1. Stream-based Geometric Algorithms
Piotr Indyk
MIT

2. Streaming Algorithms for Geometric Problems
Input: a stream S=p1…pn of points in Rd
Goal: compute certain geometric quantity and/or structure
Variations:
Dynamic case: points can be deleted
Sliding window: points disappear after some time t

3. Minimum Spanning Tree The tree has representation size (n)
We only estimate the cost of MST

4. Minimum Weight Matching

5. Minimum Weight Bichromatic Matching

6. Facility Location Goal: choose a set F of facilities to minimize the
sum of the distances to nearest facility plus
the number of facilities times f

7. K-median K is given Goal: choose K medians to minimize the sum of
the distances to the nearest median

8. Known Results Computing Lp norms of a stream (Graham’s talk)
Clustering of points in metric spaces
Charikar et al ’97, ’03; Guha et al’00:
K-center and K-median
(K) space, no deletions
Meyerson’02:
Facility location
(|F|) space, no deletions

9. More of Known Results Approximate diameter etc Convex hulls etc
Indyk’03: high dimensions
Feigenbaum et al, Hershberger et al, Cormode et al’03: low dimensions
Convex hulls etc

10. *follows Charikar’02; also Varadarajan’02 and Indyk-Thaper’02
Our Results
Problem
Type
Delete
Space
Appr.
MST
Cost
Yes
polylog(D,n)
log D
MWM
MWBM*
Fac.Loc. No
log2 D
K-median
Full
poly(K,log D,log n)
*follows Charikar’02; also Varadarajan’02 and Indyk-Thaper’02

11. Applications MST, MWM: ? MWBM: similarity of low-dim data sets
Fac. Loc. : “clusterability” of a data set
K-median: allocation of servers to clients (Muthu’03)
log D might be not so bad in practice
(1.1 in Indyk-Thaper’03)

12. Approach
Impose square grids G0…Gk, with side lengths 20,21, …, 2k , shifted at random.
For each square cell c in Gi, let nP(c) be the number of points from P in c.
The algorithms will maintain certain statistics over nP(.), which will allow it to approximately solve the problems 2 1 1 3 1 1 3

13. Estimators MST: ∑i 2i ∑c Gi [nP(c)>0]
MWM: ∑i 2i ∑c Gi [nP(c) is odd]
MWBM: ∑i 2i ∑c Gi |nG(c)-nB(c)|
Fac. Loc.: ∑i 2i ∑c Gi min[nP(c), Ti]
K-median: ∑c Bj nP(c)
for B1…Bl sampled from Gi’s with density 1/K

14. Proofs
View the grids as a probabilistic embedding of P into a tree (HST’s)
Show how to solve the problem in HST’s
Show how to express the solution using just nP(c)’s
First application of this kind of embeddings to streaming

15. Conclusions and Open Problems
Replace log D by O(1)
Other apps ?