arXiv subjects

Seth Pettie

Publications and source records attributed to Seth Pettie.

At least 55 records · Page 3Linked to original sources

arXiv2017

Scaling Algorithms for Weighted Matching in General Graphs

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in $O(m\sqrt{n}\log(nN))$ time, $O(m\sqrt{n})$ per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs. Here $m,n,$ and $N$ bound the number of edges, vertices, and magnitude of any edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in $O(m\sqrt{n\log nα(m,n)} \log(nN))$ time.

cs.DS↗

arXiv2017

The Energy Complexity of Broadcast

Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits/listens; idle time and computation are free. In this paper we investigate the energy complexity of fundamental communication primitives such as broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include: 1. The energy complexity of broadcast in a multi-hop network is intimately connected to the time complexity of leader election in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need $Ω(\log n)$ and $Ω(\log^2 n)$ energy, respectively. 2. The energy lower bounds above can almost be achieved, given sufficient ($Ω(n)$) time. In the CD and No-CD models we can solve broadcast using $O(\frac{\log n\log\log n}{\log\log\log n})$ energy and $O(\log^3 n)$ energy, respectively. 3. The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant $ε>0$, broadcast can be solved in $O(D^{1+ε}\log^{O(1/ε)} n)$ time with $O(\log^{O(1/ε)} n)$ energy, where $D$ is the diameter of the network.

cs.DC↗

arXiv2017

Connectivity Oracles for Graphs Subject to Vertex Failures

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_{\star} m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge-failure connectivity oracle. Using space $O(n\log^2 n)$, $d$ edge failures are processed in $O(d\log d\log\log n)$ time and thereafter, connectivity queries are answered in $O(\log\log n)$ time, which are correct w.h.p. Our data structures are based on a new decomposition theorem for an undirected graph $G=(V,E)$, which is of independent interest. It states that for any terminal set $U\subseteq V$ we can remove a set $B$ of $|U|/(s-2)$ vertices such that the remaining graph contains a Steiner forest for $U-B$ with maximum degree $s$.

cs.DS↗

arXiv2017

A Resource-Competitive Jamming Defense

Consider a scenario where Alice wishes to send a message $m$ to Bob in a time-slotted wireless network. However, there exists an adversary, Carol, who aims to prevent the transmission of $m$ by jamming the communication channel. There is a per-slot cost of $1$ to send, receive or jam $m$ on the channel, and we are interested in how much Alice and Bob need to spend relative to Carol in order to guarantee communication. Our approach is to design an algorithm in the framework of resource-competitive analysis where the cost to correct network devices (i.e., Alice and Bob) is parameterized by the cost to faulty devices (i.e., Carol). We present an algorithm that guarantees the successful transmission of $m$ and has the following property: if Carol incurs a cost of $T$ to jam, then both Alice and Bob have a cost of $O(T^{φ- 1} + 1)=O(T^{.62}+1)$ in expectation, where $φ= (1+ \sqrt{5})/2$ is the golden ratio. In other words, it possible for Alice and Bob to communicate while incurring asymptotically less cost than Carol. We generalize to the case where Alice wishes to send $m$ to $n$ receivers, and we achieve a similar result. Our findings hold even if (1) $T$ is unknown to either party; (2) Carol knows the algorithms of both parties, but not their random bits; (3) Carol can jam using knowledge of past actions of both parties; and (4) Carol can jam reactively, so long as there is sufficient network traffic in addition to $m$.

cs.DC↗

arXiv2017

Thorup-Zwick Emulators are Universally Optimal Hopsets

A $(β,ε)$-$\textit{hopset}$ is, informally, a weighted edge set that, when added to a graph, allows one to get from point $a$ to point $b$ using a path with at most $β$ edges ("hops") and length $(1+ε)\mathrm{dist}(a,b)$. In this paper we observe that Thorup and Zwick's $\textit{sublinear additive}$ emulators are also actually $(O(k/ε)^k,ε)$-hopsets for every $ε>0$, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made $O(n^{1+\frac{1}{2^{k+1}-1}})$. As corollaries, we also shave "$k$" factors off the size of Thorup and Zwick's sublinear additive emulators and the sparsest known $(1+ε,O(k/ε)^{k-1})$-spanners, due to Abboud, Bodwin, and Pettie.

cs.DS↗

arXiv2017

A Time Hierarchy Theorem for the LOCAL Model

The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as $O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}$, etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical $2\frac{1}{2}$-Coloring}. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the $k$-level Hierarchical $2\frac{1}{2}$-Coloring problem is $Θ(n^{1/k})$, for $k\in\mathbb{Z}^+$. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized $n^{o(1)}$-time algorithm solving the LCL can be transformed into a deterministic $O(\log n)$-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges $ω(\log^* n)$---$o(\log n)$ or $ω(\log n)$---$n^{o(1)}$. 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in $O(T_{LLL})$ time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be $Ω(\log\log n)$ and $O(\log n)$.

cs.DC↗

arXiv2017

A Hierarchy of Lower Bounds for Sublinear Additive Spanners

Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say $\tilde{O}(n^{1+δ})$ bits. There is an inherent tradeoff between the sparsity parameter $δ$ and the stretch function $f$ of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. In this paper we show that the recent additive spanner lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that fully characterize the asymptotic behavior of the optimal stretch function $f$ as a function of $δ\in (0,1/3)$. Specifically, for any integer $k\ge 2$, any compression scheme with size $O(n^{1+\frac{1}{2^k-1} - ε})$ has a sublinear additive stretch function $f$: $$f(d) = d + Ω(d^{1-\frac{1}{k}}).$$ This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's $(1+ε,β)$-spanners have an essentially optimal tradeoff between $δ,ε,$ and $β$, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of $(1+ε, O(k/ε)^{k-1})$-spanners with size $O((k/ε)^{h_k} kn^{1+\frac{1}{2^{k+1}-1}})$, where $h_k < 3/4$. This size bound improves on the spanners of Elkin and Peleg (2004), Thorup and Zwick (2006), and Pettie (2009). According to our lower bounds neither the size nor stretch function can be substantially improved.

cs.DS↗

arXiv2016

Lower Bounds on Davenport-Schinzel Sequences via Rectangular Zarankiewicz Matrices

An order-$s$ Davenport-Schinzel sequence over an $n$-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length $s+2$. The main problem is to determine the maximum length of such a sequence, as a function of $n$ and $s$. When $s$ is fixed this problem has been settled but when $s$ is a function of $n$, very little is known about the extremal function $λ(s,n)$ of such sequences. In this paper we give a new recursive construction of Davenport-Schinzel sequences that is based on dense 0-1 matrices avoiding large all-1 submatrices (aka Zarankiewicz's Problem.) In particular, we give a simple construction of $n^{2/t} \times n$ matrices containing $n^{1+1/t}$ 1s that avoid $t\times 2$ all-1 submatrices. Our lower bounds on $λ(s,n)$ exhibit three qualitatively different behaviors depending on the size of $s$ relative to $n$. When $s \le \log\log n$ we show that $λ(s,n)/n \ge 2^s$ grows exponentially with $s$. When $s = n^{o(1)}$ we show $λ(s,n)/n \ge (\frac{s}{2\log\log_s n})^{\log\log_s n}$ grows faster than any polynomial in $s$. Finally, when $s=Ω(n^{1/t}(t-1)!)$, $λ(s,n) = Ω(n^2 s/(t-1)!)$ matches the trivial upper bound $O(n^2s)$ asymptotically, whenever $t$ is constant.

math.CO↗

arXiv2016

An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model

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 bounds of Brandt et al., we prove that the randomized complexity of $Δ$-coloring a tree with maximum degree $Δ\ge 55$ is $Θ(\log_Δ\log n)$, whereas its deterministic complexity is $Θ(\log_Δn)$ for any $Δ\ge 3$. This also establishes a large separation between the deterministic complexity of $Δ$-coloring and $(Δ+1)$-coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in $O(1)+o(\log_Δn)$ rounds can be transformed to run in $O(\log^*n-\log^*Δ+1)$ rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires $Ω(\log_Δn)$ time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size $n$ is at least its deterministic complexity on instances of size $\sqrt{\log n}$. This shows that a deterministic $Ω(\log_Δn)$ lower bound for any problem implies a randomized $Ω(\log_Δ\log n)$ lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model.

cs.CC↗

arXiv2015

Faster Worst Case Deterministic Dynamic Connectivity

We present a deterministic dynamic connectivity data structure for undirected graphs with worst case update time $O\left(\sqrt{\frac{n(\log\log n)^2}{\log n}}\right)$ and constant query time. This improves on the previous best deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time $O(\sqrt{n})$. All other algorithms for dynamic connectivity are either randomized (Monte Carlo) or have only amortized performance guarantees.

cs.DS↗

arXiv2015

Mind the Gap

We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following. Preprocess a dictionary $D$ of $d$ patterns, where each pattern contains a special gap symbol that can match any string, so that given a text that arrives online, a character at a time, we can report all of the patterns from $D$ that are suffixes of the text that has arrived so far, before the next character arrives. In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings. Finding efficient algorithmic solutions for (online) DMOG has proven to be a difficult algorithmic challenge. We demonstrate that the difficulty in obtaining efficient solutions for the DMOG problem even, in the offline setting, can be traced back to the infamous 3SUM conjecture. Interestingly, our reduction deviates from the known reduction paths that follow from 3SUM. In particular, most reductions from 3SUM go through the set-disjointness problem, which corresponds to the problem of preprocessing a graph to answer edge-triangles queries. We use a new path of reductions by considering the complementary, although structurally very different, vertex-triangles queries. Using this new path we show a conditional lower bound of $Ω(δ(G_D)+op)$ time per text character, where $G_D$ is a bipartite graph that captures the structure of $D$, $δ(G_D)$ is the degeneracy of this graph, and $op$ is the output size. We also provide matching upper-bounds (up to sub-polynomial factors) for the vertex-triangles problem, and then extend these techniques to the online DMOG problem. In particular, we introduce algorithms whose time cost depends linearly on $δ(G_D)$. Our algorithms make use of graph orientations, together with some additional techniques. Finally, when $δ(G_D)$ is large we are able to obtain even more efficient solutions.

cs.DS↗

arXiv2015

A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $Π$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $Ω(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^ε)$, for an arbitrarily small $ε> 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(β(k))$, for a function $β(\cdot)$, space $O(n^{1+1/k} \cdot β)$, and query time $O(n^ε)$, for an arbitrarily small constant $ε>0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $β(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+ε,β)$-spanners due to Elkin and Peleg. However, so far $(1+ε,β)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}.

cs.DS↗

arXiv2015

Dynamic Set Intersection

Consider the problem of maintaining a family $F$ of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given $S,S'\in F$, report every member of $S\cap S'$ in any order. We show that in the word RAM model, where $w$ is the word size, given a cap $d$ on the maximum size of any set, we can support set intersection queries in $O(\frac{d}{w/\log^2 w})$ expected time, and updates in $O(\log w)$ expected time. Using this algorithm we can list all $t$ triangles of a graph $G=(V,E)$ in $O(m+\frac{mα}{w/\log^2 w} +t)$ expected time, where $m=|E|$ and $α$ is the arboricity of $G$. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in $O(m α)$ time. We provide an incremental data structure on $F$ that supports intersection {\em witness} queries, where we only need to find {\em one} $e\in S\cap S'$. Both queries and insertions take $O\paren{\sqrt \frac{N}{w/\log^2 w}}$ expected time, where $N=\sum_{S\in F} |S|$. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using $M$ words of space, each update costs $O(\sqrt {M \log N})$ expected time, each reporting query costs $O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1})$ expected time where $op$ is the size of the output, and each witness query costs $O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N)$ expected time.

cs.DS↗

arXiv2015

The Locality of Distributed Symmetry Breaking

Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes - An MIS algorithm running in $O(\log^2Δ+ 2^{O(\sqrt{\log\log n})})$ time, where $Δ$ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when $\log n \ll Δ\ll 2^{\sqrt{\log n}}$, and comes close to the $Ω(\log Δ)$ lower bound of Kuhn, Moscibroda, and Wattenhofer. - A maximal matching algorithm running in $O(\logΔ+ \log^4\log n)$ time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on $Δ$ is provably optimal. - A method for reducing symmetry breaking problems in low arboricity/degeneracy graphs to low degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an $O(\sqrt{\log n})$-time maximal matching algorithm for graphs with arboricity up to $2^{\sqrt{\log n}}$ and an $O(\log^{2/3} n)$-time MIS algorithm for graphs with arboricity up to $2^{(\log n)^{1/3}}$. Each of our algorithms is based on a simple, but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly$(\log n)$-size graph.

cs.DC↗

arXiv2014

Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time

We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in $O(m\logα(m,n))$ time, where $α$ is the inverse-Ackermann function. This improves upon a long standing bound of $O(mα(m,n))$ established by Tarjan. Our algorithms are based on an efficient split-findmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our split-findmin algorithm is the first with superlinear but sub-inverse-Ackermann complexity. We also give a reduction from MST sensitivity to the MST problem itself. Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algoritm.

cs.DS↗

arXiv2014

Threesomes, Degenerates, and Love Triangles

The 3SUM problem is to decide, given a set of $n$ real numbers, whether any three sum to zero. It is widely conjectured that a trivial $O(n^2)$-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies $Ω(n^2)$ lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is $O(n^{3/2}\sqrt{\log n})$ and give two subquadratic 3SUM algorithms, a deterministic one running in $O(n^2 / (\log n/\log\log n)^{2/3})$ time and a randomized one running in $O(n^2 (\log\log n)^2 / \log n)$ time with high probability. Our results lead directly to improved bounds for $k$-variate linear degeneracy testing for all odd $k\ge 3$. The problem is to decide, given a linear function $f(x_1,\ldots,x_k) = α_0 + \sum_{1\le i\le k} α_i x_i$ and a set $A \subset \mathbb{R}$, whether $0\in f(A^k)$. We show the decision tree complexity of this problem is $O(n^{k/2}\sqrt{\log n})$. Finally, we give a subcubic algorithm for a generalization of the $(\min,+)$-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-$O(n^{5/2}\sqrt{\log n})$ decision tree for this problem, as well as an algorithm running in time $O(n^3 (\log\log n)^2/\log n)$.

cs.DS↗

arXiv2014

Three Generalizations of Davenport-Schinzel Sequences

We present new, and mostly sharp, bounds on the maximum length of certain generalizations of Davenport-Schinzel sequences. Among the results are sharp bounds on order-$s$ {\em double DS} sequences, for all $s$, sharp bounds on sequences avoiding {\em catenated permutations} (aka formation free sequences), and new lower bounds on sequences avoiding {\em zig-zagging} patterns.

math.CO↗

arXiv2013

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $λ_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $λ_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $λ_s(n)$ behaves essentially like $λ_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).

cs.DM↗