Estimating the weight of metric minimum spanning trees in sublinear time

In this paper we present a sublinear-time $(1+\varepsilon)$-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an $n$-point metric space. The running time of the algorithm is $\widetilde{\mathcal{O}}(n/\varepsilon^{\mathcal{O}(1)})$. Since the full description of an $n$-point metric space is of size $\Theta(n^2)$, the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in $o(n)$ time the weight of the minimum spanning tree to within any factor. We also show that no deterministic algorithm can achieve a $B$-approximation in $o(n^2/B^3)$ time. Furthermore, it has been previously shown that no $o(n^2)$ algorithm exists that returns a spanning tree whose weight is within a constant times the optimum

Dynamic Graph Stream Algorithms in o(n)o(n) Space

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require $\Omega(n)$ space, where $n$ is the number of vertices, existing works mainly focused on designing $\tilde{O}(n)$ space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. $n$ is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present $o(n)$ space algorithms for estimating the number of connected components with additive error $\varepsilon n$ and $(1+\varepsilon)$-approximating the weight of minimum spanning tree, for any small constant $\varepsilon>0$. The latter improves previous $\tilde{O}(n)$ space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are $\varepsilon$-far from having the property. We consider the problem of testing $k$-edge connectivity, $k$-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly $\tilde{O}(n^{1-\varepsilon})$ space, which is $o(n)$ for any constant $\varepsilon$. To complement our algorithms, we present $\Omega(n^{1-O(\varepsilon)})$ space lower bounds for these problems, which show that such a dependence on $\varepsilon$ is necessary.Comment: ICALP 201

Streaming Verification of Graph Properties

Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate) well in a streaming setting. The most notable of these problems is finding maximum matchings, which has received intense interest in recent years but has strong lower bounds even for constant factor approximations. In this paper, we present efficient streaming interactive proofs that can verify maximum matchings exactly. Our results cover all flavors of matchings (bipartite/non-bipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP. In particular, these are the first efficient results for weighted matchings and for metric TSP in any streaming verification model.Comment: 26 pages, 2 figure, 1 tabl

Sublinear time approximation of the cost of a metric k-nearest neighbor graph

Let (X, d) be an n-point metric space. We assume that (X, d) is given in the distance oracle model, that is, X = {1, …, n} and for every pair of points x, y from X we can query their distance d(x, y) in constant time. A k-nearest neighbor (k-NN) graph for (X, d) is a directed graph G = (V, E) that has an edge to each of v's k nearest neighbors. We use cost(G) to denote the sum of edge weights of G. In this paper, we study the problem of approximating cost(G) in sublinear time, when we are given oracle access to the metric space (X, d) that defines G. Our goal is to develop an algorithm that solves this problem faster than the time required to compute G. We first present an algorithm that in Õ∊(n2/k) time with probability at least approximates cost(G) to within a factor of 1 + ∊. Next, we present a more elaborate sublinear algorithm that in time Õϵ(min{nk3/2, n2/k}) computes an estimate of cost(G) that satisfies with probability at least where mst(X) denotes the cost of the minimum spanning tree of (X, d). Further, we complement these results with near matching lower bounds. We show that any algorithm that for a given metric space (X, d) of size n, with probability at least estimates cost(G) to within a 1 + ∊ factor requires Ω(n2/k) time. Similarly, any algorithm that with probability at least estimates cost(G) to within an additive error term ϵ · (mst(X) + cost(X)) requires Ωϵ(min{nk3/2, n2/k}) time

Query Complexity of the Metric Steiner Tree Problem

We study the query complexity of the metric Steiner Tree problem, where we are given an $n \times n$ metric on a set $V$ of vertices along with a set $T \subseteq V$ of $k$ terminals, and the goal is to find a tree of minimum cost that contains all terminals in $T$. The query complexity for the related minimum spanning tree (MST) problem is well-understood: for any fixed $\varepsilon > 0$, one can estimate the MST cost to within a $(1+\varepsilon)$-factor using only $\tilde{O}(n)$ queries, and this is known to be tight. This implies that a $(2 + \varepsilon)$-approximate estimate of Steiner Tree cost can be obtained with $\tilde{O}(k)$ queries by simply applying the MST cost estimation algorithm on the metric induced by the terminals. Our first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - \varepsilon)$-factor requires $\Omega(n^2)$ queries, even if $k$ is a constant. This lower bound is in sharp contrast to an upper bound of $O(nk)$ queries for computing a $(5/3)$-approximate Steiner Tree, which follows from previous work by Du and Zelikovsky. Our second main result, and the main technical contribution of this work, is a sublinear query algorithm for estimating the Steiner Tree cost to within a strictly better-than-$2$ factor, with query complexity $\tilde{O}(n^{12/7} + n^{6/7}\cdot k)=\tilde{O}(n^{13/7})=o(n^2)$. We complement this result by showing an $\tilde{\Omega}(n + k^{6/5})$ query lower bound for any algorithm that estimates Steiner Tree cost to a strictly better than $2$ factor. Thus $\tilde{\Omega}(n^{6/5})$ queries are needed to just beat $2$-approximation when $k = \Omega(n)$; a sharp contrast to MST cost estimation where a $(1+o(1))$-approximate estimate of cost is achievable with only $\tilde{O}(n)$ queries