k-Center Clustering with Outliers in Sliding Windows
Abstract
:1. Introduction
1.1. Related Work
1.2. Our Contribution
- A sliding window algorithm for general metric spaces that, at any time, is able to return a set of centers covering all but at most points of the current window W, within a radius that is an factor larger than the optimal radius for z ouliers. The algorithm requires a working memory of size and processes each point in time linear in the working memory size. By setting , the number of uncovered points becomes, at most, z;
- An improved algorithm with the same coverage guarantee as above, featuring a radius that is only a factor larger than the optimal radius, at the expense of an extra factor in both the working memory size and update time, for a suitable constant c;
- A sliding-window algorithm for streams of bounded doubling dimension that, starting from a (possibly crude) lower bound on the ratio between the -effective and the full diameter of the window W, returns upper and lower upper bounds to the -effective diameter of W. The algorithm features accuracy–space tradeoffs akin to those of the improved algorithm for 1-center with outliers;
- Experimental evidence that both the improved k-center and the effective diameter algorithms feature a good performance and provide accurate solutions.
1.3. Organization of the Paper
2. Preliminaries
2.1. Definition of the Problems
2.2. Doubling Dimension
3. -Center with Outliers
3.1. Weighted Coreset Construction
3.1.1. Algorithm
- The first pair is kept in the histogram;
- If a pair is kept in the histogram, all subsequent pairs with and are deleted, except for the last such pair, if any.
Algorithm 1: update(p, t) |
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Algorithm 2: insertAttraction(p, γ) |
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Algorithm 3: updateHistrograms(L) |
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Algorithm 4: extractCoreset( ) |
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3.1.2. Analysis
- 1.
- If , then .
- 2.
- .
- 1.
- For every , ;
- 2.
- For every , or ;
- 3.
- For every , ;
- 4.
- .
3.2. Computation of the Solution from the Coreset
Algorithm 5: outliersCluster(T, k, ρ, ε) |
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- .
Algorithm 6: computeSolution( ) |
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3.3. Obliviousness to and
3.4. Improved Approximation under Bounded Doubling Dimension
4. Effective Diameter Estimation
5. Experiments
5.1. k-Center with Outliers
5.2. Effective Diameter
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Dataset | Algorithm | Obj. Ratio | Memory ( Floats) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs (z = 10) | our-sliding | ||||||
charikar | - | ||||||
samp-charikar | |||||||
gon | |||||||
Higgs (z = 50) | our-sliding | ||||||
charikar | - | - | |||||
samp-charikar | |||||||
gon | |||||||
Cover (z = 10) | our-sliding | ||||||
charikar | 1.02 * | ||||||
samp-charikar | |||||||
gon | |||||||
Cover (z = 50) | our-sliding | ||||||
charikar | - | ||||||
samp-charikar | |||||||
gon | |||||||
Higgs+ (z = 10) | our-sliding | ||||||
charikar | - | ||||||
samp-charikar | |||||||
gon | |||||||
Cover+ (z = 10) | our-sliding | ||||||
charikar | 1.02 * | ||||||
samp-charikar | |||||||
gon |
Dataset | Algorithm | Update Time (ms) | Query Time (s) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs (z = 10) | our-sliding | ||||||
charikar | - | ||||||
samp-charikar | |||||||
Higgs (z = 50) | our-sliding | ||||||
charikar | – | – | |||||
samp-charikar | |||||||
Cover (z = 10) | our-sliding | ||||||
charikar | * | ||||||
samp-charikar | |||||||
Cover (z = 50) | our-sliding | ||||||
charikar | - | ||||||
samp-charikar | |||||||
Higgs+ (z = 10) | our-sliding | ||||||
charikar | – | ||||||
samp-charikar | |||||||
Cover+ (z = 10) | our-sliding | ||||||
charikar | * | ||||||
samp-charikar |
Dataset | Algorithm | Clustering Radius | Memory ( Floats) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs (z = 10) | - | - | |||||
Cover (z = 10) | |||||||
Dataset | Algorithm | Update Time (ms) | Query Time (s) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs (z = 10) | - | - | |||||
Cover (z = 10) | |||||||
Dataset | Algorithm | Diameter Ratio | Memory ( floats) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs-eff | eff-sliding | ||||||
eff-sequential | - |
Dataset | Algorithm | Update Time (ms) | Query Time (s) | ||||
---|---|---|---|---|---|---|---|
Window Size | Window Size | ||||||
Higgs-eff | eff-sliding | ||||||
eff-sequential | - |
Dataset | Eff. Diameter | Memory ( floats) | Update Time (ms) | Query Time (s) |
---|---|---|---|---|
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Pellizzoni, P.; Pietracaprina, A.; Pucci, G. k-Center Clustering with Outliers in Sliding Windows. Algorithms 2022, 15, 52. https://doi.org/10.3390/a15020052
Pellizzoni P, Pietracaprina A, Pucci G. k-Center Clustering with Outliers in Sliding Windows. Algorithms. 2022; 15(2):52. https://doi.org/10.3390/a15020052
Chicago/Turabian StylePellizzoni, Paolo, Andrea Pietracaprina, and Geppino Pucci. 2022. "k-Center Clustering with Outliers in Sliding Windows" Algorithms 15, no. 2: 52. https://doi.org/10.3390/a15020052