Kfir Barhum, Oded Goldreich and Adi Shraibman

1. Kfir Barhum, Oded Goldreich and Adi Shraibman On Approximating the Average Distance Between Points Weizmann Institute of Science

2. General Metric: Average Distance Between Points Given: Approximate:

3. Two Approaches 1. Manipulating the object itself 2. Sampling and averaging

4. First Approach: Projection on a random direction • Project all points on a random direction • Use a simple algorithm for the 1-D case • The expected value is a fraction of the average • To get a -approximation repeat times Yields a -approximation in time

5. Second Approach: Using a random sample • General metric • Sample a pair of points selected uniformly • Use a sample of size to get a -approximation

6. Comparison for the Euclidean Case Time Randomness 1st Projection 2nd Averaging

7. A Universal Approximator Definition:A multi-set of pairs is called auniversal (L,U)-approximatorif for every metric it holds that:

8. Universal Approximator: A construction The k-dimensional hypercube on n vertices • Vertices: • Edges: + add k self-loops to each vertex Thm 1:This is a -Approximator #edges: Lower factor: Use k-long Canonical Paths to connect each pair of vertices Upper factor: Use regularity of the graph.

9. Universal Approximator: A lower bound Thm 2: A -universal approximator must have edges. Idea: * Out degrees * Consider the (distance) metric induced by the graph itself

10. Universal Approximators, revisited Definition:A multi-set of pairs is called a(L,U)-approximator for class Mif for every it holds that:

11. Thm 3:There exists a -Approximator with edges for the Euclidean Metric Reduction to the 1-D case:

12. A -regular expander with parameter is a -strong expander. Strong Expanders Definition:An (undirected) graph G=([n],E) is called a -strong expanderif for every it holds that: where . Lemma:

13. A -Approximator for the line • Consider “Sorting permutation” For every :

14. Thank You!