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The Freeze Tag Problem: How to wake up a swarm of robots

The Freeze Tag Problem: How to wake up a swarm of robots. B88505002 資訊三 曾宏偉 B88506043 資訊三 林明鴻 B88506065 資訊三 林宗茂. What ’ s is the freeze tag problem?. A set of n robots, modeled as points on vertices in some metric space. Initially, only one “ awake ” robot, and all others are asleep.

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The Freeze Tag Problem: How to wake up a swarm of robots

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  1. The Freeze Tag Problem: How to wake up a swarm of robots B88505002 資訊三 曾宏偉 B88506043 資訊三 林明鴻 B88506065 資訊三 林宗茂

  2. What’s is the freeze tag problem? • A set of n robots, modeled as points on vertices in some metric space. • Initially, only one “awake” robot, and all others are asleep. • Goal: to wake up all the robots a.s.a.p.minimize the “makespan”. • To wake a robot, an awake must go to its location; once awake, the new awaked robot can wake up other robots.

  3. Input: • A graph G=(V,E) l(ei) stands for the weight(length) of edge ei belongs to EThe cost of a robot passing ei; n(vi) stands for the number of robots on vertex vi belongs to V.

  4. Output: • Minimize to makespan of traversal all vertices using awakened robots.

  5. Facts • NP-hard, even for star graphs with an equal number of robots at each vertex.

  6. Greedy Algorithm for Star Graph • Scenario: Simplify the graph into star graph, with one awake robot a central node, and only one robot on each leaf. • Sort edges into l(e1)< l(e2)<……< l(en) • Go to the shortest edge first, and after visit a node, a new robot can awake other robots. • If all nodes are visited, done.

  7. P-time? Feasible Solution? • P-time? A: Of course! Since this is a greedy algorithm. • Feasible Solution? A: Sure, it really visits all node, and wake up all robots. The algorithm seems perfect, however…

  8. Consider the graph… 2k-1 edges with length 1 2k edges with length k 1 edge with length 3k

  9. What Greedy will do? • Go to the edge with length 1 first, and wake up all robots with distance to central node as 1, cost 2*lg2k=2k • Then the 2k to the edge with kcost k. • Finally go to the edge with length 3kcost k(for a r go back to central)+3k4k • Sum up to 7K. • However, this can be finish at 3k+4wake up a robot at edge length 1, at time 2, wake up the node with length 3k. At time 4, a robot go to edge with length k to wake robotssum up only to 3k+4.

  10. Wow! The ratio comes up to 7/3 • Is 7/3 the approximation ratio?

  11. Wow! The ratio comes up to 7/3(Cont.)

  12. A PTAS • Divide the edges into long and short sets L stands for the set of long edges. S stands for the set of short edges. T stands for (3/7)*Opt. Sol. of Greedy Alg.

  13. A PTAS (cont.) • 1. Completely Enumeration for L, to find the best wake up location for edges in L. • 2. Greedy for fill short edges into long edges. • result in an 1+ -approximation alg.

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