Tag: algorithms

The Symposium on Theory of Computing (STOC) covers research within theoretical computer science, such as algorithms and computation theory. This year, four papers from CS researchers and collaborators from various institutions made it into the conference.

Local Decodability of the Burrows-Wheeler Transform
Sandip Sinha Columbia University and Omri Weinstein Columbia University

The researchers were interested in the problem of compressing texts with local context, like texts in which there is some correlation between nearby characters. For example, the letter ‘q’ is almost always followed by ‘u’ in an English text.

It is a reasonable goal to design compression schemes that exploit local context to reduce the length of the string considerably. Indeed, the FM-Index and other such schemes, based on a transformation called the Burrows-Wheeler transform followed by Move-to-Front encoding, have been widely used in practice to compress DNA sequences etc. “I think it’s interesting that compression schemes have been known for nearly 20 years in the pattern-matching and bioinformatics community but there has not been satisfactory theoretical guarantees of the compression achieved by these algorithms,” said Sandip Sinha, a PhD student in the Theory Group.

Moreover, these schemes are inherently non-local – in order to extract a character or a short substring at a particular position of the original text, one needs to decode the entire string, which requires time proportional to the length of the original string. This is prohibitive in many applications. The team designed a data structure which matches almost exactly the space bound of such compression schemes, while also supporting highly efficient local decoding queries (alluded to above), as well as certain pattern-matching queries. In particular, they were able to design a succinct “locally-decodable” Move-to-Front (MTF) code, that reduces the decoding time per character (in the MTF encoding) from n to around log(n), where n is the length of the string. Shared Sinha, “We also show a lower bound showing that for a wide class of strings, one cannot hope to do much better using any data structure based on the above transform.”

“Hopefully our paper draws wider attention of the theoretical CS community to similar problems in these fields,” said Sinha. To that end, they have made a conscious effort to make the paper accessible across research domains. “I also think there is no significant mathematical knowledge required to understand the paper, beyond some basic notions in information theory.”

Fooling Polytopes
Ryan O’Donnell Carnegie Mellon University, Rocco A. Servedio Columbia University, Li-Yang Tan Stanford University

The paper is about “getting rid of the randomness in random sampling”.  

Suppose you are given a complicated shape on a blackboard and you need to estimate what fraction of the blackboard’s area is covered by the shape.  One efficient way to estimate this fraction is by doing random sampling: throw darts randomly at the blackboard and count the fraction of the darts that land inside the shape. If you throw a reasonable number of darts, and they land uniformly at random inside the blackboard, the fraction of darts that land inside the shape will be a good estimate of the actual fraction of the blackboard’s area that is contained inside the shape. (This is analogous to surveying a small random sample of voters to try and predict who will win an election.)  

“This kind of random sampling approach is very powerful,” said Rocco Servedio, professor and chair of the computer science department. “In fact, there is a sense in which every randomized computation can be viewed as doing this sort of random sampling.”  

It is a fundamental goal in theoretical computer science to understand whether randomness is really necessary to carry out computations efficiently.  The point of this paper is to show that for an important class of high-dimensional estimation problems of the sort described above, it is actually possible to come up with the desired estimates efficiently without using any randomness at all.

In this specific paper, the “blackboard” is a high-dimensional Boolean hypercube and the “shape on the blackboard” is a subset of the hypercube defined by a system of high-dimensional linear inequalities (such a subset is also known as a polytope).  Previous work had tried to prove this result but could only handle certain specialized types of linear inequalities. By developing some new tools in high dimensional geometry and probability, in this paper the researchers were able to get rid of those limitations and handle all systems of linear inequalities.

Static Data Structure Lower Bounds Imply Rigidity
Zeev Dvir Princeton University, Alexander Golovnev Harvard University, Omri Weinstein Columbia University

The paper shows an interesting connection between the task of proving time-space lower bounds on data structure problems (with linear queries), and the long-standing open problem of constructing “stable” (rigid) matrices — a matrix M whose rank remains very high unless a lot of entries are modified. Constructing rigid matrices is one of the major open problems in theoretical computer science since the late 1970s, with far-reaching consequences on circuit complexity.

The result shows a real barrier for proving lower bounds on data structures: If one can exhibit any “hard” data structure problem with linear queries (the canonical example being Range Counting queries: given n points in d dimensions, report the number of points in a given rectangle), then this problem can be essentially used to construct “stable” (rigid) matrices.

“This is a rather surprising ‘threshold’ result, since in slightly weaker models of data structures (with small space usage), we do in fact have very strong lower bounds on the query time,” said Omri Weinstein, an assistant professor of computer science. “Perhaps surprisingly, our work shows that anything beyond that is out of reach with current techniques.”

Testing Unateness Nearly Optimally
Xi Chen Columbia University, Erik Waingarten Columbia University

The paper is about testing unateness of Boolean functions on the hypercube.

For this paper the researchers set out to design highly efficient algorithms which, by evaluating very few random inputs of a Boolean function, can “test” whether the function is unate (meaning that every variable is either non-increasing or non-decreasing or is pretty non-unate).

Referring to a previous paper the researchers set out to create an algorithm which is optimal (up to poly-logarithmic factors), giving a lower bound on the complexity of these testing algorithms.

An example of a Boolean function which is unate is a halfspace, i.e., for some values w₁, …, w_n, θ ∈ ℝ, the function f : {0,1}ⁿ → {0,1} is given by f(x) = 1 if ∑ w_i x_i≥ θ and 0 otherwise. Here, every variable i ∈ [n] is either non-decreasing, when w_i ≥ 0, or non-increasing, when w_i ≤ 0.

“One may hope that such an optimal algorithm could be non-adaptive, in the sense that all evaluations could be done at once,” said Erik Waingarten, an algorithms and computational complexity PhD student. “These algorithms tend to be easier to analyze and have the added benefit of being parallelize-able.”

However, the algorithm they developed is crucially adaptive, and a surprising thing is that non-adaptive algorithms could never achieve optimal complexity. A highlight of the paper is a new analysis of a very simple binary search procedure on the hypercube.

“This procedure is the ‘obvious’ thing one would do for these kinds of algorithms, but analyzing it has been very difficult because of its adaptive nature,” said Waingarten. “For us, this is the crucial component of the algorithm.”