present important properties of this transform, as well as its application to solve some classes of recurrences, find inverse This work was done while the first two authors were at the University of Waterloo. Correspondence to: A. Viola
ObjectiveTo evaluate the safety and assess the different symptom improvements found after a combined low-frequency primary motor cortex and high-frequency prefrontal cortex (PFC) stimulation using the deep TMS (dTMS) H-coil, as an add-on treatment for Parkinson’s disease (PD).MethodsForty-five PD patients underwent 14 dTMS sessions; each consisting of 1 Hz stimulation of the primary motor cortex for 15 min, followed by 10 Hz stimulation of the PFC for 15 min. Clinical assessments were performed, BEFORE, at the MIDDLE, and END of therapy as well as at FOLLOW-UP after 30 days, using Movement Disorder Society-Unified Parkinson’s Disease Rating Scale, TINETTI, UP&GO, SCOPA, HDRS21, Beck Depression Inventory, and self-applied daily motor assessment scales.ResultsTreatment was well-tolerated, without serious adverse effects. dTMS-induced significant PD symptom improvements at END and at FOLLOW-UP, in all subscales of the UPDRS, gait speed, depressive symptoms, balance, autonomic symptoms, and a 73% increase in daily ON time.ConclusionIn the cohort of PD patients treated, dTMS was well-tolerated with only minor adverse effects. The dTMS-induced significant improvements in motor, postural, and motivational symptoms of PD patients and may potentiate concurrent levodopa treatment.SignificanceThe present study demonstrates that dTMS may have a much wider spectrum of beneficial effects than previously reported for TMS, including enhancement of levodopa effects, suggesting that future clinical trials with dTMS should include a broader range of symptom measurements.
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