CMU Randomized Algorithms

Randomized Algorithms, Carnegie Mellon: Spring 2011

Lecture #16: Odds and ends

Some examples for the CKR decomposition scheme

Varun asked why the CKR algorithm differs in so many ways from Bartal’s procedure:

  • it chooses a single radius instead of choosing an independent radius for each piece it carves out,
  • it chooses the radius from a uniform distribution,
  • it picks the next vertex randomly (instead of arbitrarily), and moreover,
  • it also grows regions from vertices that have been captured in previously carved out pieces.

Here are some examples that might clarify the situation:

Suppose we choose a random radius R uniformly from [r/4,r/2], but independently for each center. Then in the following example, each time a leaf is picked, the unit-length edge (u,v) will be cut with probability 2/r. If there are n leaves, the edge will be cut with probability about Theta(n/r) >> O(log n)/r.

Example 1

Now, suppose we choose a single radius R uniformly from [r/4,r/2], but choose the centers in arbitrary (adversarial) order. In the following example, if we choose the centers in the order l_{r/2-1}, …, l_2, l_1, then we cut the edge (u,v) for sure (with probability 1). Note that in this case, the random order means there’s a good chance that early on in the process, we pick some vertex l_i with some small value of $i$. Since this leaf is close to the edge (u,v), it will put both u and v in the same set.

Example 2

So yeah, once we go with a uniformly random choice of R (instead of choosing it from a geometric distribution), we run into trouble if we re-sample R independently each time time we grow a region, or if we choose the next vertex to grow from in an worst-case fashion.

Finally, what about the growing balls from centers that have already been carved out? That is important for the analysis, because it makes the process depend very mildly on the past evolution. Would the algorithm break down if we did not do that? I am blanking on a bad example right now, but I think there should be one. Let me know if you see it.

Some citations for k-server

In STOC 2008, Cote, Meyerson, and Poplawski gave a randomized algorithm for the k-server problem on certain special kinds of HSTs that achieved a poly-logarithmic competitive ratio. In SODA 2009, Bansal, Buchbinder, and Naor abstracted out certain “convexity” properties, which if we could prove, would give a polylogarithmic competitive ratio for general HSTs, and hence for all metric spaces using the Theorems we saw in class. (See also their ICALP 2009 paper.) It’d be great to make progress on this problem.

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