Non-crossing shortest paths lengths in planar graphs in linear time

Comparison of weighted and unweighted network analysis in the case of a pig trade network in Northern Germany

Approximate tree decompositions of planar graphs in linear time

Let G be a finite undirected graph with edge set E. An edge set E′⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge e∈E∖E′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is ${\mathbb{NP}}$ -complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for P k -free graphs for any k≥5; P k denotes a chordless path with k vertices and k−1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7-free graphs in a robust way.

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent tau in (2,3), up to value n^{{beta _n}} for some {beta _n}gg (log n)^{-gamma } and gamma in (0,1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n^{{beta _n}}. These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log (n^{{beta _n}}) / |log (tau -2)| + 1/({beta _n}(3-tau )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/{beta _n}=o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log -periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n^{f_n{beta _n}}, where f_n in (0,1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two ‘end-segments’ starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log (n^{{beta _n}}) / |log (tau -2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/({beta _n}(3-tau ))+tight, and it contains only vertices with degree at least of order n^{(1-f_n){beta _n}}, thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n^{{beta _n}}, and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.

Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct anO(nlogn) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product, we give a yet simpler proof of Grötzsch's theorem.

Grotzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grotzsch's theorem.

Shortest paths in Sierpiński graphs

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of O(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.

We consider the following problem. Let G=(V, E) be an undirected planar graph and let s, t∈V, s≠t. The problem is to find a set of pairwise edge-disjoint paths, each connecting s with t, with maximum cardinality. In other words, the problem is to find a maximum unit flow from s to t. The fastest algorithm in the literature has running time O(|V|log|V|)(¦V¦ log ¦V¦). In this paper now, we give a linear time algorithm.KeywordsPlanar GraphLinear TimeUnit FlowOuter FaceSearch PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Rainbow colouring of split graphs

Complex networks describe a broad range of systems in nature and society. As a fundamental concept of graph theory, the path connecting nodes and edges plays a crucial role in network science, where the computation of shortest path lengths and numbers has garnered substantial focus. It is well known that powers of the adjacency matrix can calculate the number of walks, specifying their corresponding lengths. However, developing methodologies to quantify both the number and length of shortest paths through the adjacency matrix remains a challenge. Here, we extend powers of the adjacency matrix from walks to shortest paths. We address the all-pairs shortest path count problem and propose a fast algorithm based on powers of the adjacency matrix that counts both the number and the length of all shortest paths. Numerous experiments on synthetic and real-world networks demonstrate that our algorithm is significantly faster than the classical algorithms across various network types and sizes. Moreover, we verified that the time complexity of our proposed algorithm significantly surpasses that of the current state-of-the-art algorithms. The superior property of the algorithm allows for rapid calculation of all shortest paths within large-scale networks, offering significant potential applications in traffic flow optimization and social network analysis.

The ℓ1 -embedding problem of a graph is the problem to find a map from its vertex set to ℝ d such that the length of the shortest path between any two vertices is equal to the ℓ1-distance between the mapping of the two vertices in ℝ d . The ℓ1-embedding problem partially contains the shortest path problem since an ℓ1-embedding provides the all-pairs shortest paths. While Höfting and Wanke showed that the shortest path problem is NP-hard, Chepoi, Deza, and Grishukhin showed a polynomial-time algorithm for the ℓ1-embedding of planar 2-dimensional periodic graphs. In this paper, we study the ℓ1-embedding problem on ℓ1 -rigid 2-dimensional periodic graphs, for which there are finite representations of the ℓ1-embedding. The periodic graphs form a strictly larger class than planar ℓ1-embeddable 2-dimensional periodic graphs. Using the theory of geodesic fiber, which was originally proposed by Eon as an invariant of a periodic graph, we show an exponential-time algorithm for the ℓ1-embedding of ℓ1-rigid 2-dimensional periodic graphs, including the non-planar ones. Through Höfting and Wanke’s formulation of the shortest path problem as an integer program, our algorithm also provides an algorithm for solving a special class of parametric integer programming.

The recent line of research into topology design focuses on lowering network diameter. Many low-diameter topologies such as Slim Fly or Jellyfish that substantially reduce cost, power consumption, and latency have been proposed. A key challenge in realizing the benefits of these topologies is routing. On one hand, these networks provide shorter path lengths than established topologies such as Clos or torus, leading to performance improvements. On the other hand, the number of shortest paths between each pair of endpoints is much smaller than in Clos, but there is a large number of non-minimal paths between router pairs. This hampers or even makes it impossible to use established multipath routing schemes such as ECMP. In this article, to facilitate high-performance routing in modern networks, we analyze existing routing protocols and architectures, focusing on how well they exploit the diversity of minimal and non-minimal paths. We first develop a taxonomy of different forms of support for multipathing and overall path diversity. Then, we analyze how existing routing schemes support this diversity. Among others, we consider multipathing with both shortest and non-shortest paths, support for disjoint paths, or enabling adaptivity. To address the ongoing convergence of HPC and “Big Data” domains, we consider routing protocols developed for both HPC systems and for data centers as well as general clusters. Thus, we cover architectures and protocols based on Ethernet, InfiniBand, and other HPC networks such as Myrinet. Our review will foster developing future high-performance multipathing routing protocols in supercomputers and data centers.

Construction of a simple elimination scheme for a chordal comparability graph in linear time

Semantic understanding is an essential research issue for many applications, such as social network analysis, collective intelligence and content computing, which tells the inner meaning of language form. Recently, Abstract Meaning Representation (AMR) is attracted by many researchers for its semantic representation ability on an entire sentence. However, due to the non-projectivity and reentrancy properties of AMR graphs, they lose some important semantic information in parsing from sentences. In this paper, we propose a general AMR parsing model which utilizes a two-stack-based transition algorithm for both Chinese and English datasets. It can incrementally parse sentences to AMR graphs in linear time. Experimental results demonstrate that it is superior in recovering reentrancy and handling arcs while is competitive with other transition-based neural network models on both English and Chinese datasets.