Tuesday, November 20, 2012

Posts from FOCS (part 5)

[Again, edited by Justin Thaler, who organized all this.  The following was written by Michael Forbes of MIT.]

Title: Higher Cell Probe Lower Bounds for Evaluating Polynomials
Author: Kasper Green Larsen

Larsen gave a very understandable talk about data structure lower bounds, where he managed to survey the two relevant prior techniques and introduce his new technique. The lower bounds he discusses are in Yao's cell-probe model. On a basic level, this model is to data structures as communication complexity is to algorithmic complexity. That is, the cell-probe model (and communication complexity) are really only concerned with the ability of a data structure (or algorithm) to move information around. These models ignore computational factors. This makes their results (as lower bounds) hold generally (even when we do care about computation), and also allows us to actually derive results (since getting unconditional lower bounds on computation is hard, but lower bounds for information are possible).

Specifically, the cell-probe model thinks of a data structure as a random-access array of cells, each with a word of some fixed size. Each word can store some sum, or some sort of pointer to another cell. To allow pointers to be meaningful, we allow the word size to be at least log(n), and typically O(log(n)) is the word size considered, as this models real-life usage of numbers. A data structure then has two parts: (1) a preprocessing step for storing information into the cells, and (2) a method for probing these cells to support queries on the original information. In this model, we will only charge for (possibly adaptive) probes to cells. Once we have a cell's information, we can perform arbitrary computation on it to determine the next probe.

The cost of any conventional data structure (as designed, say, in the word RAM model) can only decrease in the cell-probe model, as we consider computation as free. Thus, any lower bounds in the cell-probe model will necessarily apply to any reasonable model for data structures one could consider.

The first technique (by Miltersen-Nisan-Safra-Wigderson) used to prove cell-probe lower bounds was based in communication complexity. As mentioned above, the data structure can be seen as divided: the cells storing the information, and the probes used to answer queries on this information. Given this division, it seems natural to divide these two parts between two players in a communication game, albeit one that is fairly asymmetric in the inputs the players have. That is, Alice has the query that we are trying to answer, and Bob has original information. They are tasked with outputting the answer to Alice's query on Bob's data. Notice that there are two trivial protocols: Alice could send her query to Bob, or Bob could send all of the original information to Alice. These are asymmetric, as typically Alice's query will have a very small description, while Bob has many bits of information.

A data structure can then be converted into a communication protocol as follows: Bob will alone construct the cells of the data structure, and Alice will send indices of the cells she wishes to probe. Bob will return the values of those cells, and Alice will generate the next probe, and so on, until the query can be answered. With this transformation, we can now apply the techniques of communication complexity to get the needed lower bounds.

While the above idea is a good one, it is not good enough for the regime that Larsen is considering. This regime is that of data structures that only have a poly(n) number of possible queries. A good example of this is where the original information is a set of n points in the two-dimensional grid [n]x[n], where each point has a weight at most n. The data structure seeks to return the sum of the weights of the points contained in a given geometric rectangle SxT, where S,T are intervals in [n]. There are n^4 possible rectangles and thus n^4 many queries. To compare, encoding the points takes O(n log(n)) bits. Clearly, in this regime a data structure using n^4 cells can answer queries in O(1) probes. The more interesting question is the time/space trade-off: how many probes are needed when we allow the data structure to only use as many bits as needed to describe the original information?

It turns out that in this regime, the above lower bound technique cannot give lower bounds better than a constant number of probes. This is provable, in the sense that we cannot prove good communication lower bounds because there are good upper bounds: the above trivial protocols are too efficient in this setting.

A second technique (by Patrascu-Thorup), more geared to this regime, uses again the above communication framework, but now gives Alice a harder problem. Alice is given d queries to answer on the single instance Bob holds. The reason this becomes interesting is because given a data structure for this problem, we can now develop relatively better communication protocols: Alice can run all d queries in parallel. Doing this naively will not gain much, but Patrascu-Thorup observed that in one round of Alice sending the indices of the cells she wishes to probe, the order of these cells is irrelevant, as Bob will simply return their contents anyways. Thus, Alice can then send fewer bits by encoding her messages appropriately. This approach turns out to yield lower bounds of the form lg(n)/lg(lg(n)), and cannot do better for the same reason as with the first technique: the trivial protocols are too efficient.

Finally, the technique Larsen introduces diverges from the above themes. Panigrahy-Talwar-Wieder introduces a cell-sampling technique for proving lower bounds, and Larsen takes this further. So far, he can only apply the idea to the polynomial evaluation problem. In this problem, we are given a degree n polynomial over a finite field of size n^2. We will support queries to the evaluation of this polynomial to any of the n^2 points of the field. As before, we can do this in O(1) probes if we have n^2 space, and rather we wish to study the question in the O(n * polylog(n)) space regime. Larsen shows a lower bound of log(n) probes in this regime, which notably is better than any of the previous techniques could hope to prove.

To prove this lower bound, consider any small probe data structure. If we subsample the cells of the data structure (delete any given cell independently and at random) then because any query can be answered by few probes, there will be many queries that can still be answered from the sub-sampled cells. As degree n polynomials can be recovered from any n+1 evaluations, if the polynomial evaluation data structure can still answer n+1 queries with the sub-sampled cells then this means that the sub-sampled cells contain at least as much information as the space of degree n polynomials. However, if the number of probes is small enough, then we can subsample so few cells that this will reach a contradiction, as the number of sub-sampled cells will be too small to recover an arbitrary degree n polynomial. Setting the parameters correctly then yields the lower bound.

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