1. Clustering Motion or Crunching Clusters, Hiding Logs
Sa r iel Ha r -Peled
UIUC

2. Introduction k-center clustering Input: P - set of n points in
Target: Cluster the points into k clusters.
Useful for learning, data-mining, databases, etc.
Huge field. A lot of variants.
k-center clustering
Cover with k balls of min-max radius.
𝑅 𝑑

3. K-Center clustering [Gonzalez, 85] Always pick the furthest point.
2-approximation with the triangle inequality.

4. Clustering Motion Demo
How to do k-center clustering for moving points?
Demo

5. Clustering Motion - cont'
Problem: Staying competitive with optimal (changing) optimal k clustering.
Idea:
In any point in time, stop and cluster when needed.
Slow...
Kinetic Data Structures
Large number of events
Payment even if we dont use clusters.
Cons: General, no assumptions.

6. Static Competitive Clustering
P(t) - set of moving points
Partition P(t) into "small" number of sets
s.t.
(c,m,k)-static clustering
Q: Is there (const, small val, k)-static clustering?
𝑆 1 𝑡 , 𝑆 2 𝑡 ,... 𝑆 𝑚 𝑡
𝑟𝑎𝑑𝑖𝑢𝑠 𝑆 𝑖 𝑡 ≤𝑐∗ 𝑟 𝑜𝑝𝑡 𝑃 𝑡 ,𝑘

7. Result P(t) - points move with motion of degreeμ
P(t) - points move with motion of degree
There exists a clustering.
Extension:
static clustering.
Boils down to a strange clustering problem that can be solved efficiently using new technique.
2 μ+1 −1, 𝑘 μ+1 ,𝑘
ϵ,𝑂 𝑘 ϵ 𝑑 μ+1 ,𝑘

8. Reduction Points moving linearly on the real line -> line in 2d
Clustering: Covering lines with short vertical segments
𝑟 𝑜𝑝𝑡 𝑃,𝑡 𝑡
Time axis

9. Static clustering Q: Find partition of lines? Time axis

10. Algorithm - cont' 𝑡′ 𝑡′ 𝑡′ 𝑡

11. Result P(t) - points moving linearly One can partition P(t) into sets
Q: How to compute this partition?
# of clusters is not crucial.
A: Find time t' that minimizes optimal k-center clustering
𝑘 2
𝑃 1, ..., 𝑃 𝑘 2
𝑟𝑎𝑑𝑖𝑢𝑠 𝑃 𝑖 ,𝑡 ≤3 𝑟 𝑜𝑝𝑡 𝑃,𝑡,𝑘

12. 2-Approx K-Center Clustering
[Gonzalez, 85]
Requires only the triangles inequality.
Can be implemented in O(nk) time.
[Feder, Greene, 88]
Points in cover by k balls of min-max radius
Running time
Optimal in the comparison model.
Better than approx NP-Complete.
𝑅 𝑑
𝑂 𝑛log 𝑘

13. Result
2-approx k-center clustering in linear time for geometric settings. (hiding logs...)
Algorithm - floor, randomized, hashing
Extends to other clustering variants
Covering by min-max width strips in the plane in time.
[Agarwal, Procopiuc, 00]
𝑂 𝑘log𝑘
𝑂 𝑛log 𝑘
𝑂 𝑛 𝑘 2 log 4 𝑘

14. Lemma S -sample of size m from P If -> Any k clustering of S
-good with high probability.
Known: [Alon, Dar, Parnas, Ron, 00]
[Mishra, Oblinger, Pitt, 01]
𝑚≥ logρ+μ𝑘log𝑛+βlog𝑛 ϵ ≈Ω 𝑘log𝑛 ϵ ϵ

15. Algorithm S: Sample of size C: opt/approx k-clustering of S
𝑂 𝑘log𝑛 ϵ
S: Sample of size
C: opt/approx k-clustering of S
Compute the set P' not covered by C.
By lemma:
C': opt/approx k-clustering of P'
Return as resulting clustering
∣𝑃′∣≤ϵ𝑛
𝐶∪𝐶′

16. Point-Location Idea: Use point-location data-structure.
𝑂 𝑛 𝑘 3 log 𝑛 + 𝑘 2 +𝑛log 𝑘 =𝑂 𝑛log𝑘

17. Result Compute 2k clusters Radius is Running time:
Same running time as [Feder and Greene, 88]
≤ 2r 𝑜𝑝𝑡
𝑂 𝑛log𝑘

18. Clustering algorithm needs...
Two black boxes:
A way to do k-clustering
A fast way to do point-location queries in clusters
Result:
2k-clusters comparable to k-opt clustering in linear / near-linear time. Generic.
Bottom line: approx clustering is easy if clusters # is not crucial.

19. 2-approx clustering with k clusters
Uses grid for fast O(1) point-location
Long sequence of simple observations.
Uses ideas from [Feder, Greene, 88]
Details are technical but straightforward...
Overall result:
2-approx k-center clustering in
Expected linear running time.
𝑅 𝑑

20. Conclusions
Proof that small static clustering of moving points exists.
New trick for fast clustering using point-location
Linear time algorithm for 2-approx k-center clustering.
Efficient algorithm for computing the static clustering of moving points.

21. Quote
To calm my nerves I calculated till evening the components of its trajectory, as well as the orbital perturbation caused by the presence of the lost wrench. I figured out that for the next six million years the sirloin, rotating about the ship in circular path, would lead the wrench, then catch up with it from behind and pass it again.
The Star Diaries, Stanislaw Lem.