arXiv subjects

Christian Wulff-Nilsen

Publications and source records attributed to Christian Wulff-Nilsen.

At least 37 records · Page 2Linked to original sources

arXiv2017

Minor-free graphs have light spanners

We show that every $H$-minor-free graph has a light $(1+ε)$-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is \[O\left(\frac{σ_H}{ε^3}\log \frac{1}ε\right)\] where $σ_H = |V(H)|\sqrt{\log |V(H)|}$ is the sparsity coefficient of $H$-minor-free graphs. That is, it has a practical dependency on the size of the minor $H$. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in $H$-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is $2^{O_H\left(\frac{1}{ε^4}\log \frac{1}ε\right)}n^{O(1)}$ where $O_H$ ignores dependencies on the size of $H$. Our techniques significantly deviate from existing lines of research on spanners for $H$-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.

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arXiv2017

Dynamic Minimum Spanning Forest with Subpolynomial Worst-case Update Time

We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an $n$-node graph undergoing edge insertions and deletions. Our algorithm guarantees an $O(n^{o(1)})$ worst-case update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen [STOC'17] with update time $O(n^{0.5-ε})$ for some constant $ε>0$ and, independently, by Nanongkai and Saranurak [STOC'17] with update time $O(n^{0.494})$ (the latter works only for maintaining a spanning forest). Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from $O(n^{0.5-ε})$ in Wulff-Nilsen [STOC'17] to $O(n^{o(1)})$ for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the "contraction technique" by Henzinger and King [1997] and Holm et al. [STOC'98], we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most $(1+o(1))n$) edges. This significantly improves the previous approach in [Wulff-Nilsen STOC'17] and [Nanongkai and Saranurak STOC'17] which is based on Frederickson's 2-dimensional topology tree and illustrates a new application to this old technique.

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arXiv2017

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

We present an $O(n^{1.5})$-space distance oracle for directed planar graphs that answers distance queries in $O(\log n)$ time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses $O(n^{5/3})$-space and answers queries in $O(\log n)$ time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any $S\in [n,n^2]$, we show an oracle of size $S$ that answers queries in $\tilde O(\max\{1,n^{1.5}/S\})$ time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of $S$ and improves by polynomial factors over all the previously known tradeoffs for the range $S \in [n,n^{5/3}]$.

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arXiv2017

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - ε})$ for some constant $ε> 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time.

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arXiv2016

Fully-Dynamic Minimum Spanning Forest with Improved Worst-Case Update Time

We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n^{1/2 - c}) expected worst-case time for some constant c > 0 and this worst-case bound holds with probability at least 1 - n^{-d} where d is a constant that can be made arbitrarily large. This is the first data structure achieving an improvement over the O(n^{1/2}) deterministic worst-case update time of Eppstein et al., a bound that has been standing for nearly 25 years. In fact, it was previously not even known how to maintain a spanning forest of an unweighted graph in worst-case time polynomially faster than Theta(n^{1/2}). Our result is achieved by first giving a reduction from fully-dynamic to decremental minimum spanning forest preserving worst-case update time up to logarithmic factors. Then decremental minimum spanning forest is solved using several novel techniques, one of which involves keeping track of low-conductance cuts in a dynamic graph. An immediate corollary of our result is the first Las Vegas data structure for fully-dynamic connectivity where each update is handled in worst-case time polynomially faster than Theta(n^{1/2}) w.h.p.; this data structure has O(1) worst-case query time.

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arXiv2016

Approximate Distance Oracles for Planar Graphs with Improved Query Time-Space Tradeoff

We consider approximate distance oracles for edge-weighted n-vertex undirected planar graphs. Given fixed epsilon > 0, we present a (1+epsilon)-approximate distance oracle with O(n(loglog n)^2) space and O((loglog n)^3) query time. This improves the previous best product of query time and space of the oracles of Thorup (FOCS 2001, J. ACM 2004) and Klein (SODA 2002) from O(n log n) to O(n(loglog n)^5).

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arXiv2015

All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$.

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arXiv2015

Near-optimal adjacency labeling scheme for power-law graphs

An adjacency labeling scheme is a method that assigns labels to the vertices of a graph such that adjacency between vertices can be inferred directly from the assigned label, without using a centralized data structure. We devise adjacency labeling schemes for the family of power-law graphs. This family that has been used to model many types of networks, e.g. the Internet AS-level graph. Furthermore, we prove an almost matching lower bound for this family. We also provide an asymptotically near- optimal labeling scheme for sparse graphs. Finally, we validate the efficiency of our labeling scheme by an experimental evaluation using both synthetic data and real-world networks of up to hundreds of thousands of vertices.

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arXiv2014

Space-Efficient Path-Reporting Approximate Distance Oracles

We consider approximate {\em path-reporting} distance oracles, distance labeling and labeled routing with extremely low space requirement, for general undirected graphs. For distance oracles, we show how to break the n\log n space bound of Thorup and Zwick if approximate {\em paths} rather than distances need to be reported. For approximate distance labeling and labeled routing, we break the previously best known space bound of O(log n) words per vertex. The cost for such space efficiency is an increased stretch.

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arXiv2014

Faster Fully-Dynamic Minimum Spanning Forest

We give a new data structure for the fully-dynamic minimum spanning forest problem in simple graphs. Edge updates are supported in $O(\log^4n/\log\log n)$ amortized time per operation, improving the $O(\log^4n)$ amortized bound of Holm et al. (STOC'98, JACM'01). We assume the Word-RAM model with standard instructions.

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arXiv2014

Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and $n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has a separator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can be found in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+ε}/\ell)$ for any constant $ε> 0$ was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time $O(\mbox{poly}(h)\ell m^{1+ε})$. This is a significant improvement for small $h$ and $\ell$. If $\ell = Ω(n^{ε'})$ for an arbitrarily small chosen constant $ε' > 0$, we get a time bound of $O(\mbox{poly}(h)\ell n^{1+ε})$. The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on $h$) and running time $O(\mbox{poly}(h)(\sqrt\ell n^{1+ε} + n^{2+ε}/\ell^{3/2}))$ when $\ell = Ω(n^{ε'})$. Our third algorithm has running time $O(\mbox{poly}(h)\sqrt\ell n^{1+ε})$ when $\ell = Ω(n^{ε'})$. It finds a separator of size $O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n)$ which is no worse than previous bounds when $h$ is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.

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arXiv2013

Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected $n$-vertex planar graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a min $st$-cut in $G$? We show how to answer such queries in constant time with $O(n\log^4n)$ preprocessing time and $O(n\log n)$ space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in $O(n\log^4n)$ time and $O(n\log n)$ space. Additionally, an explicit representation can be obtained in $O(C)$ time and space where $C$ is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead, or deterministically with an additional $\log^2 n$ factor in the preprocessing times.

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arXiv2012

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.

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arXiv2012

Approximate Distance Oracles with Improved Query Time

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 2$, we show that a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ and with $O(\log k)$ query time can be constructed in $O(\min\{kmn^{1/k},\sqrt km + kn^{1 + c/\sqrt k}\})$ time for some constant $c$. This improves the $O(k)$ query time of Thorup and Zwick. Furthermore, for any $0 < ε\leq 1$, we give an oracle of size $O(kn^{1 + 1/k})$ that answers $((2 + ε)k)$-approximate distance queries in $O(1/ε)$ time. At the cost of a $k$-factor in size, this improves the $128k$ approximation achieved by the constant query time oracle of Mendel and Naor and approaches the best possible tradeoff between size and stretch, implied by a widely believed girth conjecture of Erdős. We can match the $O(n^{1 + 1/k})$ size bound of Mendel and Naor for any constant $ε> 0$ and $k = O(\log n/\log\log n)$.

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arXiv2012

Faster Deterministic Fully-Dynamic Graph Connectivity

We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in $O(\log^2n/\log\log n)$ amortized time and connectivity queries in $O(\log n/\log\log n)$ worst-case time, where $n$ is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have $O(\log^2n)$ amortized update time and $O(\log n/\log\log n)$ worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer machine with standard $AC^0$ instructions.

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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 $O(\log^2 n)$.

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arXiv2011

Approximate Distance Oracles with Improved Preprocessing Time

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 1$, we show that for some universal constant $c$, a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ can be constructed in $O(\sqrt km + kn^{1 + c/\sqrt k})$ time and can answer queries in $O(k)$ time. We also give an oracle which is faster for smaller $k$. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all $k\geq 6$ and improve the $O(kmn^{1/k})$ time bound of Thorup and Zwick except for very sparse graphs and small $k$. When $m = Ω(n^{1 + c/\sqrt k})$ and $k = O(1)$, our oracle is optimal w.r.t.\ both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erdős.

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arXiv2011

Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plotkin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawarabayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + ε})$ time for any constant $ε> 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in $O(\poly(h)n^{5/4 + ε})$ time for any constant $ε> 0$. We also present the first $O(\poly(h)n)$ time algorithm to find a separator of size $O(n^c)$ for a constant $c < 1$. As corollaries of our results, we get improved algorithms for shortest paths and maximum matching. Furthermore, for integers $\ell$ and $h$, we give an $O(m + n^{2 + ε}/\ell)$ time algorithm that either produces a $K_h$-minor of depth $O(\ell\log n)$ or a separator of size at most $O(n/\ell + \ell h^2\log n)$. This improves the shallow minor algorithm of Plotkin, Rao, and Smith when $m = Ω(n^{1 + ε})$. We get a similar running time improvement for an approximation algorithm for the problem of finding a largest $K_h$-minor in a given graph.

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