An analysis of Atkinson's algorithm

We consider the implementation of meldable priority queues with integer keys in the RAM model. We present two new general techniques for transforming non-meldable priority queues into meldable ones. These transformations can be described symbolically as:non-meldable priority queue +union-find ← meldable priority queuenon-meldable priority queue +slow meldable priority queue ←faster meldable priority queueUsing the first transformation to combine a recent non-meldable RAM priority queue of Thorup with the classical union-find data structure we obtain a meldable RAM priority queue with an amortized cost of O(log log n·α(n)) per operation, where α(n) = α(n, n) is the inverse Ackermann function. Using instead a randomized priority queue of Han and Thorup we obtain an expected amortized cost of O(√(log log n) · α(n)) per operation. The second transformation yields slower meldable priority queues, but the obtained queues can support the insert, find-min and decrease-key operations in constant time. In particular, by combining a randomized atomic-heap of Thorup with, e.g., the classical Fibonacci heaps of Fredman and Tarjan, we obtain, for every fixed e > 0, a meldable priority queue with an expected amortized cost of O(1) for each insert, find-min and decrease-key operation, and an expected amortized cost of O((log n)1/2+e) for each delete or meld operation.Using the meldable priority queues of the first type, we obtain improved algorithms for finding minimum directed spanning trees in graphs with integer edge weights: a deterministic O(m · log log n · α(n)) time algorithm and a randomized O(m · √(log log n) · α(n)) expected time algorithm. These bounds improve, for very sparse graphs, on the O(m + n log n) running time of an algorithm by Gabow, Galil, Spencer and Tarjan that works for arbitrary edge weights.

Path compression techniques are efficient on-demand routing optimizing techniques for mobile ad hoc networks. But the existing path compression techniques have some shortages such as blindfold compression and unstable path. It is necessary to achieve the effective tradeoff of compression efficiency and path stability. This paper calculated the dynamic, real-time stable compression period probabilistically based on a dynamic path compression model and proposed a stability module of path compression technique (SMPC), which can restrain the blindness of path compression efficiently and improve the stability of routes. SMPC introduces GPS based SMPC-ES and compression coverage based SMPC-MS to lessens the ephemeral short-cuts and multiple short-cuts. The simulation results show that SMPC based path compression algorithms have good performance in merits of original RREQ, end-to-end delay and delivery rate compared to AODV and PCA. It is a generic and efficient stability module of path compression technique for on-demand ad-hoc routing protocols.

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

Isomorphism of graphs of bounded valence can be tested in polynomial time

Strong bias of group generators: an obstacle to the “product replacement algorithm”