Morten Stöckel

The Input/Output Complexity of Sparse Matrix Multiplication

We consider the problem of multiplying sparse matrices (over a semiring) where the number of non-zero entries is larger than main memory. In the classical paper of Hong and Kung (STOC ’81) it was shown that to compute a product of dense U ×U matrices, \(\Theta \left( U^3 / (B \sqrt{M}) \right)\) I/Os are necessary and sufficient in the I/O model with internal memory size M and memory block size B.

Fast Output-Sensitive Matrix Multiplication

We consider the problem of multiplying two U ×U matrices A and C of elements from a field \(\mathcal{F}\). We present a new randomized algorithm that can use the known fast square matrix multiplication algorithms to perform fewer arithmetic operations than the current state of the art for output matrices that are sparse.

Graph Reconstruction with a Betweenness Oracle.

Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes. We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem.

Quicksort, Largest Bucket, and Min-Wise Hashing with Limited Independence

Randomized algorithms and data structures are often analyzed under the assumption of access to a perfect source of randomness. The most fundamental metric used to measure how “random” a hash function or a random number generator is, is its independence: a sequence of random variables is said to be k-independent if every variable is uniform and every size k subset is independent.

Randomized Primitives for Big Data Processing

A basic question on two pieces of data is: “What is the similarity of the data?” In this extended abstract we give an overview of new developments in randomized algorithms and data structures that relate to this question. In particular we provide new state of the art methods in three particular settings, that all relate to the computation of intersection sizes:1.We give a new space-efficient summary data structure for answering set intersection size queries. The new summaries are based on one-permutation min-wise hashing, and we provide a lower bound that nearly matches our new upper bound.2.For sparse matrix multiplication, we give new tight bounds in the I/O model, settling the I/O complexity a natural parameterization of the problem—namely where the complexity depends on the input sparsity N, the output sparsity Z and the parameters of the I/O model. In the RAM model we give a new algorithm that exploits output sparsity and which beats previous known results for most of the parameter space.3.We give a new I/O efficient algorithm to compute the similarity join between two sets: two elements are members of this join if they are close according to a specified metric. Our new algorithm is based on locality-sensitive hashing and strictly improves on previous work.

I/O-Efficient Similarity Join

We present an I/O-efficient algorithm for computing similarity joins based on locality-sensitive hashing (LSH). In contrast to the filtering methods commonly suggested our method has provable sub-quadratic dependency on the data size. Further, in contrast to straightforward implementations of known LSH-based algorithms on external memory, our approach is able to take significant advantage of the available internal memory: Whereas the time complexity of classical algorithms includes a factor of

I/O-Efficient Similarity Join.

N^\rho

New Subquadratic Approximation Algorithms for the Girth.

Nρ, where