Showing papers by "Rolf Fagerberg published in 2013"
06 Jan 2013
TL;DR: This paper shows how to compute the triplet distance in time O(n log n) and the quartet distance inTime O(dn log n), where d is the maximal degree of any node in the two trees, and presents efficient algorithms for computing these distances.
Abstract: The triplet and quartet distances are distance measures to compare two rooted and two unrooted trees, respectively. The leaves of the two trees should have the same set of n labels. The distances are defined by enumerating all subsets of three labels (triplets) and four labels (quartets), respectively, and counting how often the induced topologies in the two input trees are different. In this paper we present efficient algorithms for computing these distances. We show how to compute the triplet distance in time O(n log n) and the quartet distance in time O(dn log n), where d is the maximal degree of any node in the two trees. Within the same time bounds, our framework also allows us to compute the parameterized triplet and quartet distances, where a parameter is introduced to weight resolved (binary) topologies against unresolved (non-binary) topologies. The previous best algorithm for computing the triplet and parameterized triplet distances have O(n2) running time, while the previous best algorithms for computing the quartet distance include an O(d9n log n) time algorithm and an O(n2.688) time algorithm, where the latter can also compute the parameterized quartet distance. Since d ≤ n, our algorithms improve on all these algorithms.
26 citations
TL;DR: This paper shows that for odd k ≥ 5, the spanning ratio of Yk is at most 1/(1−2sin(3θ/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/( 1−2 sin(θ)/2) for even k ≥ 7.
Abstract: For a set of points in the plane and a fixed integer $k > 0$, the Yao graph $Y_k$ partitions the space around each point into $k$ equiangular cones of angle $\theta=2\pi/k$, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of $Y_5$, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd $k \geq 5$, the spanning ratio of $Y_k$ is at most $1/(1-2\sin(3\theta/8))$, which gives the first constant upper bound for $Y_5$, and is an improvement over the previous bound of $1/(1-2\sin(\theta/2))$ for odd $k \geq 7$. We further reduce the upper bound on the spanning ratio for $Y_5$ from $10.9$ to $2+\sqrt{3} \approx 3.74$, which falls slightly below the lower bound of $3.79$ established for the spanning ratio of $\Theta_5$ ($\Theta$-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and $\Theta$-graph with the same number of cones. We also give a lower bound of $2.87$ on the spanning ratio of $Y_5$. Finally, we revisit the $Y_6$ graph, which plays a particularly important role as the transition between the graphs ($k > 6$) for which simple inductive proofs are known, and the graphs ($k \le 6$) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of $Y_6$ from $17.6$ to $5.8$, getting closer to the spanning ratio of 2 established for $\Theta_6$.
24 citations
TL;DR: A series of algorithmic improvements that have been used during the last decade to develop more efficient algorithms are reviewed by exploiting two different strategies for this; one based on dynamic programming and another based on coloring leaves in one tree and updating a hierarchical decomposition of the other.
Abstract: Distance measures between trees are useful for comparing trees in a systematic manner, and several different distance measures have been proposed. The triplet and quartet distances, for rooted and unrooted trees, respectively, are defined as the number of subsets of three or four leaves, respectively, where the topologies of the induced subtrees differ. These distances can trivially be computed by explicitly enumerating all sets of three or four leaves and testing if the topologies are different, but this leads to time complexities at least of the order n3 or n4 just for enumerating the sets. The different topologies can be counte dimplicitly, however, and in this paper, we review a series of algorithmic improvements that have been used during the last decade to develop more efficient algorithms by exploiting two different strategies for this; one based on dynamic programming and another based oncoloring leaves in one tree and updating a hierarchical decomposition of the other.
18 citations
21 Jan 2013
TL;DR: An algorithm is presented that computes the triplet distance between two rooted binary trees in time O (n log2n) and it is shown that this algorithm can be implemented to give a competitive wall-time running time.
Abstract: The triplet distance is a distance measure that compares two rooted trees on the same set of leaves by enumerating all sub-sets of three leaves and counting how often the induced topologies of the tree are equal or different. We present an algorithm that computes the triplet distance between two rooted binary trees in time O (n log2 n). The algorithm is related to an algorithm for computing the quartet distance between two unrooted binary trees in time O (n log n). While the quartet distance algorithm has a very severe overhead in the asymptotic time complexity that makes it impractical compared to O (n2) time algorithms, we show through experiments that the triplet distance algorithm can be implemented to give a competitive wall-time running time.
10 citations
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TL;DR: This paper addresses the problem of reconstructing a hypergraph from its species and reaction graph and shows NP-completeness of the problem in its Boolean formulation and studies the problem empirically on random and real world instances in order to investigate its computational limits in practice.
Abstract: The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes. Such networks are well described as hypergraphs. However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, e.g.\ on the so-called species and reaction graphs. However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique. In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation. Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice.
2 citations
TL;DR: In this article, the problem of reconstructing a hypergraph from its species and reaction graph was studied empirically on random and real world instances in order to investigate its computational limits in practice.
Abstract: The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes Such networks are well described as hypergraphs However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, eg on the so-called species and reaction graphs However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice
2 citations