Linear time constant factor approximation algorithm for the Euclidean “Freeze - Tag” robot awakening problem Mohammad Javad Namazifar 1 , Alireza Bagheri 2 and Keivan Borna 3 1 Computer Engineering & IT Department, Qazvin Branch, Islamic Azad University Qazvin, Iran MJ.Namazifar@qiau.ac.ir 2 Computer Engineering & IT Department, Amirkabir University of Technology Tehran, Iran ar.bagheri@aut.ac.ir 3 Department of Computer Science, Kharazmi University, Faculty of Mathematics and Computer Science Tehran, Iran borna@khu.ac.ir Abstract The Freeze-Tag Problem (FTP) arises in the study of swarm robotics. The FTP is a combinatorial optimization problem that starts by locating a set of robots in a Euclidean plane. Here, we are given a swarm of n asleep (frozen or inactive) robots and a single awake (active) robot. In order to activate an inactive robot in FTP, the active robot should either be in the physical proximity to the inactive robot or "touch" it. The new activated robot starts moving and can wake up other inactive robots. The goal is to find an optimal activating schedule with the minimum time required for activating all robots. In general, FTP is an NP- Hard problem and in the Euclidean space is an open problem. In this paper, we present a recursive approximation algorithm with a constant approximation factor and a linear running time for the Euclidean Freeze-Tag Problem Keywords: Freeze Tag Problem, Recursive Algorithm, Optimization, Swarm Robotics, Computational Geometry. 1. Introduction FTP was first introduced by Joseph Mitchell at the Fall Workshop on Computational Geometry. FTP is a combinatorial optimization problem that arises in the field of swarm robotics. At the start, in this swarm, there is only one “awake” or active robot )   , and all the other robots are “asleep”. The goal is to awaken all of the inactive robots by the first robot in the shortest possible time. Activating the robots occurs only by touching or being in the physical proximity to the asleep robots. Once a robot is awakened, it can assist in activating other sleeping robots. The problem will only be ended when all the robots are awakened [1]. FTP, has its name from a children’s game called “Freeze Tag”, where a player is chosen to be “it” who tries to tag )touch) other players. Once players are tagged by “it” they must “freeze” in place until an uncaptured player tag )un- freeze) them and thereby return them to the game. The traditional FTP takes the special case where there is only one uncaptured player remaining and ‘it’ is incapacitated and will not be attempting to re-freeze players. The one remaining uncaptured player must release all the frozen players in the shortest possible time (in scheduling terminology, this is called the makespan). Once a player is unfrozen, he or she assists by unfreezing other frozen players. As in Arkin et al’s [2] research on FTP the frozen players are replaced by stationary robots and ‘it’ is removed from the problem. This revised terminology is used in this research. The problem can be demonstrated by a directed graph in which robots sleep at the vertices and the edges show the direction of an awake robot’s movement in order to awaken an asleep robot. Since, each robot can be awakened just once, and each awake robot can wake up only one asleep robot at a time, the solution to the problem would be a directed binary tree, with   as its root [2], which is obtained as follows: In order to awaken an asleep robot from a set of asleep robots  , robot   moves to the location of the asleep robot   . Once the robot   reaches to the proximity of the robot   , robot   is awakened then both robots   and   start to move from the location of robot   towards the other asleep robots. The process of awakening continues until the last robot is awakened. According to the structure of this tree, obtaining a solution to the FTP requires finding a minimum spanning tree in a complete weighted graph, provided that the out- degree of the root node must be equal to 1. The resulted tree determines the schedule and order of the process of awakening the robots, and is therefore called an “awakening tree” [2]. The objective of the FTP is to find an awakening tree that makes the “makespan” (i.e. the ACSIJ Advances in Computer Science: an International Journal, Vol. 4, Issue 5, No.17 , September 2015 ISSN : 2322-5157 www.ACSIJ.org 87 Copyright (c) 2015 Advances in Computer Science: an International Journal. All Rights Reserved.