| viXra.org > Data Structures and Algorithms > 2509 |
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| viXra.org > Data Structures and Algorithms > 2509 |
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[4] viXra:2509.0135 [pdf] replaced on 2026-06-11 04:20:30
Authors: Nathan O. Schmidt
Comments: 111 Pages.
The Tri-Quarter Framework (TQF) unleashes a radial dual triangular lattice graph with unified complex-Cartesian-polar coordinates, structured orientation phase pair assignments for directional labeling, and topological zones to build exact bijective mappings without approximations. By modifying the Eisenstein integer lattice and establishing combinatorial duality for radial separation, Escher reflective duality for zone swapping, and bijective self-duality for reversible transformations, the discretized framework leverages the lattice graph’s order-6 rotational symmetry to natively support angular sectors, modular decompositions, equivariant encodings, and trihexagonal six-coloring for conflict-free parallel algorithms. At this discretized framework’s core is the Tri-Quarter Inversive Hexagonal Dihedral Symmetry Group T24—the order-24 direct product D6 × Z2, with ιr as a central involution—which exploits rotational, reflective, and inversive symmetries to unlock these bijective transformations with exact precision. As an abstract group, T24 = D6 × Z2 is the classical centrosymmetric hexagonal point group D6h; our contribution is its concrete realization as a circle inversion action on Λr, together with a radial dual construction whose zone bijections, constant-size balanced separator, and equivariant encodings hold exactly by design rather than by approximation. We provide formal proofs of these dualities, along with numerous step-by-step examples, and demonstrate practical efficiency through benchmarked simulations. For inversion-based path mirroring via bijections, we achieve measured speedups of approximately 1.6–1.8x with bitwise-exact agreement against full recomputation. For symmetry-reduced clustering, computed in exact rational arithmetic, the measured speedups are a steady 3.5–4.0x (peaking near 4.0x), below the idealized Z6-symmetry ceiling of approximately 6x—the gap reflecting the uniform per-vertex cost of exact rational arithmetic and orbit bookkeeping—while the orbit-reduced coefficient reproduces the full graph computation exactly, as the same rational number, at every R. Most significantly, the equivariant trihexagonal six-coloring partitions the lattice into six conflict-free independent sets that map directly onto data parallel hardware: on a consumer NVIDIA GeForce RTX 4060 laptop GPU, a color-ordered relaxation sweep achieves a median ~8x (ranging 6.6–8.6x over five sessions) over a single threaded CPU baseline at ~290k vertices, with the GPU runtime remaining nearly flat as the graph grows while the CPU cost scales linearly—so the advantage widens with scale. This work advances scalable computations on symmetric structures, with applications in computational geometry, graph traversals, tiling, clustering, and conflict-free data parallel computation. This work aims to solidify a practical mathematical and computational foundation for both classical and non-classical computing paradigms—targeting future integrations in complex emergent systems that harness intricate “superposition-like” symmetries to advance symmetry-aware algorithms and data structures across diverse computing architectures.
Category: Data Structures and Algorithms
[3] viXra:2509.0106 [pdf] submitted on 2025-09-17 03:20:55
Authors: Andy Wang
Comments: 7 Pages.
This paper proposes a distributed blogging system [1] [2] [3] [4] [5] [6] based on independent HTML fragments, which we term TOM (Time Object Model) [7]. The core characteristics of this blogging system are: "individual authors have full autonomy over managing their own blog content, using independent HTML fragments (in JSON format, with unique HTTP addresses) as the content carrier, and supporting other blogs to embed these fragments without notifying the original author."
Category: Data Structures and Algorithms
[2] viXra:2509.0101 [pdf] submitted on 2025-09-16 03:48:16
Authors: Ran Qin
Comments: 6 Pages.
Microservices architecture, which enhances large-scale distributed systems by breaking applications into small, independent service units, offers significant benefits in scalability, fault tolerance, and the ability to integrate diverse technologies. This paper details the characteristics of microservices architecture and its practical implementation in distributed systems. Furthermore, it analyzes cutting-edge applications, such as service mesh, edge computing, and intelligent observability, considering the convergence of AI and cloud-native technologies. The paper concludes by proposing future development directions, including modular design, automated operations, and ecosystem-based toolchains.
Category: Data Structures and Algorithms
[1] viXra:2509.0043 [pdf] submitted on 2025-09-08 01:00:28
Authors: Mario Stöckli
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references)
We prove P ≠ NP by contradiction by showing that there is no polynomial-time algorithm to solve the set partition problem.
Category: Data Structures and Algorithms
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