Spanning Forest Research Articles

Counting spanning trees of multiple complete split-like graph containing a given spanning forest

The multiple complete split-like graph MCSb,sa is the join of an empty graph K‾a and s copies of complete graph Kb. In this article, we obtain the formulas for the number of spanning trees of MCSb,sa containing a given spanning forest when s=1 and 2. Particularly, when s=1, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

Percolation transition for random forests in d⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d\\geqslant 3$\\end{document}

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta >0$\\end{document} per edge. It arises as the q→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\ o 0$\\end{document} limit of the q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document}-state random cluster model with p=βq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p=\\beta q$\\end{document}. We prove that in dimensions d⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d\\geqslant 3$\\end{document} the arboreal gas undergoes a percolation phase transition. This contrasts with the case of d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d=2$\\end{document} where no percolation transition occurs.The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane H0|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{H}^{0|2}$\\end{document}. This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the H0|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{H}^{0|2}$\\end{document} model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

Hyperbolic radial spanning tree

We define and analyze an extension to the d-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019). First, the hyperbolic metric allows us to obtain fine control of the branches’ fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude thanks to the precise estimates mentioned before.

Generating functions and counting formulas for spanning trees and forests in hypergraphs

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15].

A Career Recommendation Method for College Students Based on Occupational Values

The numerous recruitment information and the information asymmetry between college majors and work posts make it difficult for college students to seize the job opportunities that conform to their occupational values, and the employment success rate is always at a low level. The effect of current college student career recommendation systems is usually unsatisfactory, and the existing systems haven’t fully considered the role of college students’ occupational values in instructing their employment. To fill in this research blank, this paper studied a career recommendation method for college students based on occupational values. At first, the paper proposed a collaborative filtering algorithm based on the features of collect students’ occupational values, introduced a few features that can affect their occupational values, assigned weight values to these features, and gave the method for determining the weight. Then, based on the principle of the Kruskal’s algorithm (the minimum spanning forest), this paper modified the K-Means algorithm and clustered the features of occupational values. At last, the paper elaborated in detail the principle of generating career recommendation results for college students based on occupational values, and used experimental results to verify the correctness of the proposed career recommendation algorithm.

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either the open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb gas technique and conformal field theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with the closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

Spectral-spatial classification of hyperspectral images by combining hierarchical and marker-based Minimum Spanning Forest algorithms

Spectral-spatial classification of hyperspectral images by combining hierarchical and marker-based Minimum Spanning Forest algorithms

Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of (1pm epsilon ). This paper considers computing cut sparsifiers of weighted graphs of size O(nlog (n)/epsilon ^2). Our algorithm computes such a sparsifier in time O(mcdot min (alpha (n)log (m/n),log (n))), both for graphs with polynomially bounded and unbounded integer weights, where alpha (cdot ) is the functional inverse of Ackermann’s function. This improves upon the state of the art by Benczúr and Karger (SICOMP, 2015), which takes O(mlog ^2 (n)) time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP, 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi–Ibaraki forest packing. MSF packings have previously been used by Abraham et al. (FOCS, 2016) in the dynamic setting, and are defined as follows: an M-partial MSF packing of G is a set mathcal {F}={F_1, ldots , F_M}, where F_i is a maximum spanning forest in G{setminus } bigcup _{j=1}^{i-1}F_j. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (Probability on Trees and Networks (2016) Cambridge Univ. Press) in the negative. A version of our argument gives an example of a nonunimodular transitive graph where WUSF≠FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).

A new spectral and spatial framework for detecting buildings with special roofing in hyperspectral images

One of the analyses in hyperspectral images is target detection. With recent developments and the creation of high spatial resolution images, the need to use both spectral and spatial information in the target detection has increased. Recently, an effective approach for spectral and spatial classification has been proposed using minimum spanning forest (MSF) algorithm. It was attempted to improve this approach for target detection in hyperspectral images. In the proposed method, the spectral image was primarily segmented using the watershed algorithm. Afterwards, for the objects resulting from segmentation, five spatial properties of area, environment, strength, meaning intensity, and entropy were extracted. Finally, the detection operation was performed utilizing the marker-based MSF algorithm. The above-mentioned techniques were applied to three hyperspectral images, first Toulouse, second Toulouse and Quebec. The results of quantitative and qualitative evaluations showed that the proposed method improved the kappa coefficient by 40, 34 and 23% in comparison with the spectral angle measurement (SAM) algorithm in the first Toulouse, second Toulouse and Quebec images, respectively.

Counting spanning trees in a complete bipartite graph which contain a given spanning forest

AbstractIn this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let be the bipartition of the complete bipartite graph with and . We prove that for any given spanning forest of with components , the number of spanning trees in which contain all edges in is equal to where and for .

Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. We consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. We consider four classical graph problems: For Maximum Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, for Minimum Vertex Cover the Late Accept model is sufficient, and for Minimum Spanning Forest the Late Reject model is sufficient. The Maximum Matching problem admits constant competitive ratios in all cases. We also consider Maximum Acyclic Subgraph and Maximum Planar Subgraph, which exhibit patterns similar to Maximum Independent Set.

Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] that the Free Uniform Spanning Forest (FSF) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for a wide family of H and T, the FSF is disconnected when w is small (relying on [4]). Finally, we prove that when H is the graph of one edge, then for any w, the FSF is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

A family of matrix-tree multijections

For a natural class of r×n integer matrices, we construct a non-convex polytope which periodically tiles ℝ n . From this tiling, we provide a family of geometrically meaningful maps from a generalized sandpile group to a set of generalized spanning trees which give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be applied to graphs, regular matroids, cell complexes with a torsion-free spanning forest, and representable arithmetic matroids with a multiplicity one basis. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

Uniform spanning forests on biased Euclidean lattices

La forêt couvrante uniforme (notée USF) sur un graphe connexe, infini et localement fini avec conductance c(·), est définie comme la loi limite de l’arbre couvrant uniforme de sous-graphes finis. Suivant le graphe et les conductances, la mesure USF ne porte pas nécessairement sur l’ensemble des arbres couvrants. Pemantle (Ann. Probab. 19 (1991) 1559–1574) a démontré que sur Zd muni de la conductance unité c=1, USF porte sur les arbres couvrants si et seulement si d≤4. Dans ce travail, nous étudions USF associée aux conductances c(e)=λ−|e|, où |e| est la distance entre l’arête e et l’origine, et λ∈]0,1[ est un paramètre fixé. Notre résultat principal montre que dans ce cas, USF porte sur un nombre fini d’arbres si et seulement si d=2 ou 3. Plus précisément, nous prouvons que la forêt couvrante uniforme contient 2d arbres si d=2 ou 3, et contient une infinité d’arbres si d≥4. Notre approche s’appuie sur une analyse du rayon spectral et de la vitesse de la marche aléatoire λ-biaisée sur Zd.

Edge-disjoint spanning trees and forests of graphs

The spanning tree packing number of a graph G is the maximum number of edge-disjoint spanning trees in G, and the arboricity of G is the minimum number of edge-disjoint forests needed to partition the edge set of G. In this paper, we give bounds on the spanning tree packing number and the arboricity of graphs in terms of effective resistances. As applications, we show that equiarboreal graphs are uniformly dense, and determine the maximum number of edge-disjoint spanning c-forests of equiarboreal graphs, including edge-transitive graphs, 1-walk-regular graphs, distance-regular graphs and so on. The arboricity of a regular graph can be derived from our effective resistance bounds directly.

Centralized Multi-Hop Routing Based on Multi-Start Minimum Spanning Forest Algorithm in the Wireless Sensor Networks.

Wireless sensor networks (WSNs) are widely applied in environmental monitoring, target tracking, military, and industrial fields. However, the battery energy of sensor nodes in WSNs is limited, which limits its development. Previous studies have shown that clustering protocols and multi-hop communication are beneficial to reduce nodes energy consumption. The multi-hop protocol based on low energy adaptive clustering hierarchy (LEACH) has been proven to significantly reduce energy dissipation. However, LEACH-based multi-hop protocols generally have the problem of unbalanced energy dissipation and data conflicts. In this paper, we propose a centralized multi-hop routing based on multi-start minimum spanning forest (LEACH-CMF) to optimize LEACH. In order to realize multi-hop communication, we introduced a multi-start minimum spanning tree algorithm to select relay nodes with the minimum relay cost and generate appropriate multi-hop paths. To avoid data collision in multi-hop communication and make nodes including the cluster heads sleep as much as possible in the non-working state, we design a bottom-up continuous time slot allocation method to improve the time division multiple access (TDMA) cycle. We performed simulation in NS2. The simulation results show that the network lifetime is approximately doubled compared to LEACH and centralized low energy adaptive clustering hierarchy (LEACH-C). The simulation results show that the proposed protocol can effectively balance the energy dissipation of nodes and prolong network lifetime.

Infinite-step stationarity of rotor walk and the wired spanning forest

We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).

Weights of uniform spanning forests on nonunimodular transitive graphs

Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.

Graph Tikhonov Regularization and Interpolation Via Random Spanning Forests

Novel Monte Carlo estimators are proposed to solve both the Tikhonov regularization (TR) and the interpolation problems on graphs. These estimators are based on random spanning forests (RSF), the theoretical properties of which enable to analyze the estimators' theoretical mean and variance. We also show how to perform hyperparameter tuning for these RSF-based estimators. TR is a component in many well-known algorithms, and we show how the proposed estimators can be easily adapted to avoid expensive intermediate steps in generalized semi-supervised learning, label propagation, Newton's method and iteratively reweighted least squares. In the experiments, we illustrate the proposed methods on several problems and provide observations on their run time.