We present a deterministic algorithm that given a tree T with n vertices, a starting vertex v and a slackness parameter epsilon > 0, estimates within an additive error of epsilon the cover and return time, namely, the expected time it takes a simple random walk that starts at v to visit all vertices of T and return to v. The running time of our algorithm is polynomial in n/epsilon, and hence remains polynomial in n also for epsilon = 1/n^{O(1)}. We also show how the algorithm can be extended to estimate the expected cover (without return) time on trees.
Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and ill-suited against adaptive adversaries. In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing $O(\log n)$ copies of each vertex while preserving the connectivity structure of every subgraph and $O(\log^2 n)$-approximating the cost of every subgraph. Using this embeddi…
We consider network design problems with deadline or delay. All previous results for these models are based on randomized embedding of the graph into a tree (HST) and then solving the problem on this tree. We show that this is not necessary. In particular, we design a deterministic framework for these problems which is not based on embedding. This enables us to provide deterministic $\text{poly-log}(n)$-competitive algorithms for Steiner tree, generalized Steiner tree, node weighted Steiner tree, (non-uniform) facility location and directed Steiner tree with deadlines or with delay (where $n$ …
We present a new overlay, called the {\em Deterministic Decentralized tree} ($D^2$-tree). The $D^2$-tree compares favourably to other overlays for the following reasons: (a) it provides matching and better complexities, which are deterministic for the supported operations; (b) the management of nodes (peers) and elements are completely decoupled from each other; and (c) an efficient deterministic load-balancing mechanism is presented for the uniform distribution of elements into nodes, while at the same time probabilistic optimal bounds are provided for the congestion of operations at the node…
Hyper-minimization is a state reduction technique that allows a finite change in the semantics. The theory for hyper-minimization of deterministic weighted tree automata is provided. The presence of weights slightly complicates the situation in comparison to the unweighted case. In addition, the first hyper-minimization algorithm for deterministic weighted tree automata, weighted over commutative semifields, is provided together with some implementation remarks that enable an efficient implementation. In fact, the same run-time O(m log n) as in the unweighted case is obtained, where m is the s…
In this work, we propose D3-Tree, a dynamic distributed deterministic structure for data management in decentralized networks. We present in brief the theoretical algorithmic analysis, in which our proposed structure is based on, and we describe thoroughly the key aspects of the implementation. Conducting experiments, we verify that the implemented structure outperforms other well-known hierarchical tree-based structures, since it provides better complexities regarding load-balancing operations. More specifically, the structure achieves a logarithmic amortized bound, using an efficient determi…
Data compression plays a crucial part in the cloud based systems of today. One the fundaments of compression is quasi-periodicity, for which there are several models. We build upon the most popular quasi-periodicity model for strings, i.e., covers, generalizing it to trees. We introduce a new type of cover automata, which we call \textbf{D}eterministic \textbf{T}ree \textbf{A}utomata. Then, we formulate a cover problem on these DTA and study its complexity, in both sequential and parallel settings. We obtain bounds for the Cover Minimization Problem. Along the way, we uncover an interesting ap…
An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers i…
Jet Propulsion Laboratory, California Institute of Technology（喷气推进实验室，加州理工学院）
K-Nearest Neighbors (KNN) search is a fundamental algorithm in artificial intelligence software with applications in robotics, and autonomous vehicles. These wide-ranging applications utilize KNN either directly for simple classification or combine KNN results as input to other algorithms such as Locally Weighted Learning (LWL). Similar to binary trees, kd-trees become unbalanced as new data is added in online applications which can lead to rapid degradation in search performance unless the tree is rebuilt. Although approximate methods are suitable for graphics applications, which prioritize q…
A {\em tree cover} of a metric space $(X,d)$ is a collection of trees, so that every pair $x,y\in X$ has a low distortion path in one of the trees. If it has the stronger property that every point $x\in X$ has a single tree with low distortion paths to all other points, we call this a {\em Ramsey} tree cover. Tree covers and Ramsey tree covers have been studied by \cite{BLMN03,GKR04,CGMZ05,GHR06,MN07}, and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed …
We present the current fastest deterministic algorithm for $k$-SAT, improving the upper bound $(2-2/k)^{n + o(n)}$ dues to Moser and Scheder [STOC'11]. The algorithm combines a branching algorithm with the derandomized local search, whose analysis relies on a special sequence of clauses called chain, and a generalization of covering code based on linear programming. We also provide a more ingenious branching algorithm for $3$-SAT to establish the upper bound $1.32793^n$, improved from $1.3303^n$.
We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/ε, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\…
We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Ste…
We consider a new Steiner tree problem, called vertex-cover-weighted Steiner tree problem. This problem defines the weight of a Steiner tree as the minimum weight of vertex covers in the tree, and seeks a minimum-weight Steiner tree in a given vertex-weighted undirected graph. Since it is included by the Steiner tree activation problem, the problem admits an O(log n)-approximation algorithm in general graphs with n vertices. This approximation factor is tight up to a constant because it is NP-hard to achieve an o(log n)-approximation for the vertex-cover-weighted Steiner tree problem on genera…
In the Connected Vertex Cover problem we are given an undirected graph G together with an integer k and we are to find a subset of vertices X of size at most k, such that X contains at least one end-point of each edge and moreover X induces a connected subgraph. For this problem we present a deterministic algorithm running in O(2^k n^O(1)) time and polynomial space, improving over previously best O(2.4882^k n^O(1)) deterministic algorithm and O(2^k n^O(1)) randomized algorithm. Furthermore, when usage of exponential space is allowed, we present an O(2^k k(n+m)) time algorithm that solves a mor…
We present a deterministic distributed $2$-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is $O(\log n \log Δ/ \log^2 \log Δ)$. This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the $(2+ε)$-approximation algorithm of [BCS17], improving the dependency on $ε^{-1}$ from linear to logarithmic. In addition, for every $ε=(\log Δ)^{-c}$, where $c\geq 1$ is a constant, our algorithm computes a $(2+ε)$-approximation in $O(\log Δ/ \log \log Δ)$~rounds (which is asymptotic…
Given a graph $G=(V,E)$ with non-negative real edge lengths and an integer parameter $k$, the Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The objective is to minimize the maximum length among all the trees. We give the first constant factor approximation for the hard uniform capacitated version of this problem, where, an input parameter $λ$ upper bounds the number of vertices that can be covered by any of the trees. Our result extends to the rooted version of the problem, where we are given a set of $k…
We revisit the design of self-adjusting single-source tree networks. The problem can be seen as a generalization of the classic list update problem to trees, and finds applications in reconfigurable datacenter networks. We are given a fixed balanced binary tree T connecting n nodes V = {v_1, ... , v_n}. A source node v_0, attached to the root of the tree, issues communication requests to nodes in V, in an online and adversarial manner; the access cost of a request to a node v, is given by the current depth of v in T. The online algorithm can try to reduce the access cost by performing swap ope…
We consider the Modular Subset Sum problem: given a multiset $X$ of integers from $\mathbb{Z}_m$ and a target integer $t$, decide if there exists a subset of $X$ with a sum equal to $t \pmod{m}$. Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in $O(m \log m)$ time, while Axiotis et al. gave a deterministic algorithm that runs in $O(m \text{ polylog } m)$ time. Both results work by reduction to a text problem, which is solved using a dyn…
We introduce the Non-commutative Subset Convolution - a convolution of functions useful when working with determinant-based algorithms. In order to compute it efficiently, we take advantage of Clifford algebras, a generalization of quaternions used mainly in the quantum field theory. We apply this tool to speed up algorithms counting subgraphs parameterized by the treewidth of a graph. We present an $O^*((2^ω+ 1)^{tw})$-time algorithm for counting Steiner trees and an $O^*((2^ω+ 2)^{tw})$-time algorithm for counting Hamiltonian cycles, both of which improve the previously known upper bounds. T…
It is widely assumed that $O(m+\lg σ)$ is the best one can do for finding a pattern of length $m$ in a compacted trie storing strings over an alphabet of size $σ$, if one insists on linear-size data structures and deterministic worst-case running times [Cole et al., ICALP'06]. In this article, we first show that a rather straightforward combination of well-known ideas yields $O(m+\lg\lg σ)$ deterministic worst-case searching time for static tries. Then we move on to dynamic tries, where we achieve a worst-case bound of $O(m+\frac{\lg^{2}\lgσ}{\lg\lg\lgσ})$ per query or update, which should aga…
\textproc{Weighted Vertex Cover} is a variation of an extensively studied NP-complete problem, \textproc{Vertex Cover}, in which we are given a graph, $G = (V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$ and a parameter $k$. The objective is to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$. In our work, we first study the hardness of \textproc{Weighted Vertex Cover} and then examine this problem under parameterization by $l$ and $k$, where $l$ is the number of vertices with fractional weights. Then, we study the \textproc{Red-Blue Weighted Verte…
It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the …
Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a rooted directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be $NP$-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combi…
This paper is motivated by the k-nearest neighbors search: given an arbitrary metric space, and its finite subsets (a reference set R and a query set Q), design a fast algorithm to find all k-nearest neighbors in R for every point q in Q. In 2006, Beygelzimer, Kakade, and Langford introduced cover trees to justify a near-linear time complexity for the neighbor search in the sizes of Q,R. Section 5.3 of Curtin's PhD (2015) pointed out that the proof of this result was wrong. The key step in the original proof attempted to show that the number of iterations can be estimated by multiplying the le…
This paper studies a fundamental algorithmic problem related to the design of demand-aware networks: networks whose topologies adjust toward the traffic patterns they serve, in an online manner. The goal is to strike a tradeoff between the benefits of such adjustments (shorter routes) and their costs (reconfigurations). In particular, we consider the problem of designing a self-adjusting tree network which serves single-source, multi-destination communication. The problem has interesting connections to self-adjusting datastructures. We present two constant-competitive online algorithms for thi…
Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast $Δ$-coloring of trees requires random bits: Building on the recent lower b…
This paper studies the important problem of finding all $k$-nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. Our previous work defined compressed cover trees and corrected the key arguments in several past papers for challenging datasets. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a time complexity for $k=1$. The current work fills all previous gaps and improves the near…
It is well-known that the Vertex Cover problem is in P on bipartite graphs, however; the computational complexity of the Partial Vertex Cover problem on bipartite graphs is open. In this paper, we first show that the Partial Vertex Cover problem is NP-hard on bipartite graphs. We then identify an interesting special case of bipartite graphs, for which the Partial Vertex Cover problem can be solved in polynomial-time. We also show that the set of acyclic bipartite graphs, i.e., forests, and the set of bipartite graph where the degree of each vertex is at most 3 fall into that special case. Ther…
Given point sets $A$ and $B$ in $\mathbb{R}^d$ where $A$ and $B$ have equal size $n$ for some constant dimension $d$ and a parameter $\varepsilon>0$, we present the first deterministic algorithm that computes, in $n\cdot(\varepsilon^{-1} \log n)^{O(d)}$ time, a perfect matching between $A$ and $B$ whose cost is within a $(1+\varepsilon)$ factor of the optimal under any $\smash{\ell_p}$-norm. Although a Monte-Carlo algorithm with a similar running time is proposed by Raghvendra and Agarwal [J. ACM 2020], the best-known deterministic $\varepsilon$-approximation algorithm takes $Ω(n^{3/2})$ time.…
