*I missed one whole day of the workshop because of classes, and also missed a half day because of an intense burst of slide-making. While I wouldn't apologize for missing talks at a conference, it feels worse to miss them at a small focused workshop. At any rate, the usual disclaimers apply: omissions are not due to my not liking a presentation, but because of having nothing even remotely intelligent to say about it*.

Jeff Phillips led off with his work on mergeable summaries. The idea is that you have a distributed collection of nodes, each with their own data. The goal is to compute some kind of summary from all the nodes, with the caveat that each node only transmits a fixed size summary to other nodes (or the parent in an implied hierarchy). What's tricky about this is keeping the error down. It's easy to see for example that $\epsilon$-samples compose - you could take two $\epsilon$-samples and take an $\epsilon$-sample of that, giving you a $2\epsilon$-sample over the union. But you want to keep the error fixed AND the size the sample fixed. He showed a number of summary structures that could be maintained in this mergeable fashion, and there are a number of interesting questions that remain open, including how to do clustering in a mergeable way.

In the light of what I talked about earlier, you could think of the 'mergeable' model as a restricted kind of distributed computation, where the topology is fixed, and messages are fixed size. The topology is a key aspect, because nodes don't encounter data more than once. This is good, because otherwise the lack of idempotence of some of the operators could be a problem: indeed, it would be interesting to see how to deal with non-idempotent summaries in a truly distributed fashion.

Andrew McGregor talked about graph sketching problems (sorry, no abstract yet). One neat aspect of his work is that in order to build sketches for graph connectivity, he uses a vertex-edge representation that essentially looks like the cycle-basis vector in the 1-skeleton of a simplicial complex, and exploits the homology structure to compute the connected components (aka $\beta_0$). He also uses the bipartite double cover trick to reduce bipartiteness testing to connected component computation. It's kind of neat to see topological methods show up in a useful way in these settings, and his approach probably extends to other homological primitives.

Donatella Firmani and Luigi Laura talked about different aspects of graph sketching and MapReduce, studying core problems like the MST and bi/triconnectivity. Donatella's talk in particular had a detailed experimental study of various MR implementations for these problems, and had interesting (but preliminary) observations about tradeoff between the number of reducers and the amount of communication needed.

This theme was explored further by Jeff Ullman in his talk on one-pass MR algorithms (

*the actual talk title was slightly different, since the unwritten rule at the workshop was to change the name of the title from the official listing*). Again, his argument was that one should be combining both the communication cost and the overall computation cost. A particularly neat aspect of his work was showing (for the problem of finding a particular shaped subgraph in a given large graph) when there was an efficient one-pass MR algorithm, given the existence of a serial algorithm for the same problem. He called such algorithms convertible algorithms: one result type is that if there's an algorithm running in time $n^\alpha m^\beta$ for finding a particular subgraph of size $s$, and $s \le \alpha + 2\beta$, then there's an efficient MR algorithm for the problem (in the sense of total computation time being comparable to the serial algorithm).

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