arXiv subjects

Seth Pettie

Publications and source records attributed to Seth Pettie.

58 records · Page 4Linked to original sources

arXiv2012

Connectivity Oracles for Planar Graphs

We consider dynamic subgraph connectivity problems for planar graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either "off" (failed) or "on" (recovered). We wish to answer connectivity queries with respect to the "on" subgraph. The model has two natural variants, one in which there are $d$ edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a $d$-failure connectivity oracle for planar graphs that processes any $d$ edge/vertex failures in $sort(d,n)$ time so that connectivity queries can be answered in $pred(d,n)$ time. (Here $sort$ and $pred$ are the time for integer sorting and integer predecessor search over a subset of $[n]$ of size $d$.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Patrascu and Thorup's lower bound on predecessor search is that no $d$-failure connectivity oracle (even on trees) can beat $pred(d,n)$ query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in $O(\log n)$ time (amortized), whereas in general planar graphs both bounds become $O(\log^2 n)$.

cs.DS↗

arXiv2011

Scaling algorithms for approximate and exact maximum weight matching

The {\em maximum cardinality} and {\em maximum weight matching} problems can be solved in time $\tilde{O}(m\sqrt{n})$, a bound that has resisted improvement despite decades of research. (Here $m$ and $n$ are the number of edges and vertices.) In this article we demonstrate that this "$m\sqrt{n}$ barrier" is extremely fragile, in the following sense. For any $ε>0$, we give an algorithm that computes a $(1-ε)$-approximate maximum weight matching in $O(mε^{-1}\logε^{-1})$ time, that is, optimal {\em linear time} for any fixed $ε$. 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. Our second contribution is a new {\em exact} maximum weight matching algorithm for integer-weighted bipartite graphs that runs in time $O(m\sqrt{n}\log N)$. This improves on the $O(Nm\sqrt{n})$-time and $O(m\sqrt{n}\log(nN))$-time algorithms known since the mid 1980s, for $1\ll \log N \ll \log n$. Here $N$ is the maximum integer edge weight.

cs.DS↗

arXiv2007

Splay Trees, Davenport-Schinzel Sequences, and the Deque Conjecture

We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a Davenport-Schinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan's deque conjecture and prove that n deque operations require O(n alpha^*(n)) time, where alpha^*(n) is the minimum number of applications of the inverse-Ackermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.

cs.DS↗

arXiv2007

Sources of Superlinearity in Davenport-Schinzel Sequences

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(σ,n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to σ. It has been proved that for every σ, Ex(σ,n) is either linear or very close to linear; in particular it is O(n 2^{α(n)^{O(1)}}), where αis the inverse-Ackermann function and O(1) depends on σ. However, very little is known about the properties of σthat induce superlinearity of \Ex(σ,n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences.

cs.DM↗