We study logarithmic integrals of the form $$\int _0^1 x^i\ln ^n(x)\ln ^m(1-x)dx$$ . They are expressed as a rational linear combination certain numbers $$(n,m)_{i}$$ , which we call tiered binomial coefficients, and products zeta values $$\zeta (2)$$ (3)$$ ,.... Various properties coefficients established. involve, amongst others, transform, truncated multiple star values, well special functio...

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL: the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, and the expe...

An algorithm, which asymptotically halves the number of comparisons made by the common HEAPSORT, is presented and analysed in the worst case. The number of comparisons is shown to be (n+ 1)(log(n+ 1) + log log(n+ 1) + 1.82) + O(log n) in the worst case to sort n elements, without using any extra space. QUICKSORT, which usually is referred to as the fastest in-place sorting method, uses 1.38n lo...

In the context of word associations, multiword units (sequences of words that co-occur more often than expected by chance) are frequently used in everyday language, usually to precisely express ideas and concepts that cannot be compressed into a single word. For instance, [Bill of Rights], [swimming pool], [as well as], [in order to], [to comply with] or [to put forward] are multiword units. As...

The normalized number of key comparisons needed to sort a list of randomly permuted items by the Quicksort algorithm is known to converge in distribution. We identify the rate of convergence to be of the order Θ(ln(n)/n) in the Zolotarev metric. This implies several ln(n)/n estimates for other distances and local approximation results as for characteristic functions, for density approximation, ...

We discuss how string sorting algorithms can be parallelized on modern multi-core shared memory machines. As a synthesis of the best sequential string sorting algorithms and successful parallel sorting algorithms for atomic objects, we propose string sample sort. The algorithm makes effective use of the memory hierarchy, uses additional word level parallelism, and largely avoids branch mispredi...

In a continuous-time setting, Fill [2] proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable Y —not even that it is nondegenerate. We establish the nondegeneracy of Y . The proof is perhaps...

We introduce several modifications of the partitioning schemes used in Hoare’s quicksort and quickselect algorithms, including ternary schemes which identify keys less or greater than the pivot. We give estimates for the numbers of swaps made by each scheme. Our computational experiments indicate that ternary schemes allow quickselect to identify all keys equal to the selected key at little add...

We revisit the method of Kirschenhofer, Prodinger and Tichy to calculate asymptotic expressions for the moments of number of comparisons used by the randomized quick sort algorithm.

Quicksort is one of the earliest and most famous algorithms. It was invented and analyzed by Tony Hoare around 1960. This was before the big-O notation was used to analyze algorithms. Hoare invented the algorithm while an exchange student at Moscow State University while studying probability under Kolmogorov—one of the most famous researchers in probability theory. The analysis we will cover is...