Graphs In Linear Time Research Articles

Space limited linear-time graph algorithms on big data

We study algorithms for graph problems in which the graphs are of extremely large size N so that super-linear time ω(N) or linear space Θ(N) would become impractical. We use a parameter k to characterize the computational power of a normal computer that can provide additional time and space bounded by polynomials of k in dealing with the large graphs. In particular, we are interested in strict linear-time algorithms using space O(kO(1)). In our case studies, as examples, we present (1) a randomized greedy algorithm of time O(N) and space O(k2) for a parameterized version of the Maximal Matching problem; and (2) randomized kernelization algorithms of time O(N) and space O(kO(1)) for a number of well-known NP-hard problems. Our kernelization algorithms have their kernel sizes match the best kernel sizes by known polynomial-time kernelization algorithms with no space constraints for the problems. We also study the relationship between our proposed model and the streaming model. This study motivates a new streaming kernelization algorithm for the famous Vertex Cover problem that has an optimal update time complexity while matches the best known space complexity of streaming algorithms for the problem.

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

Given a plane graph it is known how to compute the union of non-crossing shortest paths. These algorithms do not allow neither to list each single shortest path nor to compute length of shortest paths. Given the union of non-crossing shortest paths, we introduce the concept of shortcuts that allows us to establish whether a path is a shortest path by checking local properties on faces of the graph. By using shortcuts we can compute the length of each shortest path, given their union, in total linear time, and we can list each shortest path p in O(max{ℓ,ℓloglogkℓ}), where ℓ is the number of edges in p and k the number of shortest paths.

Finding strong components using depth-first search

We survey three algorithms that use depth-first search to find the strong components of a directed graph in linear time: (1) Tarjan’s algorithm; (2) a cycle-finding algorithm; and (3) a bidirectional search algorithm.

Partitioning into degenerate graphs in linear time

Let G be a connected graph with maximum degree Δ≥3 distinct from KΔ+1. Generalizing Brooks’ Theorem, Borodin and independently Bollobás and Manvel, proved that if p1,…,ps are non-negative integers such that p1+⋯+ps≥Δ−s, then G admits a vertex partition into parts A1,…,As such that, for 1≤i≤s, G[Ai] is pi-degenerate. Here we show that such a partition can be performed in time O(n+m). This generalizes previous results that treated subcases of a conjecture of Abu-Khzam et al. (2020) which our result settles in full.

Eulertigs: minimum plain text representation of k-mer sets without repetitions in linear time

A fundamental operation in computational genomics is to reduce the input sequences to their constituent k-mers. For maximum performance of downstream applications it is important to store the k-mers in small space, while keeping the representation easy and efficient to use (i.e. without k-mer repetitions and in plain text). Recently, heuristics were presented to compute a near-minimum such representation. We present an algorithm to compute a minimum representation in optimal (linear) time and use it to evaluate the existing heuristics. Our algorithm first constructs the de Bruijn graph in linear time and then uses a Eulerian-cycle-based algorithm to compute the minimum representation, in time linear in the size of the output.

Computing Bend-Minimum Orthogonal Drawings of Plane Series–Parallel Graphs in Linear Time

AbstractA planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of G to a polygonal chain consisting of horizontal and vertical segments. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term “plane” indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper we positively answer the question above for the widely-studied class of series–parallel graphs. Our linear-time algorithm is based on a characterization of the planar series–parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is “rolled-up” in an orthogonal drawing of the graph.

String representation of trivalent 2-stratifolds with trivial fundamental group

We give a Python program that is capable of computing and printing all distinct trivalent simple connected 2-stratifold graphs (see Gómez-Larrañaga et al. in Topol Proc 52:329–340, 2018) up to a chosen number of white vertices. Our algorithm uses the three basic operations described on (Gómez-Larrañaga et al. in Bol Soc Mat Mex 26:1301–1312, 2020) to construct new graphs from a set of graphs. We iterate this process to construct all the desired graphs. The algorithm includes an optimization that reduces repetitions at generating the graphs, this is done by recognizing equivalent white vertices of a graph under automorphism. We use a variation of the AHU algorithm to identify those vertices and as well to distinguish isomorphic graphs in linear time. The returned string from the AHU algorithm is also used as a hashing function to search for repeated graphs in amortized constant time.

Vertex partitioning problems on graphs with bounded tree width

In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The Matching Cut problem is the problem of deciding whether a given graph has a matching cut. Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an H - Free Coloring if none of the color classes contain H as an induced subgraph. The H - Free Chromatic Number of G is the minimum number of colors required for an H - Free Coloring of G . Both The Matching Cut problem and the H - Free Coloring problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of f ( | | φ | | , t ) n O ( 1 ) , where | | φ | | is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f ( | | φ | | , t ) on | | φ | | can be as bad as a tower of exponentials. In this paper, we provide explicit combinatorial FPT algorithms for Matching Cut problem and H - Free Coloring problem, parameterized by the tree-width of G . The single exponential FPT algorithm for the Matching Cut problem answers an open question posed by Kratsch and Le (2016). The techniques used in the paper are also used to provide an FPT algorithm for a variant of H -free coloring, where H is forbidden as a subgraph (not necessarily induced) in the color classes of G .

Finding Hamiltonian cycles of truncated rectangular grid graphs in linear time

The Hamiltonian cycle problem is an important problem in graph theory. For solid grid graphs, an O(n4)-time algorithm has been given. In this paper, we solve the problem for a special class of solid grid graphs, i.e. truncated rectangular grid graphs, in linear time.

Non-Crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

Given a set of terminal pairs on the external face of an undirected unweighted planar graph, we give a linear-time algorithm for computing the union of non-crossing shortest paths joining each terminal pair, if such paths exist. This allows us to compute distances between each terminal pair, within the same time bound. We also give a novel concept of incremental shortest path subgraph of a planar graph, i.e., a partition of the planar embedding in subregions that preserve distances, that can be of interest itself.

Computing Strong Roman Domination of Trees and Unicyclic Graphs in Linear Time

Computing Strong Roman Domination of Trees and Unicyclic Graphs in Linear Time

Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Non-Crossing Shortest Paths Lengths in Planar Graphs in Linear Time

Non-Crossing Shortest Paths Lengths in Planar Graphs in Linear Time

Planar rectilinear drawings of outerplanar graphs in linear time

We show how to test in linear time whether an outerplanar graph admits a planar rectilinear drawing, both if the graph has a prescribed plane embedding that the drawing has to respect and if it does not. Our algorithm returns a planar rectilinear drawing if the graph admits one.

Dominating induced matching in some subclasses of bipartite graphs

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. An edge e∈E is said to dominate itself and all other edges adjacent to it. A matching M in a graph G=(V,E) is called a dominating induced matching (d.i.m.) if every edge of G is dominated by edges of M exactly once. The dominating induced matching decide (DIM-Decide) problem asks whether a graph G contains a dominating induced matching. The dominating induced matching (DIM) problem asks to compute a dominating induced matching (d.i.m.) in a graph G that admits a dominating induced matching. The DIM-Decide problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the DIM-Decide problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-k-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m.

The connected p-median problem on complete multi-layered graphs

We address in this paper the connected [Formula: see text]-median problem on complete multi-layered graphs. We first solve the corresponding 1-median problem on the underlying graph in linear time based on the structure of the induced path. For [Formula: see text], we prove that the connected [Formula: see text]-median contains at least one vertex in the layered set corresponding to the 1-median or its adjacent vertices in the induced path. Then, we develop an [Formula: see text] algorithm that solves the problem on a complete multi-layered graph with [Formula: see text] vertices.

NetKI: A kirchhoff index based statistical graph embedding in nearly linear time

Recent advancements in learning from graph-structured data have shown promising results on the graph classification task. However, due to their high time complexities, making them scalable on large graphs, with millions of nodes and edges, remains a challenge. In this paper, we propose NetKI, an algorithm to extract sparse representation from a given graph with n nodes and m edges in O(m∊-2log4n) time. Our approach follows the notion of Kirchhoff index that encodes the structure of the graph by estimating effective resistance - relying on this approach yields nearly linear time graph representation method that allows scalability on sufficiently large graphs. Through extensive experiments, we show that NetKI provides improved results in terms of running time on large networks and the classification accuracy is within range 2% from the state-of-the-art results.

From text to graph: a general transition-based AMR parsing using neural network

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.

Recognizing generalized Petersen graphs in linear time

By identifying a local property which structurally classifies any edge, we show that the family of generalized Petersen graphs can be recognized in linear time.

Labelled packing functions in graphs

Given a positive integer k and a graph G, a k-limited packing in G is a subset B of its vertex set such that each closed vertex neighborhood of G has at most k vertices of B (Gallant et al., 2010). A first generalization of this concept deals with a subset of vertices that cannot be in the set B and also, the number k is not a constant but it depends on the vertex neighborhood (Dobson et al., 2010). As another variation, a {k}-packing function f of G assigns a non-negative integer to the vertices of G in such a way that the sum of the values of f over each closed vertex neighborhood is at most k (Hinrichsen et al., 2014). The three associated decision problems are NP-complete in the general case. We introduce L-packing functions as a unified notion that generalizes all limited packing concepts introduced up to now. We present a linear time algorithm that solves the problem of finding the maximum weight of an L-packing function in strongly chordal graphs when a strong elimination ordering is given that includes the linear algorithm for {k}-packing functions in strongly chordal graphs (2014). Besides, we show how the algorithm can be used to solve the known clique-independence problem on strongly chordal graphs in linear time (G. Chang et al., 1993).