Sublinear Algorithmic Tools 2 Alex Andoni Plan Dimension

Sublinear Algorithmic Tools 2 Alex Andoni

Plan � Dimension reduction � Application: Numerical Linear Algebra � Sketching � Application: � and 2 more… Streaming Nearest Neighbor Search

Dimension reduction in other norms/distances? �

Computational view �

� [I’ 00] [SW’ 11, MM’ 13, WZ’ 13, WW’ 18]

Today Application: Streaming 1 131. 107. 65. 14 Challenge: log statistics of the data, using small space 18. 0. 1. 12 131. 107. 65. 14 IP Frequency 131. 107. 65. 1 4 3 18. 0. 1. 12 2 80. 97. 56. 20 2 127. 0. 0. 1 9 192. 168. 0. 1 8 257. 2. 5. 7 0 80. 97. 56. 20 18. 0. 1. 12 80. 97. 56. 20 131. 107. 65. 14

Streaming statistics � IP Frequency 131. 107. 65. 1 4 3 18. 0. 1. 12 2 80. 97. 56. 20 2

2 nd frequency moment via DR �

Streaming Scenario 2 131. 107. 65. 14 80. 97. 56. 20 18. 0. 1. 12 IP Frequency 131. 107. 65. 1 4 1 18. 0. 1. 12 2 80. 97. 56. 20 1 Similar Qs: average delay/variance in a network differential statistics between logs at different servers, etc

Sketching for Difference � IP Frequency 131. 107. 65. 1 1 4 131. 107. 65. 14 1 18. 0. 1. 12 1 80. 97. 56. 20 1 2 010110 010101

Streaming 3: # distinct elements � IP Frequency 131. 107. 65. 1 1 4 18. 0. 1. 12 2

Initialize: min. Hash=1 hash function h into [0, 1] Distinct Elements [Flajolet-Martin’ 85, Alon-Matias-Szegedy’ 96] � Process(int i): if (h(i) < min. Hash) min. Hash = h(index); Output: 1/min. Hash-1 5 7 2