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Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let P and Q be two polygonal curves in R d with n and m vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, (3) a linear-time √ d-approximation algorithm, and (4) a data structure that supports O(m log 2 n)time decision queries, where m is the number of vertices of the query curve and n the number of vertices of the preprocessed curve. 23:2Fast Fréchet Distance Between Curves with Long Edges O(n 2 /φ + n log n)-time algorithm to compute a φ-approximation of the discrete Fréchet distance for any integer 1 ≤ φ ≤ n. Therefore, an n -approximation, for any > 0, can be computed in (strongly) subquadratic time. For the continuous Fréchet distance, there are also a few subquadratic algorithms known for restricted classes of curves such as κ-bounded, backbone and c-packed curves. Alt et al. [3] considered κ-bounded curves and they gave an O(n log n) time algorithm to (κ + 1)-approximate the Fréchet distance. A curve P is κbounded if for any two points x, y ∈ P , the union of the balls with radii r centered at x and y contains the whole P [x, y] where r is equal to (κ/2) times the Euclidean distance between x and y. For any > 0, Aronov et al. [4] provided a near-linear time (1 + )-approximation algorithm for the discrete Fréchet distance for so-called backbone curves that have essentially constant edge length and require a minimum distance between non-consecutive vertices. For c-packed curves a (1 + )-approximation can be computed in O(cn/ + cn log n) time [11]. A curve is c-packed if for any ball B, the length of the portion of P contained in B is at most c times the diameter of B.

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In this paper, a new method is proposed for the path planning of multi-robots in unknown environments. The method is inspired by multi-objective particle swarm optimization (MOPSO) and is named multi-robot MOPSO. It considers shortness, safety, and smoothness. Due to the obscurity of the environment, the robots should decide the moving direction based on the information gathered by sensors only such that the optimal path between the start and goal positions can be found at the end of the algorithm. Sharing knowledge among the navigating robots is necessary to achieve this aim. So, a new concept, named the probabilistic window, is introduced in this paper. It combines the current information obtained through the robot sensors and experiences of the previous robots to select the paths that seem more likely to achieve higher fitness in the mentioned objectives. The proposed method has an outstanding performance on different complex benchmarks, and the results have shown that it is more effective and efficient compared with the classic and the state-of-the-art methods.

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