The power of vertex sparsifiers in dynamic graph algorithms

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest

Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in $O(\text{polylog}(n))$ time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating the interleaved queries and updates to this randomized data structure may be of independent interest.Comment: 58 pages, 3 figures. This is a substantially revised version of arXiv:1711.08446 with an emphasis on the underlying theoretical problem

Smith Normal Form over Local Rings and Related Problems

The Smith normal form is a diagonalization of matrices with many applications in diophantine analysis, graph theory, system control theory, simplicial homology, and more recently, in topological analysis of big data. Efficient computation of Smith normal form is a well-studied area for matrices with integer and polynomial entries. Existing successful algorithms typically rely on elimination for dense matrices and iterative Krylov space methods for sparse matrices. Our interest lies in computing Smith normal form for sparse matrices over local rings, where traditional iterative methods face challenges due to the lack of unique minimal polynomials. We explore different approaches to tackling this problem for two local rings: the integers modulo a prime power, and the polynomials modulo a power of an irreducible polynomial. Over local polynomial rings, we find success in linearization into larger dimension matrices over the base field. Effectively we transform the problem of computing the Smith normal form into a small number of rank problems over the base field. The latter problem has existing efficient algorithms for sparse and dense matrices. The problem is harder over local integer rings. We take the approach of hybrid sparse-dense algorithms. We also tackle a restricted version of the problem where we detect only the first non-trivial invariant factor. We also give an algorithm to find the first few invariant factors using iterative rank-1 updates. This method becomes dense when applied to finding all the invariant factors. We digress slightly into the related problem of preconditioning. We show that linear- time preconditioners are suitable for computing Smith normal form, and computing nullspace samples. For the latter problem we design an algorithm for computing uniform samples from the nullspace. On a separate track, we focus on the properties of the Smith normal form decomposition. We relate the invariant factors to the eigenvalues. Our ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices