528 research outputs found
Deterministic Fully Dynamic SSSP and More
We present the first non-trivial fully dynamic algorithm maintaining exact
single-source distances in unweighted graphs. This resolves an open problem
stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019].
Previous fully dynamic single-source distances data structures were all
approximate, but so far, non-trivial dynamic algorithms for the exact setting
could only be ruled out for polynomially weighted graphs (Abboud and
Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main
case for which neither a subquadratic dynamic algorithm nor a quadratic lower
bound was known.
Our dynamic algorithm works on directed graphs, is deterministic, and can
report a single-source shortest paths tree in subquadratic time as well. Thus
we also obtain the first deterministic fully dynamic data structure for
reachability (transitive closure) with subquadratic update and query time. This
answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019].
Finally, using the same framework we obtain the first fully dynamic data
structure maintaining all-pairs -approximate distances within
non-trivial sub- worst-case update time while supporting optimal-time
approximate shortest path reporting at the same time. This data structure is
also deterministic and therefore implies the first known non-trivial
deterministic worst-case bound for recomputing the transitive closure of a
digraph.Comment: Extended abstract to appear in FOCS 202
Fast Deterministic Fully Dynamic Distance Approximation
In this paper, we develop deterministic fully dynamic algorithms for
computing approximate distances in a graph with worst-case update time
guarantees. In particular, we obtain improved dynamic algorithms that, given an
unweighted and undirected graph undergoing edge insertions and
deletions, and a parameter , maintain
-approximations of the -distance between a given pair of
nodes and , the distances from a single source to all nodes
("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the
distances between all nodes ("APSP").
Our main result is a deterministic algorithm for maintaining
-approximate -distance with worst-case update time
(for the current best known bound on the matrix multiplication
exponent ). This even improves upon the fastest known randomized
algorithm for this problem. Similar to several other well-studied dynamic
problems whose state-of-the-art worst-case update time is , this
matches a conditional lower bound [BNS, FOCS 2019]. We further give a
deterministic algorithm for maintaining -approximate
single-source distances with worst-case update time , which also
matches a conditional lower bound.
At the core, our approach is to combine algebraic distance maintenance data
structures with near-additive emulator constructions. This also leads to novel
dynamic algorithms for maintaining -emulators that improve
upon the state of the art, which might be of independent interest. Our
techniques also lead to improved randomized algorithms for several problems
such as exact -distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st
distances using new algebraic data structure
Algorithm and Hardness for Dynamic Attention Maintenance in Large Language Models
Large language models (LLMs) have made fundamental changes in human life. The
attention scheme is one of the key components over all the LLMs, such as BERT,
GPT-1, Transformers, GPT-2, 3, 3.5 and 4. Inspired by previous theoretical
study of static version of the attention multiplication problem [Zandieh, Han,
Daliri, and Karbasi arXiv 2023, Alman and Song arXiv 2023]. In this work, we
formally define a dynamic version of attention matrix multiplication problem.
There are matrices , they represent query,
key and value in LLMs. In each iteration we update one entry in or . In
the query stage, we receive as input, and want to
answer , where is a square matrix and is a diagonal matrix. Here denote a length- vector
that all the entries are ones.
We provide two results: an algorithm and a conditional lower bound.
On one hand, inspired by the lazy update idea from [Demetrescu and
Italiano FOCS 2000, Sankowski FOCS 2004, Cohen, Lee and Song STOC 2019, Brand
SODA 2020], we provide a data-structure that uses
amortized update time, and
worst-case query time.
On the other hand, show that unless the hinted matrix vector
multiplication conjecture [Brand, Nanongkai and Saranurak FOCS 2019] is false,
there is no algorithm that can use both amortized update time, and worst query
time.
In conclusion, our algorithmic result is conditionally optimal unless hinted
matrix vector multiplication conjecture is false
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
Training (Overparametrized) Neural Networks in Near-Linear Time
The slow convergence rate and pathological curvature issues of first-order
gradient methods for training deep neural networks, initiated an ongoing effort
for developing faster - optimization
algorithms beyond SGD, without compromising the generalization error. Despite
their remarkable convergence rate ( of the training batch
size ), second-order algorithms incur a daunting slowdown in the
(inverting the Hessian
matrix of the loss function), which renders them impractical. Very recently,
this computational overhead was mitigated by the works of [ZMG19,CGH+19},
yielding an -time second-order algorithm for training two-layer
overparametrized neural networks of polynomial width .
We show how to speed up the algorithm of [CGH+19], achieving an
-time backpropagation algorithm for training (mildly
overparametrized) ReLU networks, which is near-linear in the dimension ()
of the full gradient (Jacobian) matrix. The centerpiece of our algorithm is to
reformulate the Gauss-Newton iteration as an -regression problem, and
then use a Fast-JL type dimension reduction to the
underlying Gram matrix in time independent of , allowing to find a
sufficiently good approximate solution via -
conjugate gradient. Our result provides a proof-of-concept that advanced
machinery from randomized linear algebra -- which led to recent breakthroughs
in (ERM, LPs, Regression) -- can be
carried over to the realm of deep learning as well
Nearly Optimal Communication and Query Complexity of Bipartite Matching
We settle the complexities of the maximum-cardinality bipartite matching
problem (BMM) up to poly-logarithmic factors in five models of computation: the
two-party communication, AND query, OR query, XOR query, and quantum edge query
models. Our results answer open problems that have been raised repeatedly since
at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck,
Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan
SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21]
and Zhang [ICALP'04]. We also settle the communication complexity of the
generalizations of BMM, such as maximum-cost bipartite -matching and
transshipment; and the query complexity of unique bipartite perfect matching
(answering an open question by Beniamini [2022]). Our algorithms and lower
bounds follow from simple applications of known techniques such as cutting
planes methods and set disjointness.Comment: Accepted in FOCS 202
Flexible and stretchable electronics for wearable healthcare
Measuring the quality of human health and well-being is one of the key growth areas in our society. Preferably, these measurements are done as unobtrusive as possible. These sensoric devices are then to be integrated directly on the human body as a patch or integrated into garments. This requires the devices to be very thin, flexible and sometimes even stretchable. An overview will be given of recent technology developments in this domain and concrete application examples will be shown
- …