Priceless: seeing the look on a speaker's face when they're asked for the exact constant in their 'constant factor approximation'
Yes, it's SODA time. And having an iphone means I can leave my laptop in my room, which means blogging waits till the day is over.
It was a good first day. I chaired the session on (as I put it), "geometry, graphs, and the geometry of graphs", and really enjoyed all the papers in the session (David has a summary of the Lee-Sidiropoulos talk). The phrase of the session was 'Poincare inequality'.
Alex Andoni talked about work with T. S. Jayram and Mihai Pătraşcu on lower bounds for computing the edit distance (and related distances like the Ulam distance). The program of attack was via the use of information complexity - a technique first developed for use in streaming lower bounds, but which has general applicability to communication complexity problems. Here's roughly speaking how the argument goes:
The direct-sum family of results says that the communication complexity of a function f expressible as an AND of n functions g is at least n * the information complexity of g. The overall plan is therefore to break down the strings being compared into pieces, and lower bound the information complexity of each piece.
Now let g be a threshold distance function that correctly reports if the distance is "too small" or "too large", but not inbetween. It turns out that the information complexity of g can be lower bounded by a constant relating to the Poincare inequality. So what is this inequality ?
In a sense, it captures the difficulty of distinguishing short distances from long. Specifically, look at the average distance of pairs of points sampled over all "short" pairs, and do the same for "long pairs". If the two resulting numbers are within some constant, then that's the constant used above, and intuitively tells us that we can't tell the pairs apart distributionally speaking.
It's not easy in general to prove Poincare inequalities, although these appear to be at the heart of non-embeddability results. What the authors go on to do is show that if the distance metric being used is "complex" i.e has a reasonably large communication complexity, then this can be used to show a Poincare-type inequality.