Relations between automata and the simple k-path problem

Let $G$ be a directed graph on $n$ vertices. Given an integer $k<=n$, the SIMPLE $k$-PATH problem asks whether there exists a simple $k$-path in $G$. In case $G$ is weighted, the MIN-WT SIMPLE $k$-PATH problem asks for a simple $k$-path in $G$ of minimal weight. The fastest currently known deterministic algorithm for MIN-WT SIMPLE $k$-PATH by Fomin, Lokshtanov and Saurabh runs in time $O(2.851^k\cdot n^{O(1)}\cdot \log W)$ for graphs with integer weights in the range $[-W,W]$. This is also the best currently known deterministic algorithm for SIMPLE k-PATH- where the running time is the same without the $\log W$ factor. We define $L_k(n)\subseteq [n]^k$ to be the set of words of length $k$ whose symbols are all distinct. We show that an explicit construction of a non-deterministic automaton (NFA) of size $f(k)\cdot n^{O(1)}$ for $L_k(n)$ implies an algorithm of running time $O(f(k)\cdot n^{O(1)}\cdot \log W)$ for MIN-WT SIMPLE $k$-PATH when the weights are non-negative or the constructed NFA is acyclic as a directed graph. We show that the algorithm of Kneis et al. and its derandomization by Chen et al. for SIMPLE $k$-PATH can be used to construct an acylic NFA for $L_k(n)$ of size $O^*(4^{k+o(k)})$. We show, on the other hand, that any NFA for $L_k(n)$ must be size at least $2^k$. We thus propose closing this gap and determining the smallest NFA for $L_k(n)$ as an interesting open problem that might lead to faster algorithms for MIN-WT SIMPLE $k$-PATH. We use a relation between SIMPLE $k$-PATH and non-deterministic xor automata (NXA) to give another direction for a deterministic algorithm with running time $O^*(2^k)$ for SIMPLE $k$-PATH

Parameterized Distributed Algorithms

In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds. Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2

Efficient Summing over Sliding Windows

This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of {\Omega}(1/{\epsilon} + log W) memory bits for W{\epsilon}-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/{\epsilon} + log W) bits, indicating that the algorithm is optimal and that the bound is tight. Next, the more general problem of maintaining a sum of the last W integers, each in the range of {0,1,...,R}, is addressed. The paper shows that approximating the sum within an additive error of RW{\epsilon} can also be done using {\Theta}(1/{\epsilon} + log W) bits for {\epsilon}={\Omega}(1/W). For {\epsilon}=o(1/W), we present a succinct algorithm which uses B(1 + o(1)) bits, where B={\Theta}(Wlog(1/W{\epsilon})) is the derived lower bound. We show that all lower bounds generalize to randomized algorithms as well. All algorithms process new elements and answer queries in O(1) worst-case time.Comment: A shorter version appears in SWAT 201

Optimal Elephant Flow Detection

Monitoring the traffic volumes of elephant flows, including the total byte count per flow, is a fundamental capability for online network measurements. We present an asymptotically optimal algorithm for solving this problem in terms of both space and time complexity. This improves on previous approaches, which can only count the number of packets in constant time. We evaluate our work on real packet traces, demonstrating an up to X2.5 speedup compared to the best alternative.Comment: Accepted to IEEE INFOCOM 201