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

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

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

4. 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 First Approach:

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

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

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

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

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

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

14. Thank You!