Algorithm Visualization (AV) helps students build mental models by animating algorithm execution states. Recent LLM-based systems such as CODE2VIDEO generate AV videos in an end-to-end manner. However, this paradigm requires the system to simultaneously simulate algorithm flow and satisfy video rendering constraints (element layout, color schemes, etc.), a complex task that induces LLM hallucinations. This results in reduced execution success rates, element overlap, and inter-frame inconsistencies.To address these challenges, we propose ALGOGEN, a novel paradigm that decouples algorithm execution from rendering. We first introduce Visualization Trace Algebra (VTA), a monoid over algorithm visual states and operations. The LLM then generates a Python tracker that simulates algorithm flow and outputs VTA-JSON traces, a JSON encoding of VTA. For rendering, we define a Rendering Style Language (RSL) to templatize algorithm layouts. A deterministic renderer then compiles algorithm traces with RSL into Manim, LaTeX/TikZ, or Three.js outputs[Manim, TikZ, and Three.js are respectively a Python animation engine, a LaTeX vector graphics package, and a JavaScript 3D rendering library.].Evaluated on a LeetCode AV benchmark of 200 tasks, ALGOGEN achieves an average success rate improvement of 17.3% compared to end-to-end methods (99.8% vs. 82.5%). These results demonstrate that our decoupling paradigm effectively mitigates LLM hallucinations in complex AV tasks, providing a more reliable solution for automated generation of high-quality algorithm visualizations. Demo videos and code are available at: .