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By starting with the assumption that motion is fundamentally a decision making problem, we use the world‐line concept from Special Relativity as the inspiration for a novel multi‐agent path planning method. We have identified a particular set of problems that have so far been overlooked by previous works. We present our solution for the global path planning problem for each agent and ensure smooth local collision avoidance for each pair of agents in the scene. We accomplish this by modelling the collision‐free trajectories of the agents through 2D space and time as rods in 3D. We obtain smooth trajectories by solving a non‐linear optimization problem with a quasi‐Newton interior point solver, initializing the solver with a non‐intersecting configuration from a modified Dijkstra's algorithm. This space–time formulation allows us to simulate previously ignored phenomena such as highly heterogeneous interactions in very constrained environments. It also provides a solution for scenes with unnaturally symmetric agent alignments without the need for jittering agent positions or velocities.

In a simple pattern matching problem one has a pattern w and a text t, which are words over a finite alphabet Σ. One may ask whether w occurs in t, and if so, where? More generally, we may have a set P of patterns and a set T of texts, where P and T are regular languages. We are interested whether any word of T begins with a word of P , ends with a word of P , has a word of P as a factor, or has a word of P as a subsequence. Thus we are interested in the languageswhere is the shuffle operation. The state complexity κ(L) of a regular language L is the number of states in the minimal deterministic finite automaton recognizing L. We derive the following upper bounds on the state complexities of our patternmatching languages, where κ(P ) m, and κ(T ) n: κ((P Σ * ) ∩ T ) mn; κ((Σ * P ) ∩ T ) 2 m−1 n; κ((Σ * P Σ * ) ∩ T ) (2 m−2 + 1)n; and κ((Σ * P ) ∩ T ) (2 m−2 + 1)n. We prove that these bounds are tight, and that to meet them, the alphabet must have at least two letters in the first three cases, and at least m − 1 letters in the last case. We also consider the special case where P is a single word w, and obtain the following tight upper bounds: κ((wΣ * ) ∩ T ) m + n − 1; κ((Σ * w) ∩ T ) (m − 1)n − (m − 2); κ((Σ * wΣ * ) ∩ T ) (m − 1)n; and κ((Σ * w) ∩ T ) (m − 1)n. For unary languages, we have a tight upper bound of m + n − 2 in all eight of the aforementioned cases.

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We present a new method for computing a smooth minimum distance function based on the LogSumExp function for point clouds, edge meshes, triangle meshes, and combinations of all three. We derive blending weights and a modified Barnes-Hut acceleration approach that ensure our method approximates the true distance, and is conservative (points outside the zero isosurface are guaranteed to be outside the surface) and efficient to evaluate for all the above data types. This, in combination with its ability to smooth sparsely sampled and noisy data, like point clouds, shortens the gap between data acquisition and simulation, and thereby enables new applications such as direct, co-dimensional rigid body simulation using unprocessed lidar data.

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