Path finding and multi-robot motion planning require trajectories that are smooth, goal-directed, and collision-free in environments with complex geometric constraints. Recent diffusion-based planners have shown that trajectory generation can be cast as iterative denoising which has opened the doors to learning-based approaches that can handle multi-modal trajectory distributions and refine entire trajectories. However, a key limitation is that diffusion planners require training on large collections of feasible trajectories, rendering them map-specific, and difficult to deploy when high-quality demonstrations are unavailable. This paper introduces a training-free diffusion-based motion planner that replaces learned global trajectory scores with analytical local scores derived from obstacle, smoothness, velocity, and inter-agent feasibility terms. The proposed idea relies on a key observation: the score of a trajectory can be reconstructed by considering only local interactions between neighboring waypoints and nearby constraints. This structure exploitation yields a decomposed denoising procedure that retains the optimization structure of classical trajectory methods while inheriting the iterative refinement behavior of diffusion models. Experiments on a large collection of complex environments and large multi-agent planning tasks show that the proposed analytical score produces smooth and feasible trajectories within limited computational costs, for example in generating feasible paths for 300+ agents in environments containing 100+ obstacles in under 6 seconds on a GPU, outperforming strong learning-based and optimization baselines, while avoiding the data requirements of learned diffusion planners.

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