Narrow Results

We study the transition graph of generic Hamiltonian surface flows, whose vertices are the topological equivalence classes of generic Hamiltonian surface flows and whose edges are the generic transitions. Using the transition graph, we can describe time evaluations of generic Hamiltonian surface flows (e.g., fluid phenomena) as walks on the graph. We propose a method for constructing the complete transition graph of all generic Hamiltonian flows. In fact, we construct two complete transition graphs of Hamiltonian surface flows having three and four genus elements. Moreover, we demonstrate that a lower bound on the transition distance between two Hamiltonian surface flows with any number of genus elements can be calculated by solving an integer programming problem using vector representations of Hamiltonian surface flows.

The IOTA Tangle, a Directed Acyclic Graph (DAG)-based distributed ledger, is popular for its scalability and suitability for IoT applications, offering fee-less transactions. A critical component of IOTA’s architecture is the Cumulative Weight Calculation (CWC), essential for its tip selection mechanism. This paper introduces an optimization of the IOTA Reference Implementation (IRI) CWC process originally implemented using Breadth-First Search (BFS) by employing Depth-First Search (DFS) and Iterative Deepening Search (IDS) algorithms. We present a comparative analysis of these methods, demonstrating that DFS and IDS provide significant improvements in computational efficiency, particularly beneficial for IoT devices with limited processing capabilities. Our findings are substantiated through a series of experiments on a Tangle snapshot, highlighting the enhanced performance and reduced resource utilization of the proposed methods. This study contributes to the ongoing development of DAG-based distributed ledgers, offering insights into more efficient algorithmic solutions for large-scale, decentralized networks.