Fault Tolerant Clustering Revisited CCCG 2013 Nirman Kumar

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

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

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).

4 Minisum facility location (k-median) •

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

6 Nearest Neighbor Distance Metric •

7 Fault Tolerant k-median •

8 Analysis •

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)

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+2 c) proof: triangle inequality (Euclidean) on opt center • Best fault tolerant k-center ▫ 2 -approximation (Chaudhuri, et. al. ) (Khuller, et. al. )

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

12 New ideas •

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

14 k-median •

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

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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+{ci}) > cost(S-{ci}+{pj}) �S = S-{ci}+{pj}

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

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24 K-median •