We study the problem of parallel comparison sorting, modeled as a two-player game between a sorter and an adversary. Using a minimax algorithm enhanced with several domain-specific optimizations, we compute the exact minimal number of rounds [Formula: see text] required to sort [Formula: see text] elements using at most [Formula: see text] comparisons per round, for all [Formula: see text] and [Formula: see text]. The optimizations include canonical labelling of states, memoization, alpha–beta pruning, and an early-stopping rule exploiting the fact that candidate sorter moves can differ in value by at most one round. Together, these techniques make the computation feasible, and we measure the contribution of each of them. To our knowledge these are the first exact values of [Formula: see text] for the adaptive model. The results quantify the overhead of parallelism: no computed value exceeds the sequential-work lower bound [Formula: see text] by more than one round, yet parallel efficiency falls from [Formula: see text] at [Formula: see text] to [Formula: see text] at [Formula: see text] for [Formula: see text], because adversarial outcomes prevent perfect speedup.
The divide-and-swap [Formula: see text]-ary [Formula: see text]-cube is a recently proposed variant of the hypercube, denoted by [Formula: see text]. It has a smaller diameter and better connectivity than the hypercube. Connectivity and super connectivity are two important parameters for measuring the fault tolerance of interconnection networks. In this work, we obtain the connectivity and super connectivity of the divide-and-swap [Formula: see text]-ary [Formula: see text]-cube.
Parallel contextual array insertion deletion P system (PCAIDPS) generating 2D rectangular picture languages, introduced in 2017, has more generating capacity than the other picture language generating devices like tiling systems and context-sensitive matrix grammars etc. In 2024 another interesting device called pure 2D Eilenberg P system (P2DEPS) based on pure 2D context-free grammars has been studied. In this paper, we compare the generative powers of these two P systems PCAIDPS and P2DEPS.
With the rapid expansion of data centers driven by big data processing and artificial intelligence, the urgent need is to seek and optimize multiprocessor systems based on interconnection networks with high reliability and strong fault tolerability. The generalized K-4-hypercube H-n(4), one extended topology derived from the hypercube, is an ideal candidate for the choice of recursive interconnection networks. To quantitatively evaluate its link fault tolerance under multi-component partition scenarios, this paper investigates the (r+1)-component edge-connectivity of H-n(4). This work provides a precise quantitative metric for evaluating the structural robustness of the generalized K-4-hypercube H-n(4) with a large number of link failures.
Graph entropy is originated from Shannon entropy, which is used to quantify the complexity and uncertainty of networks and is widely applied in fields such as information theory, communication theory and statistics. As a kind of classical graph entropy, the chromatic entropy of a graph plays a key role in describing the structures of networks. Generally, a hypergraph is the extension of a graph, which is often used in characterising complex systems. In this paper, we investigate the chromatic entropy of linear p-uniform unicyclic hypergraphs based on strong vertex coloring. We determine the hypergraphs that attain the maximum and minimum chromatic entropy within this class. Using edge-moving operations, we characterize the extremal structures and derive explicit formulas for their entropy values.
