4,080 research outputs found

### Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into
$n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where
$d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round
$t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and
performs a non-backtracking random walk of length $l$ from the chosen node.Then
it allocates itself on one of the visited nodes with minimum load (ties are
broken uniformly at random). Suppose that $G$ has a sufficiently large girth
and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number
of balls at any bin after allocating $n$ balls by the algorithm, called {\it
maximum load}, in terms of $l$ with high probability. We also show that the
upper bound is at most an $O(\log\log n)$ factor above the lower bound that is
proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log
n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and
$G$ has girth at least $\omega(l)$, then the maximum load attained by the
algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we
slightly modify the algorithm to have similar results for balanced allocation
on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

### Clustering, Hamming Embedding, Generalized LSH and the Max Norm

We study the convex relaxation of clustering and hamming embedding, focusing
on the asymmetric case (co-clustering and asymmetric hamming embedding),
understanding their relationship to LSH as studied by (Charikar 2002) and to
the max-norm ball, and the differences between their symmetric and asymmetric
versions.Comment: 17 page

### Heavy Hitters and the Structure of Local Privacy

We present a new locally differentially private algorithm for the heavy
hitters problem which achieves optimal worst-case error as a function of all
standardly considered parameters. Prior work obtained error rates which depend
optimally on the number of users, the size of the domain, and the privacy
parameter, but depend sub-optimally on the failure probability.
We strengthen existing lower bounds on the error to incorporate the failure
probability, and show that our new upper bound is tight with respect to this
parameter as well. Our lower bound is based on a new understanding of the
structure of locally private protocols. We further develop these ideas to
obtain the following general results beyond heavy hitters.
$\bullet$ Advanced Grouposition: In the local model, group privacy for $k$
users degrades proportionally to $\approx \sqrt{k}$, instead of linearly in $k$
as in the central model. Stronger group privacy yields improved max-information
guarantees, as well as stronger lower bounds (via "packing arguments"), over
the central model.
$\bullet$ Building on a transformation of Bassily and Smith (STOC 2015), we
give a generic transformation from any non-interactive approximate-private
local protocol into a pure-private local protocol. Again in contrast with the
central model, this shows that we cannot obtain more accurate algorithms by
moving from pure to approximate local privacy

### The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show
that if $G$ has maximum degree at most $5$ then ${\rm ind}(G) \leq 2^{{\rm
iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$
(where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated
vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices
in one partition class and $b$ in the other), with equality if and only if each
connected component of $G$ is either a complete bipartite graph or a single
vertex. This bound (for all $G$) was conjectured by Kahn.
A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq
5$ then ${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$ with
equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of
$K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and
recently proved for all $d$ by the second author, without the characterization
of the extreme cases.
Our proof involves a reduction to a finite search. For graphs with maximum
degree at most $3$ the search could be done by hand, but for the case of
maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

### Compressibility and probabilistic proofs

We consider several examples of probabilistic existence proofs using
compressibility arguments, including some results that involve Lov\'asz local
lemma.Comment: Invited talk for CiE 2017 (full version

### On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic
linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the
latter also being known as l1-heavy hitters), norm estimation, and approximate
inner product. We focus on devising a fixed matrix A in R^{m x n} and a
deterministic recovery/estimation procedure which work for all possible input
vectors simultaneously. Our results improve upon existing work, the following
being our main contributions:
* A proof that linf/l1 sparse recovery and inner product estimation are
equivalent, and that incoherent matrices can be used to solve both problems.
Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log
n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms
by making use of the Fast Johnson-Lindenstrauss transform. Both our running
times and number of measurements improve upon previous work. We can also obtain
better error guarantees than previous work in terms of a smaller tail of the
input vector.
* A new lower bound for the number of linear measurements required to solve
l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are
required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where
x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude.
* A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of
measurements required to solve deterministic norm estimation, i.e., to recover
|x|_2 +/- eps|x|_1.
For all the problems we study, tight bounds are already known for the
randomized complexity from previous work, except in the case of l1/l1 sparse
recovery, where a nearly tight bound is known. Our work thus aims to study the
deterministic complexities of these problems

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