Seth Pettie

An optimal minimum spanning tree algorithm

We establish that the algorithmic complexity of the minimumspanning tree problem is equal to its decision-tree complexity.Specifically, we present a deterministic algorithm to find aminimum spanning tree of a graph with <i>n</i> vertices and<i>m</i> edges that runs in time<i>O</i>(<i>T</i>^*(<i>m,n</i>)) where<i>T</i>^* is the minimum number of edge-weightcomparisons needed to determine the solution. The algorithm isquite simple and can be implemented on a pointer machine.Althoughour time bound is optimal, the exact function describing it is notknown at present. The current best bounds known for<i>T</i>^* are <i>T</i>^*(<i>m,n</i>) =Ω(<i>m</i>) and <i>T</i>^*(<i>m,n</i>) =<i>O</i>(<i>m</i> ∙ α(<i>m,n</i>)), where α is acertain natural inverse of Ackermanns function.Even under theassumption that <i>T</i>^* is superlinear, we show thatif the input graph is selected from <i>G</i>_<i>n,m</i>,our algorithm runs in linear time with high probability, regardlessof <i>n</i>, <i>m</i>, or the permutation of edge weights. Theanalysis uses a new martingale for <i>G</i>_<i>n,m</i>similar to the edge-exposure martingale for<i>G</i>_<i>n,p</i>.

A new approach to all-pairs shortest paths on real-weighted graphs

We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-addition model. It runs in O(mn + n2 log log n) time, improving on the long-standing bound of O(mn + n2 log n) derived from an implementation of Dijkstras algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.Our algorithm is rooted in the so-called component hierarchy approach to shortest paths invented by Thorup for integer-weighted undirected graphs, and generalized by Hagerup to integer-weighted directed graphs. The technical contributions of this paper include a method for approximating shortest path distances and a method for leveraging approximate distances in the computation of exact ones. We also provide a simple, one line characterization of the class of hierarchy-type shortest path algorithms. This characterization leads to some pessimistic lower bounds on computing single-source shortest paths with a hierarchy-type algorithm.

The Locality of Distributed Symmetry Breaking

We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log Δ + (log log n)⁴) rounds, where Δ is the maximum degree. This improves a 25-year old randomized algorithm of Israeli and Itai that takes O(log n) rounds and is provably optimal for all log Δ in the range [(log log n)⁴, √log n]. 2) A randomized maximal independent set (MIS) algorithm requiring O(log Δ√log n) rounds, for all Δ, and only 2^O(√log log n) rounds when Δ = poly(log n). These improve on the 25-year old O(log n)-round randomized MIS algorithms of Luby and Alon, Babai, and Itai when log Δ ≫ √log n. 3) A randomized (Δ + 1)-coloring algorithm requiring O(log Δ + 2^O(^{(√log log n)}) rounds, improving on an algorithm of Schneider and Wattenhofer that takes O(log Δ + √log n) rounds. This result implies that an O(Δ)-coloring can be computed in 2^{O(√log log n)} rounds for all Δ, improving on Kothapalli et al.s O(√log n)-round algorithm. We also introduce a new technique for reducing symmetry breaking problems on low arboricity graphs to low degree graphs. Corollaries of this reduction include MM and MIS algorithms for low arboricity graphs (e.g., planar graphs and graphs that exclude any fixed minor) requiring O(√log n) and O(log^2/3 n) rounds w.h.p., respectively.

Linear-Time Approximation for Maximum Weight Matching

The <i>maximum cardinality</i> and <i>maximum weight matching</i> problems can be solved in <i>Õ</i>(<i>m</i>√<i>n</i>) time, a bound that has resisted improvement despite decades of research. (Here <i>m</i> and <i>n</i> are the number of edges and vertices.) In this article, we demonstrate that this “<i>m</i>√<i>n</i> barrier” can be bypassed by approximation. For any <i>ε</i> > 0, we give an algorithm that computes a (1 − <i>ε</i>)-approximate maximum weight matching in <i>O</i>(<i>mε</i>⁻¹ log <i>ε</i>⁻¹) time, that is, optimal <i>linear time</i> for any fixed <i>ε</i>. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error.

Low distortion spanners

A <i>spanner</i> of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say <i>H</i> ⊆ <i>G</i> is an <i>f</i>-spanner of <i>G</i> if any two vertices <i>u</i>,<i>v</i> at distance <i>d</i> in <i>G</i> are at distance at most <i>f</i>(<i>d</i>) in <i>H</i>. There is clearly some trade-off between the sparsity of <i>H</i> and the <i>distortion</i> function <i>f</i>, though the nature of the optimal trade-off is still poorly understood. In this article we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called <i>connection schemes</i>. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. [1999] and Baswana et al. [2009], and give spanners whose multiplicative distortion quickly tends toward 1. Our results rival the simplicity of all previous algorithms and provide substantial improvements (up to a doubly exponential reduction in edge density) over the comparable spanners of Elkin and Peleg [2004] and Thorup and Zwick [2006].

A simpler linear time 2/3 - ε approximation for maximum weight matching

We present two 2/3 - e approximation algorithms for the maximum weight matching problem that run in time O(m log 1e;). We give a simple and practical randomized algorithm and a somewhat more complicated deterministic algorithm. Both algorithms are exponentially faster in terms of e than a recent algorithm by Drake and Hougardy. We also show that our algorithms can be generalized to find a 1 - e approximation to the maximum weight matching, for any e > 0.

A Shortest Path Algorithm for Real-Weighted Undirected Graphs

We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log

Approximating Maximum Weight Matching in Near-Linear Time

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Additive spanners and (α, β)-spanners

) time, where

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

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