Verified Analysis of Functional Data Structures

In recent work the author has analyzed a number of classical functional search tree and priority queue implementations with the help of the theorem prover Isabelle/HOL. The functional correctness proofs of AVL trees, red-black trees, 2-3 trees, 2-3-4 trees, 1-2 brother trees, AA trees and splay trees could be automated. The amortized logarithmic complexity of skew heaps, splay trees, splay heaps and pairing heaps had to be proved manually

A Complexity O(1) Priority Queue for Event Driven Molecular Dynamics Simulations

We propose and implement a priority queue suitable for use in event driven molecular dynamics simulations. All operations on the queue take on average O(1) time per collision. In comparison, previously studied queues for event driven molecular dynamics simulations require O(log $N$) time per collision for systems of $N$ particles.Comment: Accepted for publication in Journal of Computational Physic

Efficiently Computing Directed Minimum Spanning Trees

Computing a directed minimum spanning tree, called arborescence, is a fundamental algorithmic problem, although not as common as its undirected counterpart. In 1967, Edmonds discussed an elegant solution. It was refined to run in $O(\min(n^2, m\log n))$ by Tarjan which is optimal for very dense and very sparse graphs. Gabow et al.~gave a version of Edmonds' algorithm that runs in $O(n\log n + m)$, thus asymptotically beating the Tarjan variant in the regime between sparse and dense. Despite the attention the problem received theoretically, there exists, to the best of our knowledge, no empirical evaluation of either of these algorithms. In fact, the version by Gabow et al.~has never been implemented and, aside from coding competitions, all readily available Tarjan implementations run in $O(n^2)$. In this paper, we provide the first implementation of the version by Gabow et al.~as well as five variants of Tarjan's version with different underlying data structures. We evaluate these algorithms and existing solvers on a large set of real-world and random graphs

On Deciding Local Theory Extensions via E-matching

Satisfiability Modulo Theories (SMT) solvers incorporate decision procedures for theories of data types that commonly occur in software. This makes them important tools for automating verification problems. A limitation frequently encountered is that verification problems are often not fully expressible in the theories supported natively by the solvers. Many solvers allow the specification of application-specific theories as quantified axioms, but their handling is incomplete outside of narrow special cases. In this work, we show how SMT solvers can be used to obtain complete decision procedures for local theory extensions, an important class of theories that are decidable using finite instantiation of axioms. We present an algorithm that uses E-matching to generate instances incrementally during the search, significantly reducing the number of generated instances compared to eager instantiation strategies. We have used two SMT solvers to implement this algorithm and conducted an extensive experimental evaluation on benchmarks derived from verification conditions for heap-manipulating programs. We believe that our results are of interest to both the users of SMT solvers as well as their developers

A Priority-based Fair Queuing (PFQ) Model for Wireless Healthcare System

Healthcare is a very active research area, primarily due to the increase in the elderly population that leads to increasing number of emergency situations that require urgent actions. In recent years some of wireless networked medical devices were equipped with different sensors to measure and report on vital signs of patient remotely. The most important sensors are Heart Beat Rate (ECG), Pressure and Glucose sensors. However, the strict requirements and real-time nature of medical applications dictate the extreme importance and need for appropriate Quality of Service (QoS), fast and accurate delivery of a patient’s measurements in reliable e-Health ecosystem. As the elderly age and older adult population is increasing (65 years and above) due to the advancement in medicine and medical care in the last two decades; high QoS and reliable e-health ecosystem has become a major challenge in Healthcare especially for patients who require continuous monitoring and attention. Nevertheless, predictions have indicated that elderly population will be approximately 2 billion in developing countries by 2050 where availability of medical staff shall be unable to cope with this growth and emergency cases that need immediate intervention. On the other side, limitations in communication networks capacity, congestions and the humongous increase of devices, applications and IOT using the available communication networks add extra layer of challenges on E-health ecosystem such as time constraints, quality of measurements and signals reaching healthcare centres. Hence this research has tackled the delay and jitter parameters in E-health M2M wireless communication and succeeded in reducing them in comparison to current available models. The novelty of this research has succeeded in developing a new Priority Queuing model ‘’Priority Based-Fair Queuing’’ (PFQ) where a new priority level and concept of ‘’Patient’s Health Record’’ (PHR) has been developed and integrated with the Priority Parameters (PP) values of each sensor to add a second level of priority. The results and data analysis performed on the PFQ model under different scenarios simulating real M2M E-health environment have revealed that the PFQ has outperformed the results obtained from simulating the widely used current models such as First in First Out (FIFO) and Weight Fair Queuing (WFQ). PFQ model has improved transmission of ECG sensor data by decreasing delay and jitter in emergency cases by 83.32% and 75.88% respectively in comparison to FIFO and 46.65% and 60.13% with respect to WFQ model. Similarly, in pressure sensor the improvements were 82.41% and 71.5% and 68.43% and 73.36% in comparison to FIFO and WFQ respectively. Data transmission were also improved in the Glucose sensor by 80.85% and 64.7% and 92.1% and 83.17% in comparison to FIFO and WFQ respectively. However, non-emergency cases data transmission using PFQ model was negatively impacted and scored higher rates than FIFO and WFQ since PFQ tends to give higher priority to emergency cases. Thus, a derivative from the PFQ model has been developed to create a new version namely “Priority Based-Fair Queuing-Tolerated Delay” (PFQ-TD) to balance the data transmission between emergency and non-emergency cases where tolerated delay in emergency cases has been considered. PFQ-TD has succeeded in balancing fairly this issue and reducing the total average delay and jitter of emergency and non-emergency cases in all sensors and keep them within the acceptable allowable standards. PFQ-TD has improved the overall average delay and jitter in emergency and non-emergency cases among all sensors by 41% and 84% respectively in comparison to PFQ model

Why some heaps support constant-amortized-time decrease-key operations, and others do not

A lower bound is presented which shows that a class of heap algorithms in the pointer model with only heap pointers must spend Omega(log log n / log log log n) amortized time on the decrease-key operation (given O(log n) amortized-time extract-min). Intuitively, this bound shows the key to having O(1)-time decrease-key is the ability to sort O(log n) items in O(log n) time; Fibonacci heaps [M.L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this through the use of bucket sort. Our lower bound also holds no matter how much data is augmented; this is in contrast to the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] who showed a tradeoff between the number of augmented bits and the amortized cost of decrease-key. A new heap data structure, the sort heap, is presented. This heap is a simplification of the heap of Elmasry [SODA 2009: 471-476] and shares with it a O(log log n) amortized-time decrease-key, but with a straightforward implementation such that our lower bound holds. Thus a natural model is presented for a pointer-based heap such that the amortized runtime of a self-adjusting structure and amortized lower asymptotic bounds for decrease-key differ by but a O(log log log n) factor