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Register a new userExact Distance Oracles for Planar Graphs
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Christian Sommer
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation $Sin[nlglg n,n^2]$, we show how to construct in $tilde O(S)$ time a data structure of size $O(S)$ that answers distance queries in $tilde O(n/sqrt S)$ time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever $kin[sqrt n,n)$. * We provide a linear-space exact distance oracle for planar graphs with query time $O(n^{1/2+eps})$ for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space $tilde O(n)$ such that for any pair of nodes at distance D the query time is $tilde O(min {D,sqrt n})$. Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with $c = O(sqrt n)$ nodes, we can preprocess G in $tilde O(n)$ time to produce a data structure of size $O(n lglg c)$ that can answer the following queries in $tilde O(c)$ time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA06), Djidjev (WG96), and Fakcharoenphol and Rao (FOCS01). For $sigmain(1,4/3)$ and space $S=n^sigma$, we essentially improve the query time from $n^2/S$ to $sqrt{n^2/S}$.
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A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself.
Distance oracles are data structures that provide fast (possibly approximate) answers to shortest-path and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS01, Thorup introduced approximate distance oracles for planar graphs. He proved that, for any eps>0 and for any planar graph on n nodes, there exists a (1+eps)-approximate distance oracle using space O(n eps^{-1} log n) such that approximate distance queries can be answered in time O(1/eps). Ten years later, we give the first improvements on the space-querytime tradeoff for planar graphs. * We give the first oracle having a space-time product with subquadratic dependency on 1/eps. For space ~O(n log n) we obtain query time ~O(1/eps) (assuming polynomial edge weights). The space shows a doubly logarithmic dependency on 1/eps only. We believe that the dependency on eps may be almost optimal. * For the case of moderate edge weights (average bounded by polylog(n), which appears to be the case for many real-world road networks), we hit a sweet spot, improving upon Thorups oracle both in terms of eps and n. Our oracle uses space ~O(n log log n) and it has query time ~O(log log log n + 1/eps). (Asymptotic notation in this abstract hides low-degree polynomials in log(1/eps) and log*(n).)
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-fragile graph classes, a property that is true for many natural graph classes with sublinear separators. We also provide quasipolynomial-time approximation schemes for these problems in all classes with sublinear separators.
In the sensitive distance oracle problem, there are three phases. We first preprocess a given directed graph $G$ with $n$ nodes and integer weights from $[-W,W]$. Second, given a single batch of $f$ edge insertions and deletions, we update the data structure. Third, given a query pair of nodes $(u,v)$, return the distance from $u$ to $v$. In the easier problem called sensitive reachability oracle problem, we only ask if there exists a directed path from $u$ to $v$. Our first result is a sensitive distance oracle with $tilde{O}(Wn^{omega+(3-omega)mu})$ preprocessing time, $tilde{O}(Wn^{2-mu}f^{2}+Wnf^{omega})$ update time, and $tilde{O}(Wn^{2-mu}f+Wnf^{2})$ query time where the parameter $muin[0,1]$ can be chosen. The data-structure requires $O(Wn^{2+mu} log n)$ bits of memory. This is the first algorithm that can handle $fgelog n$ updates. Previous results (e.g. [Demetrescu et al. SICOMP08; Bernstein and Karger SODA08 and FOCS09; Duan and Pettie SODA09; Grandoni and Williams FOCS12]) can handle at most 2 updates. When $3le flelog n$, the only non-trivial algorithm was by [Weimann and Yuster FOCS10]. When $W=tilde{O}(1)$, our algorithm simultaneously improves their preprocessing time, update time, and query time. In particular, when $f=omega(1)$, their update and query time is $Omega(n^{2-o(1)})$, while our update and query time are truly subquadratic in $n$, i.e., ours is faster by a polynomial factor of $n$. To highlight the technique, ours is the first graph algorithm that exploits the kernel basis decomposition of polynomial matrices by [Jeannerod and Villard J.Comp05; Zhou, Labahn and Storjohann J.Comp15] developed in the symbolic computation community. [...]
Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem. We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used. We present an algorithm which, for any $delta > 0$, given streaming access to an array of length $n$ provides a $(1+delta)$-multiplicative approximation to the emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((log^2 n)/delta)$ space. The previous best known approximation using polylogarithmic space was a multiplicative 2-factor. Our algorithm can be used to estimate the length of the LIS to within an additive $delta n$ for any $delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$. Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time. It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS. We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$. Our algorithm works in the asymmetric setting (inspired by cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$. More precisely, it gives an additive-$delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(log^2 n)/delta)$, where $k$ is the maximum number of times any one symbol appears in $y$.
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