Showing papers by "Liam Roditty published in 2018"

Towards tight approximation bounds for graph diameter and eccentricities

Abstract: Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. [SODA 2014] showed that the diameter can be approximated within a multiplicative factor of 3/2 in O(m3/2) time. Furthermore, Roditty and Vassilevska W. [STOC 13] showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no O(n2−e) time algorithm can achieve an approximation factor better than 3/2 in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than 3/2. It was, however, completely plausible that a 3/2-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no O(m3/2−e) time algorithm can achieve an approximation factor better than 5/3. Another fundamental set of graph parameters are the Eccentricities. The Eccentricity of a vertex v is the distance between v and the farthest vertex from v. Chechik et al. [SODA 2014] showed that the Eccentricities of all vertices can be approximated within a factor of 5/3 in O(m3/2) time and Abboud et al. [SODA 2016] showed that no O(n2−e) algorithm can achieve better than 5/3 approximation in sparse graphs. We show that the runtime of the 5/3 approximation algorithm is also optimal by proving that under SETH, there is no O(m3/2−e) algorithm that achieves a better than 9/5 approximation. We also show that no near-linear time algorithm can achieve a better than 2 approximation for the Eccentricities. This is the first lower bound in fine-grained complexity that addresses near-linear time computation. We show that our lower bound for near-linear time algorithms is essentially tight by giving an algorithm that approximates Eccentricities within a 2+δ factor in O(m/δ) time for any 0 To establish the above lower bounds we study the S-T Diameter problem: Given a graph and two subsets S and T of vertices, output the largest distance between a vertex in S and a vertex in T. We give new algorithms and show tight lower bounds that serve as a starting point for all other hardness results. Our lower bounds apply only to sparse graphs. We show that for dense graphs, there are near-linear time algorithms for S-T Diameter, Diameter and Eccentricities, with almost the same approximation guarantees as their O(m3/2) counterparts, improving upon the best known algorithms for dense graphs.

Towards Tight Approximation Bounds for Graph Diameter and Eccentricities

Abstract: Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\tilde{O}(m^{3/2})$ time. Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$. It was, however, completely plausible that a $3/2$-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$. Another fundamental set of graph parameters are the Eccentricities. The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$. Chechik et al. showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\tilde{O}(m^{3/2})$ time and Abboud et al. showed that no $O(n^{2-\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs. We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH. We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\delta$ factor in $\tilde{O}(m/\delta)$ time for any $0<\delta<1$. This beats all Eccentricity algorithms in Cairo et al.

Approximating cycles in directed graphs: fast algorithms for girth and roundtrip spanners

Abstract: The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted m-edge and n-node graphs require Ω(min{nω, mn}) time (for 2 ≤ ω > 2.373). In this paper, we drastically improve these runtimes as follows: • Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in O(m) time computes an O(n)-multiplicative approximation of the girth as well as an O(n)-multiplicative roundtrip spanner with O(n) edges with high probability (w.h.p). • Nearly Tight Additive Approximations: For unweighted graphs and any a ∈ (0, 1) we give an algorithm that in O(mn1−a) time computes an 0(na)-additive approximation of the girth, w.h.p. We show that the run-time of our algorithm cannot be significantly improved without a breakthrough in combinatorial boolean matrix multiplication. We also show that if the girth is 0(na), then the same guarantee can be achieved via a deterministic algorithm. Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than Ω(mn) time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.

Routing in Unit Disk Graphs

Abstract: Let $$S \subset \mathbb {R}^2$$ be a set of n sites. The unit disk graph $${{\mathrm{UD}}}(S)$$ on S has vertex set S and an edge between two distinct sites $$s,t \in S$$ if and only if s and t have Euclidean distance $$|st| \le 1$$ . A routing scheme R for $${{\mathrm{UD}}}(S)$$ assigns to each site $$s \in S$$ a label $$\ell (s)$$ and a routing table $$\rho (s)$$ . For any two sites $$s, t \in S$$ , the scheme R must be able to route a packet from s to t in the following way: given a current site r (initially, $$r = s$$ ), a header h (initially empty), and the label $$\ell (t)$$ of the target, the scheme R consults the routing table $$\rho (r)$$ to compute a neighbor $$r'$$ of r, a new header $$h'$$ , and the label $$\ell (t')$$ of an intermediate target $$t'$$ . (The label of the original target may be stored at the header $$h'$$ .) The packet is then routed to $$r'$$ , and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in $${{\mathrm{UD}}}(S)$$ , over all pairs of distinct sites in S. For any given $$\varepsilon > 0$$ , we show how to construct a routing scheme for $${{\mathrm{UD}}}(S)$$ with stretch $$1+\varepsilon $$ using labels of $$O(\log n)$$ bits and routing tables of $$O(\varepsilon ^{-5}\log ^2 n \log ^2 D)$$ bits, where D is the (Euclidean) diameter of $${{\mathrm{UD}}}(S)$$ . The header size is $$O(\log n \log D)$$ bits.

Algorithms and Hardness for Diameter in Dynamic Graphs

Abstract: The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: - Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. - Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly $(3/2+\epsilon)$-approximation to Diameter in directed or undirected graphs can be maintained decrementally in total time $m^{1+o(1)}\sqrt{n}/\epsilon^2$. This nearly matches the static $3/2$-approximation algorithm for the problem that is known to be conditionally optimal.

Stabbing pairwise intersecting disks by five points

Abstract: Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P . We present a deterministic algorithm that takes O ( n ) time to find five points that stab D . Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. Moreover, we present a simple argument showing that eight disks can be stabbed by at most three points. This provides a simple – albeit slightly weaker – algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal

Abstract: Let $G$ be a directed graph with $n$ vertices, $m$ edges, and a designated source vertex $s$. We address the problem of single-source reachability (SSR) from $s$ in the presence of failures of vertices/edges. We show that for every $k\geq1$, there is a subgraph $H$ of $G$ with at most $2^kn$ edges that preserves the reachability from $s$ even after the failure of any $k$ edges. Formally, given a set $F$ of $k$ edges, a vertex $v\in V(G)$ is reachable from $s$ in $G\setminus F$ if and only if $v$ is reachable from $s$ in $H\setminus F$. We call $H$ a $k$-fault tolerant reachability subgraph ($\textsc{$k$-FTRS}$). We also prove a matching lower bound of $\Omega(2^kn)$ edges for such subgraphs that holds for all $n,k$ with $2^k\leq n$. Our results extend to vertex failures without any extra overhead. The construction of ${$k$-FTRS}$ is interesting from several different perspectives. From the Graph theory perspective it reveals a separation between SSR and single-source shortest paths (SSSP) in directed grap...

Approximate single source fault tolerant shortest path

Abstract: Let G = (V, E) be an n-vertices m-edges directed graph with edge weights in the range [1, W] and L = log(W). Let s ∈ V be a designated source. In this paper we address several variants of the problem of maintaining the (1 + ∈)-approximate shortest path from s to each v ∈ V \ {s} in the presence of a failure of an edge or a vertex. From the graph theory perspective we show that G has a subgraph H with O(nL/∈) edges such that for any x, v ∈ V, the graph H \ x contains a path whose length is a (1 + ∈)-approximation of the length of the shortest path from s to v in G \ x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/∈) edges. Demetrescu, Thorup, Chowdhury and Ramachandran [12] showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω(m). Parter and Peleg [18] showed that even in the restricted case of unweighted undirected graphs the size of any subgraph for the exact shortest path is at least Ω(n1.5). Therefore, a (1 + ∈)-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an O(nL/∈) size oracle that for any v ∈ V reports a (1 + ∈)-approximate distance of v from s on a failure of any x ∈ V in O(log log1+∈(n W)) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/∈ log n). Finally, we present two distributed algorithms. We present a single source routing scheme that can route on a (1 + ∈)-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of O(L/∈) bits. We present also a labeling scheme that assigns each vertex a label of O(L/∈) bits. For any two vertices x, v ∈ V the labeling scheme outputs a (1 + ∈)-approximation of the distance from s to v in G \ x using only the labels of x and v.

Spanners for Directed Transmission Graphs

Abstract: Let $P \subset \mathbb{R}^2$ be a planar $n$-point set such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) \leq r_p$. Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H \subseteq G$ with vertex set $P$ so that for any two vertices $p,q \in P$, we have $d_H(p, q) \leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths). We show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (\log n + \log \Psi))$ time, where $\Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures, we obtain a construction that runs in $O(n \log^5 n)$ time, independent of $\Psi$. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in $G$ from any given star...