arXiv subjects

Seth Pettie

Publications and source records attributed to Seth Pettie.

At least 19 recordsLinked to original sources

arXiv2026

Deterministic Edge-Fault-Tolerant Connectivity Labeling Schemes with Nearly Optimal Label Size

For an undirected graph $G = (V,E)$ and a fault bound $f$, an edge-fault-tolerant connectivity labeling scheme assigns short labels to vertices and edges, so that for any vertex pair $(s,t)$ and failed edge set $F\subseteq E$ with $|F|\leq f$, the connectivity between $s$ and $t$ in $G-F$ can be answered by inspecting only the labels of $s$, $t$ and edges in $F$. In this paper, we present a labeling scheme that uses $O(\log^{2}n)$-bit labels that can be computed in deterministic polynomial time. This improves upon the previous $\tilde{O}(\sqrt{f})$ deterministic bound of [Long, Pettie, Saranurak'25], and even slightly improves the $O(\min\{f+\log n,\log^{2}n\log f\})$ randomized bound of [Dory, Parter'21] and [Long, Pettie, Saranurak'25] when $f = \Omega(\log^{2}n)$. Moreover, for a general $f$, this is the first labeling scheme that produces an $\tilde{O}(1)$-size labeling which is simultaneously correct across all queries. Our approach combines the cycle-space-based labeling scheme from Dory and Parter with a recent result by [Knauer'26] on sparse cycle bases.

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arXiv2026

A Simple Analysis of Quadratic Probing and Other Open Addressing Schemes

In open addressed hashing, quadratic probing is attractive for striking a nice balance between having a high locality of reference and a low number of probes per search. However, these are empirical observations, not theoretical guarantees. Indeed, until recently, it was not known whether quadratic probing had constant expected insertion cost under any positive load factor $\alpha > 0$, even with uniformly random hash functions. In a recent breakthrough---albeit a numerically understated breakthrough---Kuszmaul and Xi (2024) proved that any fixed offset sequence (including quadratic probing) does, in fact, have constant expected insertion cost for load factors $\alpha \leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $\alpha < 1-\epsilon$ bounded away from 1. In this paper, we prove that open addressed hashing with any fixed offset sequence has constant expected insertion cost for load factors up to $35.74\%$, and that for quadratic probing in particular, we can increase the load factor to $37.61\%$. Our main innovation is a new type of witness forest for recording collisions among the probe sequences.

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arXiv2026

The Greedy Binary Search Tree is Non-trivially Competitive

We prove that the $\textsf{Greedy}$ binary search tree is $2^{O(\sqrt{\log\log n})}$-competitive. It is widely conjectured that $\textsf{Greedy}$ is $O(1)$-competitive, but before this work it was not known to be $f$-competitive, for any non-trivial $f(n)=o(\log n)$. Our analysis differs from prior analyses of binary search trees. It takes what might be called a "scaling" approach, where the cost at a refined scale is related to the cost at a coarser scale, and Wilber's interleave lower bound.

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arXiv2026

Contention Resolution, With and Without a Global Clock

In the Contention Resolution problem $n$ parties each wish to have exclusive use of a shared resource for one unit of time. The problem has been studied since the early 1970s, under a variety of assumptions on feedback given to the parties, how the parties wake up, knowledge of $n$, and so on. The most consistent assumption is that parties do not have access to a global clock, only their local time since wake-up. This is surprising because the assumption of a global clock is both technologically realistic and algorithmically interesting. It enriches the problem, and opens the door to entirely new techniques. Our primary results are: [1] We design a new Contention Resolution protocol that guarantees latency $$O\left(\left(n\log\log n\log^{(3)} n\log^{(4)} n\cdots \log^{(\log^* n)} n\right)\cdot 2^{\log^* n}\right) \le n(\log\log n)^{1+o(1)}$$ in expectation and with high probability. This already establishes at least a roughly $\log n$ complexity gap between randomized protocols in GlobalClock and LocalClock. [2] Prior analyses of randomized ContentionResolution protocols in LocalClock guaranteed a certain latency with high probability, i.e., with probability $1-1/\text{poly}(n)$. We observe that it is just as natural to measure expected latency, and prove a $\log n$-factor complexity gap between the two objectives for memoryless protocols. The In-Expectation complexity is $\Theta(n \log n/\log\log n)$ whereas the With-High-Probability latency is $\Theta(n\log^2 n/\log\log n)$. Three of these four upper and lower bounds are new. [3] Given the complexity separation above, one would naturally want a ContentionResolution protocol that is optimal under both the In-Expectation and With-High-Probability metrics. This is impossible! It is even impossible to achieve In-Expectation latency $o(n\log^2 n/(\log\log n)^2)$ and With-High-Probability latency $n\log^{O(1)} n$ simultaneously.

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arXiv2025

Space Complexity of Vertex Connectivity Oracles

A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, reports $\min\{k,κ(u,v)\}$, where $κ(u,v)$ is the pairwise vertex connectivity between $u,v$. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov shows that a data structure of total size $\tilde{O}(kn)$ can even be encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity queries can be answered in $\tilde{O}(k)$ time. The construction time is polynomial, but unspecified. In this paper we address the top three complexity measures: Space, Query Time, and Construction Time. We give an $Ω(kn)$-bit lower bound on any vertex connectivity oracle. We construct an optimal-space connectivity oracle in max-flow time that answers queries in $O(\log n)$ time, independent of $k$.

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arXiv2025

The Squishy Grid Problem

In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E\to \mathbb{R}_{\ge 0}$, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is, for all grid points $u,v$, $\mathrm{dist}_w(u,v) = (1\pm o(1))\|u-v\|_2$. We give three methods for solving this problem, each attractive in its own way. * Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is $(1+1/Θ(\log^ξ\log D))$, where $D$ is the Euclidean distance and $ξ=Θ(1)$. The pinwheel tiling approach is conceptually simple, but can be improved quantitatively. * Our second construction is based on a hierarchical arrangement of "highways." It is simple, achieving stretch $(1 + 1/Θ(D^{1/9}))$, which converges doubly exponentially faster than the pinwheel tiling approach. * The first two methods are deterministic. An even simpler approach is to sample the edge weights independently from a common distribution $\mathscr{D}$. Whether there exists a distribution $\mathscr{D}^*$ that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when $\mathscr{D}$ is a Fisher distribution, grid distances are within 1\% of Euclidean. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44\% and 56\%, respectively.

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arXiv2025

Reviving Thorup's Shortcut Conjecture

We aim to revive Thorup's conjecture [Thorup, WG'92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph $G=(V,E)$ with $m$ edges, we can add $m^{1+o(1)}$ ``shortcut'' edges $E_+$ from the transitive closure $E^*$ of $G$ so that $\text{dist}_{G_+}(u,v) \leq m^{o(1)}$ for all $(u,v)\in E^*$, where $G_+=(V,E\cup E_+)$. The conjecture was refuted by Hesse [Hesse, SODA'03], followed by significant efforts in the last few years to optimize the lower bounds. In this paper we observe that although Hesse refuted the letter of Thorup's conjecture, his work~[Hesse, SODA'03] -- and all followup work -- does not refute the spirit of the conjecture, which should allow $G_+$ to contain both new (shortcut) edges and new Steiner vertices. Our results are as follows. (1) On the positive side, we present explicit attacks that break all known shortcut lower bounds when Steiner vertices are allowed. (2) On the negative side, we rule out ideal $m^{1+o(1)}$-size, $m^{o(1)}$-diameter shortcuts whose ``thickness'' is $t=o(\log n/\log \log n)$, meaning no path can contain $t$ consecutive Steiner vertices. (3) We propose a candidate hard instance as the next step toward resolving the revised version of Thorup's conjecture. Finally, we show promising implications. Almost-optimal parallel algorithms for computing a generalization of the shortcut that approximately preserves distances or flows imply almost-optimal parallel algorithms with $m^{o(1)}$ depth for exact shortcut paths and exact maximum flow. The state-of-the-art algorithms have much worse depth of $n^{1/2+o(1)}$ [Rozhoň, Haeupler, Martinsson, STOC'23] and $m^{1+o(1)}$ [Chen, Kyng, Liu, FOCS'22], respectively.

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arXiv2024

Connectivity Labeling Schemes for Edge and Vertex Faults via Expander Hierarchies

We consider the problem of assigning short labels to the vertices and edges of a graph $G$ so that given any query $\langle s,t,F\rangle$ with $|F|\leq f$, we can determine whether $s$ and $t$ are still connected in $G-F$, given only the labels of $F\cup\{s,t\}$. This problem has been considered when $F\subset E$ (edge faults), where correctness is guaranteed with high probability (w.h.p.) or deterministically, and when $F\subset V$ (vertex faults), both w.h.p.~and deterministically. Our main results are as follows. [Deterministic Edge Faults.] We give a new deterministic labeling scheme for edge faults that uses $\tilde{O}(\sqrt{f})$-bit labels, which can be constructed in polynomial time. This improves on Dory and Parter's [PODC 2021] existential bound of $O(f\log n)$ (requiring exponential time to compute) and the efficient $\tilde{O}(f^2)$-bit scheme of Izumi, Emek, Wadayama, and Masuzawa [PODC 2023]. Our construction uses an improved edge-expander hierarchy and a distributed coding technique based on Reed-Solomon codes. [Deterministic Vertex Faults.] We improve Parter, Petruschka, and Pettie's [STOC 2024] deterministic $O(f^7\log^{13} n)$-bit labeling scheme for vertex faults to $O(f^4\log^{7.5} n)$ bits, using an improved vertex-expander hierarchy and better sparsification of shortcut graphs. [Randomized Edge/Verex Faults.] We improve the size of Dory and Parter's [PODC 2021] randomized edge fault labeling scheme from $O(\min\{f+\log n, \log^3 n\})$ bits to $O(\min\{f+\log n, \log^2 n\log f\})$ bits, shaving a $\log n/\log f$ factor. We also improve the size of Parter, Petruschka, and Pettie's [STOC 2024] randomized vertex fault labeling scheme from $O(f^3\log^5 n)$ bits to $O(f^2\log^6 n)$ bits, which comes closer to their $Ω(f)$-bit lower bound.

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arXiv2024

A Unified Construction of Streaming Sketches via the L\'evy-Khintchine Representation Theorem

In the $d$-dimensional turnstile streaming model, a frequency vector $\mathbf{x}=(\mathbf{x}(1),\ldots,\mathbf{x}(n))\in (\mathbb{R}^d)^n$ is updated entry-wisely over a stream. We consider the problem of $f$-moment estimation, where one wants to estimate $$f(\mathbf{x})=\sum_{v\in[n]}f(\mathbf{x}(v))$$ with a small-space sketch. In this work we present a simple and generic scheme to construct sketches with the novel idea of hashing indices to L\'evy processes, from which one can estimate the $f$-moment $f(\mathbf{x})$ where $f$ is the characteristic exponent of the L\'evy process. The fundamental L\'evy-Khintchine representation theorem completely characterizes the space of all possible characteristic exponents, which in turn characterizes the set of $f$-moments that can be estimated by this generic scheme. The new scheme has strong explanatory power. It unifies the construction of many existing sketches and it implies the tractability of many nearly periodic functions that were previously unclassified. Furthermore, the scheme can be conveniently generalized to multidimensional cases ($d\geq 2$) by considering multidimensional L\'evy processes and can be further generalized to estimate heterogeneous moments by projecting different indices with different L\'evy processes. We conjecture that the set of tractable functions can be characterized using the L\'evy-Khintchine representation theorem via what we called the Fourier-Hahn-L\'evy method.

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arXiv2024

Universal Perfect Samplers for Incremental Streams

If $G : \mathbb{R}_+ \to \mathbb{R}_+$, the $G$-moment of a vector $\mathbf{x}\in\mathbb{R}_+^n$ is $G(\mathbf{x}) = \sum_{v\in[n]} G(\mathbf{x}(v))$ and the $G$-sampling problem is to select an index $v_*\in [n]$ according to its contribution to the $G$-moment, i.e., such that $\Pr(v_*=v) = G(\mathbf{x}(v))/G(\mathbf{x})$. Approximate $G$-samplers may introduce multiplicative and/or additive errors to this probability, and some have a non-trivial probability of failure. In this paper we focus on the exact $G$-sampling problem, where $G$ is selected from the class $\mathcal{G}$ of Laplace exponents of non-negative, one-dimensional Lévy processes, which includes several well studied classes such as $p$th moments $G(z)=z^p$, $p\in[0,1]$, logarithms $G(z)=\log(1+z)$, Cohen and Geri's soft concave sublinear functions, which are used to approximate concave sublinear functions, including cap statistics. We develop $G$-samplers for a vector $\mathbf{x} \in \mathbb{R}_+^n$ that is presented as an incremental stream of positive updates. In particular: * For any $G\in\mathcal{G}$, we give a very simple $G$-sampler that uses 2 words of memory and stores at all times a $v_*\in [n]$, such that $\Pr(v_*=v)$ is exactly $G(\mathbf{x}(v))/G(\mathbf{x})$. * We give a ``universal'' $\mathcal{G}$-sampler that uses $O(\log n)$ words of memory w.h.p., and given any $G\in \mathcal{G}$ at query time, produces an exact $G$-sample. With an overhead of a factor of $k$, both samplers can be used to $G$-sample a sequence of $k$ indices with or without replacement. Our sampling framework is simple and versatile, and can easily be generalized to sampling from more complex objects like graphs and hypergraphs.

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arXiv2024

A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices

The theory of forbidden 0-1 matrices generalizes Turan-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width. The foremost open problems in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $P\in\{0,1\}^{k\times l}$ is the bipartite incidence matrix of an acyclic graph (forest), then $\mathrm{Ex}(P,n) = O(n\log^{C_P} n)$, where $C_P$ is a constant depending only on $P$. This conjecture has been confirmed on many small patterns, specifically all $P$ with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{Ω(\sqrt{\log n})}$, where $S_0,S_1$ are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $(P_t)$, specifically that for every $t\geq 2$, $\mathrm{Ex}(P_t,n)=Θ(n(\log n/\log\log n)^t)$. This is the first proof of an asymptotically sharp bound that is $ω(n\log n)$.

math.CO↗

arXiv2023

Connectivity Labeling and Routing with Multiple Vertex Failures

We present succinct labeling schemes for answering connectivity queries in graphs subject to a specified number of vertex failures. An $f$-vertex/edge fault tolerant ($f$-V/EFT) connectivity labeling is a scheme that produces succinct labels for the vertices (and possibly to the edges) of an $n$-vertex graph $G$, such that given only the labels of two vertices $s,t$ and of at most $f$ faulty vertices/edges $F$, one can infer if $s$ and $t$ are connected in $G-F$. The primary complexity measure is the maximum label length (in bits). The $f$-EFT setting is relatively well understood: [Dory and Parter, PODC 2021] gave a randomized scheme with succinct labels of $O(\log^3 n)$ bits, which was subsequently derandomized by [Izumi et al., PODC 2023] with $\tilde{O}(f^2)$-bit labels. As both noted, handling vertex faults is more challenging. The known bounds for the $f$-VFT setting are far away: [Parter and Petruschka, DISC 2022] gave $\tilde{O}(n^{1-1/2^{Θ(f)}})$-bit labels, which is linear in $n$ already for $f =Ω(\log\log n)$. In this work we present an efficient $f$-VFT connectivity labeling scheme using $poly(f, \log n)$ bits. Specifically, we present a randomized scheme with $O(f^3 \log^5 n)$-bit labels, and a derandomized version with $O(f^7 \log^{13} n)$-bit labels, compared to an $Ω(f)$-bit lower bound on the required label length. Our schemes are based on a new low-degree graph decomposition that improves on [Duan and Pettie, SODA 2017], and facilitates its distributed representation into labels. Finally, we show that our labels naturally yield routing schemes avoiding a given set of at most $f$ vertex failures with table and header sizes of only $poly(f,\log n)$ bits. This improves significantly over the linear size bounds implied by the EFT routing scheme of Dory and Parter.

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arXiv2023

Sorting Pattern-Avoiding Permutations via 0-1 Matrices Forbidding Product Patterns

We consider the problem of comparison-sorting an $n$-permutation $S$ that avoids some $k$-permutation $π$. Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak prove that when $S$ is sorted by inserting the elements into the GreedyFuture binary search tree, the running time is linear in the extremal function $\mathrm{Ex}(P_π\otimes \text{hat},n)$. This is the maximum number of 1s in an $n\times n$ 0-1 matrix avoiding $P_π\otimes \text{hat}$, where $P_π$ is the $k\times k$ permutation matrix of $π$, $\otimes$ the Kronecker product, and $\text{hat} = \left(\begin{array}{ccc}&\bullet&\\\bullet&&\bullet\end{array}\right)$. The same time bound can be achieved by sorting $S$ with Kozma and Saranurak's SmoothHeap. In this paper we give nearly tight upper and lower bounds on the density of $P_π\otimes\text{hat}$-free matrices in terms of the inverse-Ackermann function $α(n)$. \[ \mathrm{Ex}(P_π\otimes \text{hat},n) = \left\{\begin{array}{ll} Ω(n\cdot 2^{α(n)}), & \mbox{for most $π$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))α(n)}), & \mbox{for all $π$.} \end{array}\right. \] As a consequence, sorting $π$-free sequences can be performed in $O(n2^{(1+o(1))α(n)})$ time. For many corollaries of the dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix theory. Our analysis may be useful in analyzing other classes of access sequences on binary search trees.

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arXiv2023

On the Extremal Functions of Acyclic Forbidden 0-1 Matrices

The extremal theory of forbidden 0-1 matrices studies the asymptotic growth of the function $\mathrm{Ex}(P,n)$, which is the maximum weight of a matrix $A\in\{0,1\}^{n\times n}$ whose submatrices avoid a fixed pattern $P\in\{0,1\}^{k\times l}$. This theory has been wildly successful at resolving problems in combinatorics, discrete and computational geometry, structural graph theory, and the analysis of data structures, particularly corollaries of the dynamic optimality conjecture. All these applications use acyclic patterns, meaning that when $P$ is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound $\mathrm{Ex}(P,n)$ for acyclic $P$. Prior results have only ruled out the strict $O(n\log n)$ bound conjectured by Furedi and Hajnal. It is consistent with prior results that $\forall P. \mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$, and also consistent that $\forall ε>0.\exists P. \mathrm{Ex}(P,n) \geq n^{2-ε}$. In this paper we establish a stronger lower bound on the extremal functions of acyclic $P$. Specifically, we give a new construction of relatively dense 0-1 matrices with $Θ(n(\log n/\log\log n)^t)$ 1s that avoid an acyclic $X_t$. Pach and Tardos have conjectured that this type of result is the best possible, i.e., no acyclic $P$ exists for which $\mathrm{Ex}(P,n)\geq n(\log n)^{ω(1)}$.

math.CO↗

arXiv2023

Fully Dynamic Connectivity in $O(\log n(\log\log n)^2)$ Amortized Expected Time

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with $O(\log n(\log\log n)^2)$ amortized expected update time and $O(\log n/\log\log\log n)$ worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011).

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arXiv2022

Simpler and Better Cardinality Estimators for HyperLogLog and PCSA

\emph{Cardinality Estimation} (aka \emph{Distinct Elements}) is a classic problem in sketching with many industrial applications. Although sketching \emph{algorithms} are fairly simple, analyzing the cardinality \emph{estimators} is notoriously difficult, and even today the state-of-the-art sketches such as HyperLogLog and (compressed) \PCSA{} are not covered in graduate level Big Data courses. In this paper we define a class of \emph{generalized remaining area} (\tGRA) estimators, and observe that HyperLogLog, LogLog, and some estimators for PCSA are merely instantiations of \tGRA{} for various integral values of $τ$. We then analyze the limiting relative variance of \tGRA{} estimators. It turns out that the standard estimators for HyperLogLog and PCSA can be improved by choosing a \emph{fractional} value of $τ$. The resulting estimators come \emph{very} close to the Cramér-Rao lower bounds for HyperLogLog{} and PCSA derived from their Fisher information. Although the Cramér-Rao lower bound \emph{can} be achieved with the Maximum Likelihood Estimator (MLE), the MLE is cumbersome to compute and dynamically update. In contrast, \tGRA{} estimators are trivial to update in constant time. Our presentation assumes only basic calculus and probability, not any complex analysis~\cite{FlajoletM85,DurandF03,FlajoletFGM07}.

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arXiv2022

Byzantine Agreement with Optimal Resilience via Statistical Fraud Detection

Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively \emph{corrupt} up to $f<n/3$ parties. Moreover, the problem is insoluble with $f\geq n/3$ corruptions. However, Bracha's 1984 protocol achieved $f<n/3$ resilience at the cost of exponential expected latency $2^{Θ(n)}$, a bound that has never been improved in this model with $f=\lfloor (n-1)/3 \rfloor$ corruptions. In this paper we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt $f<n/3$ parties, while incurring only polynomial latency with high probability. Our protocol follows earlier polynomial latency protocols of King and Saia and Huang, Pettie, and Zhu, which had suboptimal resilience, namely $f \approx n/10^9$ and $f<n/4$, respectively. Resilience $f=(n-1)/3$ is uniquely difficult as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventually lets us flip a coin with an unambiguous outcome. In the beginning the influence of the Byzantine players is too powerful to overcome and they can essentially fix the coin's behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can ``blacklist'' players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection.

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arXiv2022

Wake Up and Join Me! An Energy-Efficient Algorithm for Maximal Matching in Radio Networks

We consider networks of small, autonomous devices that communicate with each other wirelessly. Minimizing energy usage is an important consideration in designing algorithms for such networks, as battery life is a crucial and limited resource. Working in a model where both sending and listening for messages deplete energy, we consider the problem of finding a maximal matching of the nodes in a radio network of arbitrary and unknown topology. We present a distributed randomized algorithm that produces, with high probability, a maximal matching. The maximum energy cost per node is $O(\log^2 n)$, where $n$ is the size of the network. The total latency of our algorithm is $O(n \log n)$ time steps. We observe that there exist families of network topologies for which both of these bounds are simultaneously optimal up to polylog factors, so any significant improvement will require additional assumptions about the network topology. We also consider the related problem of assigning, for each node in the network, a neighbor to back up its data in case of node failure. Here, a key goal is to minimize the maximum load, defined as the number of nodes assigned to a single node. We present a decentralized low-energy algorithm that finds a neighbor assignment whose maximum load is at most a polylog($n$) factor bigger that the optimum.

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