213 research outputs found
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 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
The White-Box Adversarial Data Stream Model
We study streaming algorithms in the white-box adversarial model, where the
stream is chosen adaptively by an adversary who observes the entire internal
state of the algorithm at each time step. We show that nontrivial algorithms
are still possible. We first give a randomized algorithm for the -heavy
hitters problem that outperforms the optimal deterministic Misra-Gries
algorithm on long streams. If the white-box adversary is computationally
bounded, we use cryptographic techniques to reduce the memory of our
-heavy hitters algorithm even further and to design a number of additional
algorithms for graph, string, and linear algebra problems. The existence of
such algorithms is surprising, as the streaming algorithm does not even have a
secret key in this model, i.e., its state is entirely known to the adversary.
One algorithm we design is for estimating the number of distinct elements in a
stream with insertions and deletions achieving a multiplicative approximation
and sublinear space; such an algorithm is impossible for deterministic
algorithms.
We also give a general technique that translates any two-player deterministic
communication lower bound to a lower bound for {\it randomized} algorithms
robust to a white-box adversary. In particular, our results show that for all
, there exists a constant such that any -approximation
algorithm for moment estimation in insertion-only streams with a
white-box adversary requires space for a universe of size .
Similarly, there is a constant such that any -approximation algorithm
in an insertion-only stream for matrix rank requires space with a
white-box adversary. Our algorithmic results based on cryptography thus show a
separation between computationally bounded and unbounded adversaries.
(Abstract shortened to meet arXiv limits.)Comment: PODS 202
Adversarially Robust Property-Preserving Hash Functions
Property-preserving hashing is a method of compressing a large input x into a short hash h(x) in such a way that given h(x) and h(y), one can compute a property P(x, y) of the original inputs. The idea of property-preserving hash functions underlies sketching, compressed sensing and locality-sensitive hashing.
Property-preserving hash functions are usually probabilistic: they use the random choice of a hash function from a family to achieve compression, and as a consequence, err on some inputs. Traditionally, the notion of correctness for these hash functions requires that for every two inputs x and y, the probability that h(x) and h(y) mislead us into a wrong prediction of P(x, y) is negligible. As observed in many recent works (incl. Mironov, Naor and Segev, STOC 2008; Hardt and Woodruff, STOC 2013; Naor and Yogev, CRYPTO 2015), such a correctness guarantee assumes that the adversary (who produces the offending inputs) has no information about the hash function, and is too weak in many scenarios.
We initiate the study of adversarial robustness for property-preserving hash functions, provide definitions, derive broad lower bounds due to a simple connection with communication complexity, and show the necessity of computational assumptions to construct such functions. Our main positive results are two candidate constructions of property-preserving hash functions (achieving different parameters) for the (promise) gap-Hamming property which checks if x and y are "too far" or "too close". Our first construction relies on generic collision-resistant hash functions, and our second on a variant of the syndrome decoding assumption on low-density parity check codes
Analysis and Maintenance of Graph Laplacians via Random Walks
Graph Laplacians arise in many natural and artificial contexts. They are linear systems associated with undirected graphs. They are equivalent to electric flows which is a fundamental physical concept by itself and is closely related to other physical models, e.g., the Abelian sandpile model. Many real-world problems can be modeled and solved via Laplacian linear systems, including semi-supervised learning, graph clustering, and graph embedding.
More recently, better theoretical understandings of Laplacians led to dramatic improvements across graph algorithms. The applications include dynamic connectivity problem, graph sketching, and most recently combinatorial optimization. For example, a sequence of papers improved the runtime for maximum flow and minimum cost flow in many different settings.
In this thesis, we present works that the analyze, maintain and utilize Laplacian linear systems in both static and dynamic settings by representing them as random walks. This combinatorial representation leads to better bounds for Abelian sandpile model on grids, the first data structures for dynamic vertex sparsifiers and dynamic Laplacian solvers, and network flows on planar as well as general graphs.Ph.D
A Fast Minimum Degree Algorithm and Matching Lower Bound
The minimum degree algorithm is one of the most widely-used heuristics for
reducing the cost of solving large sparse systems of linear equations. It has
been studied for nearly half a century and has a rich history of bridging
techniques from data structures, graph algorithms, and scientific computing. In
this paper, we present a simple but novel combinatorial algorithm for computing
an exact minimum degree elimination ordering in time, which improves on
the best known time complexity of and offers practical improvements
for sparse systems with small values of . Our approach leverages a careful
amortized analysis, which also allows us to derive output-sensitive bounds for
the running time of , where is
the number of unique fill edges and original edges that the algorithm
encounters and is the maximum degree of the input graph.
Furthermore, we show there cannot exist an exact minimum degree algorithm
that runs in time, for any , assuming
the strong exponential time hypothesis. This fine-grained reduction goes
through the orthogonal vectors problem and uses a new low-degree graph
construction called -fillers, which act as pathological inputs and cause any
minimum degree algorithm to exhibit nearly worst-case performance. With these
two results, we nearly characterize the time complexity of computing an exact
minimum degree ordering.Comment: 17 page
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