31 research outputs found
Spectral Sparsification via Bounded-Independence Sampling
We give a deterministic, nearly logarithmic-space algorithm for mild spectral
sparsification of undirected graphs. Given a weighted, undirected graph on
vertices described by a binary string of length , an integer , and an error parameter , our algorithm runs in space
where
and are the maximum and minimum edge
weights in , and produces a weighted graph with
edges that spectrally approximates , in
the sense of Spielmen and Teng [ST04], up to an error of .
Our algorithm is based on a new bounded-independence analysis of Spielman and
Srivastava's effective resistance based edge sampling algorithm [SS08] and uses
results from recent work on space-bounded Laplacian solvers [MRSV17]. In
particular, we demonstrate an inherent tradeoff (via upper and lower bounds)
between the amount of (bounded) independence used in the edge sampling
algorithm, denoted by above, and the resulting sparsity that can be
achieved.Comment: 37 page
Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs
We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))
The Predicted-Deletion Dynamic Model: Taking Advantage of ML Predictions, for Free
The main bottleneck in designing efficient dynamic algorithms is the unknown
nature of the update sequence. In particular, there are some problems, like
3-vertex connectivity, planar digraph all pairs shortest paths, and others,
where the separation in runtime between the best partially dynamic solutions
and the best fully dynamic solutions is polynomial, sometimes even exponential.
In this paper, we formulate the predicted-deletion dynamic model, motivated
by a recent line of empirical work about predicting edge updates in dynamic
graphs. In this model, edges are inserted and deleted online, and when an edge
is inserted, it is accompanied by a "prediction" of its deletion time. This
models real world settings where services may have access to historical data or
other information about an input and can subsequently use such information make
predictions about user behavior. The model is also of theoretical interest, as
it interpolates between the partially dynamic and fully dynamic settings, and
provides a natural extension of the algorithms with predictions paradigm to the
dynamic setting.
We give a novel framework for this model that "lifts" partially dynamic
algorithms into the fully dynamic setting with little overhead. We use our
framework to obtain improved efficiency bounds over the state-of-the-art
dynamic algorithms for a variety of problems. In particular, we design
algorithms that have amortized update time that scales with a partially dynamic
algorithm, with high probability, when the predictions are of high quality. On
the flip side, our algorithms do no worse than existing fully-dynamic
algorithms when the predictions are of low quality. Furthermore, our algorithms
exhibit a graceful trade-off between the two cases. Thus, we are able to take
advantage of ML predictions asymptotically "for free.'