Suffix trees are the most frequently used data structures in algorithms on words. In this paper, we consider the depth of a compact suffix tree, also known as the PAT tree, under some simple probabilistic assumptions. For a biased memoryless source, we prove that the limiting distribution for the depth in a PAT tree is the same as the limiting distribution for the depth in a PATRICIA trie, even though the PATRICIA trie is constructed from statistically independent strings. As a result, we show that the limiting distribution for the depth in a PAT tree built over n suffixes is normal.

Moments play an essential role in the characterization of statistical distributions and criteria such as dispersion‎, ‎skewness‎, ‎and kurtosis‎. ‎This article is a dissection of the central moments of two-point and binomial distributions‎. ‎First‎, ‎we consider the Bernoulli distribution of the population and generalize the results‎. ‎With a simple method‎, ‎we present the condition that when the sample size is large‎, ‎the structure of the sample central moment consists of random variables independent of standard normal or chi-square or a combination of both‎. ‎In the obtained results‎, ‎the role of points that have a probability of 1/2 is very influential in the limit distribution.‎

One of the obvious reasons for most disorders in network service provisioning is network path congestion. Congestion avoidance in today's networks is too costly and sometimes impossible. With the introduction of SDN, centralizing the equipment's control plane has become possible. This paper presents an enhanced method named ESV-DBRA to avoid congestion in multi-tenant SDN networks. At first, ESV-DBRA monitors the traffic load and delay of all network paths for each tenant individually. Then, by merging the parameters obtained from the monitoring, the Service Level Agreements (SLA), and a novel proposed cost function, it calculates the cost of the network paths per tenant. As a result, traffic for each tenant is routed through the path/paths at the lowest possible cost from the tenant's perspective. Next, the bandwidth quotas will be calculated and assigned to the tenants over their optimal routes. Afterward, whenever congestion is likely to occur in a path, ESV-DBRA automatically changes the route or bandwidth of the tenants' traffic related to this path to avoid congestion. Related algorithms are also proposed.Eventually, simulations show that the proposed method effectively increases bandwidth utilization by 10.76%.

Purpose: To compare the anatomical and functional results of deep vitrectomy with and without ILM peeling for full thickness macular hole.Materials & Methods: 37 eyes of 37 patients with full thickness macular hole (FTMH), confirmed by clinical examination and optical coherence tomography (OCT), enrolled in this clinical trial. Based on the stage of FTMH (stage 2, 3, 4) as described by Gass and the techniques of surgery (with or without ILM peeling) the patients were randomized into two different groups. All patients underwent preoperative Act and at least one OCT image was obtained during the postoperative follow up.After obtaining informed consent, deep vitrectomy and 20% SF6 injection was performed. The patients were unaware of their allocation status. Patients data & surgical outcome were collected and statistically analyzed. Anatomical success was defined as hole closure at postoperative OCT and functional success was referred to any improvement in visual acuity. Macular hole index (MHI) was defined as a ratio of the greatest height to the base diameter of the hole on cross-sectional OCT images.Results: 37 eyes of 37 patients (15 men & 22 women) were enrolled. The mean age of the patients was 56.4±21.94 years and FTMH were traumatic in 13 cases (35%) and Idiopathic in 24 (65%) cases. Twenty cases underwent deep vitrectomy with ILM peeling (ILM group) and the remaining cases underwent vitrectomy without ILM peeling (no ILM group).The mean preoperative 10gMAR visual acuity was 1.38±1 in the ILM group and 1.22±0.13 in the no ILM group, these improved to 0.95±1 and 0.77±0.31 postoperatively in the ILM and no ILM group, respectively (P=0.871). The overall anatomical success rate was 64%. The hole was closed in 70% of ILM peeling and 47% in the no ILM group (Chi-square test, P=0.15). In anatomically successful cases the mean of postoperative VA was significantly improved (P=0.001).In traumatic subgroup, ILM peeling seems did not affect the anatomical (P=0.2) and visual Success rates (P=0.5). There was no significant differences in hole closure rates between the traumatic and idiopathic FTMH (P=0.968). Visual acuity significantly improved after operation In MHI ³ 0.5 group compared with the MHk 0.5 group. The stage of the hole did not affect the anatomical success rate (P=0.52) or visual acuity improvement (P=0.741). Macular hole index ³ 0.5 had a prognostic value for postoperative visual acuity improvement. The hole duration was not related to anatomical success rate but there is an inverse relation between duration of the symptoms and improvement in VA.Conclusion: Deep vitrectomy and SF6 injection was a safe & effective surgical technique for full thickness macular hole and peeling of the ILM seems did not affect the anatomical or functional success rates. The hole duration, stage and origin of the hole (traumatic or idiopathic) does not related to the anatomical success or visual acuity improvement. MHI  ³ 0.5 seems to be a prognostic factor for postoperative visual improvement but not for closure rate.

Different types of contact, including contact between node pairs, any-contact of nodes, and contacts of the entire network, are used to characterize social relations in mobile social networks. Different modes of routing, from the point of view of message delivery semantics, encompass unicasting, multicasting, any-casting, and broadcasting. Studies have shown that using probability distribution functions of contact data, which is mainly assumed to be homogeneous for nodes, improves the performance of these networks. However, there exists an important challenge in studies on distributions. A lot of works apply the distribution of one type of contact to other types. Hence in routing applications, it causes to use of the distribution of one type of contact for any mode of routing. This study provides a complete solution to model each type of homogeneous contact data distribution and to use them in different modes of routing. We propose a routing algorithm that uses this new model. Results show that our solution improves the average latency of comparing methods Epidemic, TCCB, and DR about 3.5-times, 30%, and 45%, respectively. It achieves a delivery rate of about 5% and 6%, and average latency about 6% and 8% better than that of DR and TCCB, respectively.

Several voting systems are utilized to allocate both political and non-political positions across countries worldwide. Plurality voting and approval voting are among the most widely implemented electoral systems. Established statistical methods are employedto ensure the accuracy of vote counting and the validation of election results. Two fundamental approaches that significantly enhance the likelihood of identifying errors in election outcomes are Risk-limiting audit (RLA) and Bayesian audit (BA). These audit methods assess the security of elections using statistical tools, based on the random selection of castvotes and their interpretation as evidence supporting or contradicting the reported results. In this paper, we first examine the advantages of a specific form of approval voting, referred to as score voting, and then describe two types of risk-limiting audits to evaluate the accuracy of vote counting and the results. The proposed auditing method for score voting is adapted from the Ballot-polling risk-limiting audits to verify outcomes (BRAVO). Simulation results confirm the effectiveness and accuracy of our approach.