1. “Fault Tolerant Clustering Revisited” - - CCCG 2013 Nirman Kumar, Benjamin Raichel خوشه بندی مقاوم در برابر خرابی سپیده آقاملائی

2. Facility location Minimax facility location (k-center) ▫ Given n points ▫ Find k centers ▫ Minimize the maximum distance from each point to its nearest site ▫ K = 1: Minimum enclosing ball Minisum facility location (k-median) ▫ Given n points ▫ Find k centers ▫ Minimize the (weighted) sum of distances from a given set of point sites to nearest site 2

3. Minimax facility location (k-center) Exact solution: NP hard Approximation factor=approximation/optimum Approximation: also NP hard when the error is small. ▫ Approximation: NP hard when approximation factor is less than 1.822 (dimension = 2), 2 (dimension >2). 3

4. Minisum facility location (k-median) 4

5. Fault Tolerant Clustering Fault Tolerance ▫ partial failure ▫ Redundancy i fault tolerant ▫ The system can survive faults in i components and still work. Fault tolerant clustering ▫ Keep i centers instead of one 5

6. Nearest Neighbor Distance Metric 6

7. Fault Tolerant k-median 7

8. Analysis 8

9. Gonzalez’s Algorithm (k-center) “Farthest Point Clustering (FPC)” Best approximation factor for general metric spaces Total time = O(kn), n=#points, k=#clusters Algorithm: 1.C={p} (arbitrary point) 2.Find furthest point in P from C and add it to C 3.Repeat until |C|=k Implementation: keep clusters => each step O(n) 9

10. Analysis Gonzales k-center ▫ 2-approximation Fault tolerant k-center + Gonzales ▫ If i|k : 3-approximation ▫ else: 4-approximation ▫ better than 5-approximation (1+2c) ▫ proof: triangle inequality (Euclidean) on opt center Best fault tolerant k-center ▫ 2-approximation (Chaudhuri, et.al.) (Khuller, et.al.) 10

11. Future work LP-rounding (k-median) fault tolerant (Swamy, Shmoys) ▫ Needs all i-nearest servers to work Fault tolerant k-center(Chaudhuri) ▫ given a number p, we wish to place k centers so as to minimize the maximum distance of any non-center node to its pth closest center. Fault tolerant k-center(Khuller) ▫ each vertex that does not have a center placed on it is required to have at least α centers close to it. 4-approximation  2-approximation 11

12. New ideas 12

13. Based on a true story! “Fault Tolerant Clustering Revisited” CCCG 2013 By: Nirman Kumar Benjamin Raichel 13

14. k-median 14

15. Randomized rounding Yi = probability that pi is a center Assigning points to closest center: greedy 15

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17. k-median Local Search Algorithm: (3+ε)-approximation ▫ S = { k arbitrary points of P} //centers = medians ▫ Swap: while cost(S+{c i }) > cost(S-{c i }+{p j })  S = S-{c i }+{p j } 17

18. k-median Star algorithm (Pseudo approximation) ▫ (1+2/e)-approximation ▫ Create star graphs (bi-point solution)  Convex combination of 2 solutions ▫ For every star do:  Choose center as median with probability a  Otherwise choose all leaves as median 18

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24. K-median 24