Consider k robots initially located at a point inside a region T . Each robot can move anywhere in T independently of other robots with maximum speed one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T . The objective is to minimize the evacuation time, which is defined as the time the last robot reaches the exit. We consider the face-to-face communication model for the robots: a robot can communicate with another robot only when they meet in T .In this paper, we give upper and lower bounds for the face-to-face evacuation time by k robots that are initially located at the centroid of a unit-sided equilateral triangle or square. For the case of a triangle with k = 2 robots, we give a lower bound of 1 + 2/ √ 3 ≈ 2.154, and an algorithm with upper bound of 2.3367 on the worst-case evacuation time. We show that for any k, any algorithm for evacuating k ≥ 2 robots requires at least √ 3 time. This bound is asymptotically optimal, as we show that even a straightforward strategy of evacuation by k robots gives an upper bound of √ 3 + 3/k. For k = 3 and 4, we give better algorithms with evacuation times of 2.0887 and 1.9816, respectively. For the case of the square and k = 2, we give an algorithm with evacuation time of 3.4645 and show that any algorithm requires time at least 3.118 to evacuate in the worst-case. Moreover, for k = 3, and 4, we give algorithms with evacuation times 3.1786 and 2.6646, respectively. The algorithms given for k = 3 and 4 for evacuation in the triangle or the square can be easily generalized for larger values of k. * A preliminary version of this paper appeared in

Abstract. Consider a random graph process where vertices are chosen from the interval [0,1], and edges are chosen independently at random, but so that, for a given vertex x, the probability that there is an edge to a vertex y decreases as the distance between x and y increases. We call this a random graph with a linear embedding.We define a new graph parameter Γ * , which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph G, Γ * (G) = 0 if and only if G is a unit interval graph, and thus a deterministic example of a graph with a linear embedding.We show that the behaviour of Γ * is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on [0, 1] 2 . We define an operator Γ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence {Gn} converges to a function w, then {Γ * (Gn)} converges as well. Moreover, there exists a function w * arbitrarily close to w under the box distance, so that limn→∞ Γ * (Gn) is arbitrarily close to Γ(w * ).

The problem of evacuating two robots from the disk in the face-to-face model was first introduced in [16], and extensively studied (along with many variations) ever since with respect to worst case analysis. We initiate the study of the same problem with respect to average case analysis, which is also equivalent to designing randomized algorithms for the problem. First we observe that algorithm B2 of [16] with worst case cost Wrs (B2) := 5.73906 has average case cost Avg (B2) := 5.1172. Then we verify that none of the algorithms that induced worst case cost improvements in subsequent publications has better average case cost, hence concluding that our problem requires the invention of new algorithms. Then, we observe that a remarkable simple algorithm, B1, has very small average case cost Avg (B1) := 1 + π, but very high worst case cost Wrs (B1) := 1 + 2π. Motivated by the above, we introduce constrained optimization problem 2 EVAC w F 2F , in which one is trying to minimize the average case cost of the evacuation algorithm given that the worst case cost does not exceed w. The problem is of special interest with respect to practical applications, since a common objective in search-and-rescue operations is to minimize the average completion time, given that a certain worst case threshold is not exceeded, e.g. for safety or limited energy reasons. Our main contribution is the design and analysis of families of new evacuation parameterized algorithms A (p) which can solve 2 EVAC w F 2F , for every w ∈ [Wrs (B1) , Wrs (B2)]. In particular, by letting parameter(s) p vary, we obtain parametric curve (Avg (A (p)) , Wrs (A (p))) that induces a continuous and strictly decreasing function in the mean-worst case space, and whose endpoints are (Avg (B1) , Wrs (B1)) , (Avg (B2) , Wrs (B2)). Notably, the worst case analysis of the problem, since it's introduction, has been relying on technical numerical, computerassisted, calculations, following tedious robots trajectories' analysis. Part of our contribution is a novel systematic procedure, which, given any evacuation algorithm, can derive it's worst and average case performance in a clean and unified way.

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Let w : [0, 1] 2 → [0, 1] be a symmetric function, and consider the random process G(n, w), where vertices are chosen from [0, 1] uniformly at random, and w governs the edge formation probability. Such a random graph is said to have a linear embedding, if the probability of linking to a particular vertex v decreases with distance. The rate of decrease, in general, depends on the particular vertex v. A linear embedding is called uniform if the probability of a link between two vertices depends only on the distance between them. In this article, we consider the question whether it is possible to "transform" a linear embedding to a uniform one, through replacing the uniform probability space [0, 1] with a suitable probability space on R. We give necessary and sufficient conditions for the existence of a uniform linear embedding for random graphs where w attains only a finite number of values. Our findings show that for a general w the answer is negative in most cases.