Unweighted Tree Research Articles

Improved bounds for rectangular monotone Min-Plus Product and applications

In a recent breakthrough paper, Chi et al. (STOC'22) introduce an O˜(n3+ω2) time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions n×n and polynomial bounded values. This greatly improves upon the previous O˜(n12+ω5) time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: M-bounded Single Source Replacement Path, Batch Range Mode, k-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al..

Data structures for categorical path counting queries

Consider an ordinal tree T on n nodes, each of which is assigned a category from an alphabet [σ]={1,2,…,σ}. We preprocess the tree T in order to support categorical path counting queries, which ask for the number of distinct categories occurring on the path in T between two query nodes x and y. For this problem, we propose a linear-space data structure with query time O(nlg⁡lg⁡σlg⁡w), where w=Ω(lg⁡n) is the word size in the word-RAM. As shown in our proof, from the assumption that matrix multiplication cannot be solved in time polynomially faster than cubic (with only combinatorial methods), our result is optimal, save for polylogarithmic speed-ups. For a trade-off parameter 1≤t≤n, we propose an O(n+n2t2)-word, O(tlg⁡lg⁡σlg⁡w) query time data structure. We also consider c-approximate categorical path counting queries, which must return an approximation to the number of distinct categories occurring on the query path, by counting each such category at least once and at most c times. We describe a linear-space data structure that supports 2-approximate categorical path counting queries in O(lg⁡n/lg⁡lg⁡n) time.Next, we generalize the categorical path counting queries to weighted trees. Here, a query specifies two nodes x,y and an orthogonal range Q. The answer to thus formed categorical path range counting query is the number of distinct categories occurring on the path from x to y, if only the nodes with weights falling inside Q are considered. We propose an O(nlg⁡lg⁡n+(n/t)4)-word data structure with O(tlg⁡lg⁡n) query time, or an O(n+(n/t)4)-word data structure with O(tlgϵ⁡n) query time. For an appropriate choice of the trade-off parameter t, this implies a linear-space data structure with O(n3/4lgϵ⁡n) query time. We then extend the approach to the trees weighted with vectors from [n]d, where d is a constant integer greater than or equal to 2. We present a data structure with O(nlgd−1+ϵ⁡n+(n/t)2d+2) words of space and O(tlgd−1⁡n(lg⁡lg⁡n)d−2) query time. For an O(n⋅polylogn)-space solution, one thus has O(n2d+12d+2⋅polylogn) query time.The inherent difficulty revealed by the lower bound we proved motivated us to consider data structures based on sketching. In unweighted trees, we propose a sketching data structure to solve the approximate categorical path counting problem which asks for a (1±ϵ)-approximation (i.e. within 1±ϵ of the true answer) of the number of distinct categories on the given path, with probability 1−δ, where 0<ϵ,δ<1 are constants. The data structure occupies O(n+ntlg⁡n) words of space, for the query time of O(tlg⁡n). For trees weighted with d-dimensional weight vectors (d≥1), we propose a data structure with O((n+ntlg⁡n)lgd⁡n) words of space and O(tlgd+1⁡n) query time.All these problems generalize the corresponding categorical range counting problems in Euclidean space Rd+1, for respective d, by replacing one of the dimensions with a tree topology.

Reverse 1-centre problem on trees under convex piecewise-linear cost function

ABSTRACT Given a network as well as a prescribed vertex s in which a facility is located, the aim of solving reverse 1-centre problems is to decrease edge lengths under a budget constraint in a way that the maximum distance between s and the other vertices is minimized. This paper considers the reverse 1-centre problem on general tree networks with weights associated to vertices under a convex piecewise-linear cost function. It is shown that the problem can be transformed into a parametric minimum cost flow problem. A polynomial time approximation scheme (PTAS) is first developed using the bisection method. Then, the problem on unweighted trees is investigated. Using sensitivity analysis, a pseudo-polynomial time algorithm is designed. Finally, a hybrid algorithm is developed to obtain an exact solution in polynomial time.

Distance matrices of subsets of the Hamming cube

Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of n+1 points {x0,x1,…,xn} in the Hamming cube Hn=({0,1}n,ℓ1). In this article we derive a formula for the determinant of the distance matrix D of an arbitrary set of m+1 points {x0,x1,…,xm} in Hn. It follows from this more general formula that det(D)≠0 if and only if the vectors x0,x1,…,xm are affinely independent. Specializing to the case m=n provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases m<n and m=n is noted. We also show that if D is the distance matrix of an unweighted tree on n+1 vertices, then 〈D−11,1〉=2∕n where 1 is the column vector all of whose coordinates are 1. Finally, we derive a new proof of Murugan’s classification of the subsets of Hn that have strict 1-negative type.

Coalitional games induced by matching problems: Complexity and islands of tractability for the Shapley value

The paper focuses on cooperative games where the worth of any coalition of agents is determined by the optimal value of a matching problem on (possibly weighted) graphs. These games come in different forms that can be grouped in two broad classes, namely of matching and allocation games, and they have a wide spectrum of applications, ranging from two-sided markets where buyers and sellers are encoded as vertices in a graph, to allocation problems where indivisible goods have to be assigned (matched) to agents in a fair way, possibly using monetary compensations.The Shapley value and the related notion of Banzhaf value have often been identified as appropriate solution concepts for many applications of matching/allocation games, but their computation is intractable in general. It is known that these concepts can be computed in polynomial time for matching games on unweighted trees and on graphs having degree at most two. However, it was open whether or not such positive results could be extended to the more general case of graphs having bounded treewidth, and to the case of allocation problems on weighted graphs.The paper provides a positive answer to these questions, by showing that computing the Shapley value and the Banzhaf value is feasible in polynomial time for the following classes of games: matching games over unweighted graphs having bounded treewidth, allocation games over weighted graphs having bounded treewidth, and allocation games over weighted graphs and such that each good is of interest for two agents at most. Without such structural restrictions, computing these solution concepts on allocation games is instead shown to be #P-hard, even in the case of unweighted graphs.

Combinatorial Algorithms for the Uniform-Cost Inverse 1-Center Problem on Weighted Trees

Inverse 1-center problem on a network is to modify the edge lengths or vertex weights within certain bounds so that the prespecified vertex becomes an (absolute) 1-center of the perturbed network and the modifying cost is minimized. This paper focuses on the inverse 1-center problem on a weighted tree with uniform cost of edge length modification, a generalization for the analogous problem on an unweighted tree (Alizadeh and Burkard, Discrete Appl. Math. 159, 706–716, 2011). To solve this problem, we first deal with the weighted distance reduction problem on a weighted tree. Then, the weighted distances balancing problem on two rooted trees is introduced and efficiently solved. Combining these two problems, we derive a combinatorial algorithm with complexity of \(O(n^{2})\) to solve the inverse 1-center problem on a weighted tree if there exists no topology change during the edge length modification. Here, n is the number of vertices in the tree. Dropping this condition, the problem is solvable in \(O(n^{2}\mathbf {c})\) time, where \(\mathbf {c}\) is the compressed depth of the tree. Finally, some special cases of the problem with improved complexity, say linear time, are also discussed.

Genetic diversity and population structure of Urochloa grass accessions from Tanzania using simple sequence repeat (SSR) markers

Urochloa (syn.—Brachiaria s.s.) is one of the most important tropical forages that transformed livestock industries in Australia and South America. Farmers in Africa are increasingly interested in growing Urochloa to support the burgeoning livestock business, but the lack of cultivars adapted to African environments has been a major challenge. Therefore, this study examines genetic diversity of Tanzanian Urochloa accessions to provide essential information for establishing a Urochloa breeding program in Africa. A total of 36 historical Urochloa accessions initially collected from Tanzania in 1985 were analyzed for genetic variation using 24 SSR markers along with six South American commercial cultivars. These markers detected 407 alleles in the 36 Tanzania accessions and 6 commercial cultivars. Markers were highly informative with an average polymorphic information content of 0.79. The analysis of molecular variance revealed high genetic variation within individual accessions in a species (92%), fixation index of 0.05 and gene flow estimate of 4.77 showed a low genetic differentiation and a high level of gene flow among populations. An unweighted neighbor-joining tree grouped the 36 accessions and six commercial cultivars into three main clusters. The clustering of test accessions did not follow geographical origin. Similarly, population structure analysis grouped the 42 tested genotypes into three major gene pools. The results showed the Urochloa brizantha (A. Rich.) Stapf population has the highest genetic diversity (I = 0.94) with high utility in the Urochloa breeding and conservation program. As the Urochloa accessions analyzed in this study represented only 3 of 31 regions of Tanzania, further collection and characterization of materials from wider geographical areas are necessary to comprehend the whole Urochloa diversity in Tanzania.

Upgrading the 1-center problem with edge length variables on a tree

This paper discusses upgrading the 1-center problem on networks, which tries to change the lengths of the edges within certain bounds and find the best place for 1-center with respect to the new lengths so that the objective value is minimized. As this problem is NP-hard on general graphs, the problem is considered where the underlying graph is a tree. It is mentioned that this problem is solvable in polynomial time by solving a series of linear programs. A combinatorial algorithm with On2log(n) time complexity is proposed for the equal cost case, where n is the number of vertices of the tree. It is also shown that the problem is solvable in On2log(n)2 time on an unweighted tree, i.e., all vertex weights are equal to one, but the costs are arbitrary.

KERAGAMAN GENETIK VARIETAS PADI &lt;em&gt;japonica&lt;/em&gt; DAN &lt;em&gt;indica&lt;/em&gt; BERDASARKAN MARKA DNA TERKAIT MUTU RASA

PCR-based markers and evaluation of physicochemical properties should be addressed for the improvement of rice varieties with good eating dan eating quality (EQ). This study aimed to examine the genetic diversity of rice varieties based on DNA markers related to physicochemical properties determining EQ. A total of 46 rice varieties consisting of 22 japonica varieties and 24 indica varieties were examined using 43 PCR-based markers. The results showed that polymorphic information content (PIC) ranged from 0.04 to 0.38, in support of genetic diversity indices which ranged from 0.04 to 0.50 across total markers. Pairwise genetic similarity matrix ranged from 0.40 to 0.98 with the closest genetic distance was observed between two japonica varieties (Dongjin and Hwaseong) and the most distant one was between japonica and indica (Onnuri/Manmi with Cigeulis/Fatmawati). The unweighted neighbor-joining tree clustered the rice varieties into two major clades, indica and japonica, and subsequent subclades were differentiating according to the individual genetic background. The genetic diversity of rice from different subspecies and DNA markers for EQ can effectively be utilized for basic information and marker-assisted selection (MAS) for the development of improved varieties with good EQ in rice breeding program.

Locating Eigenvalues of Perturbed Laplacian Matrices of Trees

We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian matrix of a tree in a given real interval. The algorithm can be applied to weightedor unweighted trees. Using our method we characterize the trees that have up to $5$ distincteigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacencyand normalized Laplacian matrices as special cases, among others.

New Gromov-Inspired Metrics on Phylogenetic Tree Space.

We present a new class of metrics for unrooted phylogenetic X-trees inspired by the Gromov-Hausdorff distance for (compact) metric spaces. These metrics can be efficiently computed by linear or quadratic programming. They are robust under NNI operations, too. The local behaviour of the metrics shows that they are different from any previously introduced metrics. The performance of the metrics is briefly analysed on random weighted and unweighted trees as well as random caterpillars.

A generalized linear time algorithm for an optimal k-distance dominating set of a weighted tree

The new algorithm finds an optimal k-distance dominating set D in a weighted trees T=(V,E) with link-weights w(x,y)>0 in time O(|V|). It generalizes a previous linear time algorithm for an optimal k-hop dominating set for an unweighted tree. This significantly increases the applications of dominating sets in trees.

Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part II

In Part II, we study the unweighted tree augmentation problem (TAP) via the Lasserre (sum of squares) system. We prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is $$\le \frac{3}{2}+\epsilon $$ , where $$\epsilon >0$$ can be any small constant. We obtain this result by designing a polynomial-time algorithm for TAP that achieves an approximation guarantee of ( $$\frac{3}{2}+\epsilon $$ ) relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin et al. (Integer Programming and Combinatorial Optimization, LNCS, volume 6655, Springer, pp 301–314, 2011).

Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP

In Part I, we study a special case of the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. In the special case, we forbid so-called stems; these are a particular type of subtree configuration. For stemless TAP, we prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is $\leq \frac{3}{2}+\epsilon$, where $\epsilon>0$ can be any small constant. We obtain this result by designing a polynomial-time algorithm for stemless TAP that achieves an approximation guarantee of ($\frac32+\epsilon$) relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin, Mathieu and Nguyen (IPCO 2011). Also, we present an example of stemless TAP such that the approximation guarantee of $\frac32$ is tight for the algorithm. In Part II of this paper, we extend the methods of Part I to prove the same results relative to the same SDP relaxation for TAP.

Network Majority on Tree Topological Network

Let G=(V,E) be a graph, and w:V→Q>0 be a positive weight function on the vertices of G. For every subset X of V, let w(X)=∑v∈Gw(v). A non-empty subset S⊂V(G) is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G\\S, we have w(C)≥w(D) whenever there is an edge between C and D.In this paper we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlining tree is restricted to be a star, but it is polynomially solvable for paths. Then we define the concept of a parameterized infinite family of “proper central subgraphs” on trees, whose polar ends are the minimum-weight connected safe sets and the centroids. We show that each of these central subgraphs includes a centroid. We also give a linear-time algorithm to find all of these subgraphs on unweighted trees.

Linear time algorithms to solve the linear ordering problem for oriented tree based graphs

We present in this paper two simple linear algorithms that solve to optimality the linear ordering problem for unweighted tree based graphs viz. the oriented trees and the oriented divide-and-conquer graphs.

Broadcasting in weighted trees under the postal model

Given a weighted graph G=(V,E), the broadcasting problem is to find a broadcast center such that the maximum communication time from the center to all vertices is minimized. For this problem, Slater et al. [29] proposed an O(n)-time algorithm in unweighted trees and Koh and Tcha [23] designed an O(nlog⁡n)-time algorithm in unweighted trees under the telephone model. In this paper, we strengthen the results of Slater et al. by showing an O(n)-time algorithm in weighted trees under the postal model. The algorithm computes the set of broadcast centers, determines the broadcast time of the broadcast centers, and an optimal broadcast scheme from one of the centers to all vertices in the tree. We further show that the broadcast time of any vertex in the tree can also be determined in O(n) time.

Approximation algorithms for pricing with negative network externalities

We study the problems of pricing an indivisible product to consumers who are embedded in a given social network. The goal is to maximize the revenue of the seller by the so-called iterative pricing that offers consumers a sequence of prices over time. The consumers are assumed to be impatient in that they buy the product as soon as the seller posts a price not greater than their valuations of the product. The product's value for a consumer is determined by two factors: a fixed consumer-specified intrinsic value and a variable externality that is exerted from the consumer's neighbors in a linear way. We focus on the scenario of negative externalities, which captures many interesting situations, but is much less understood in comparison with its positive externality counterpart. Assuming complete information about the network, consumers' intrinsic values, and the negative externalities, we prove that it is NP-hard to find an optimal iterative pricing, even for unweighted tree networks with uniform intrinsic values. Complementary to the hardness result, we design a 2-approximation algorithm for general weighted networks with (possibly) nonuniform intrinsic values. We show that, as an approximation to optimal iterative pricing, single pricing works fairly well for many interesting cases, such as forests, Erdźs---Renyi networks and Barabasi---Albert networks, although its worst-case performance can be arbitrarily bad in general networks.

On The Inverse Of A Class Of Bipartite Graphs With Unique Perfect Matchings

Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)−1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries ±1 such that DA(G)−1D is nonnegative. The graph associated to the matrix DA(G)−1D is called the inverse of G and it is denoted by G+. The graph G+ is an undirected, weighted, connected, bipartite graph with a unique perfect matching. Nonsingular, unweighted trees are contained inside the class G. We first give a constructive characterization of the class of weighted graphs Hw that can occur as the inverse of some graph G∈\mathcal{ G}. This generalizes Theorem 2.6 of Neumann and Pati[13], where the authors have characterized graphs that occur as inverses of nonsingular, unweighted trees. A weighted graph Gw is said to have the property (R) if for each eigenvalue λ of A(Gw), 1⁄λ is also an eigenvalue of A(Gw). If further, the multiplicity of λ and 1⁄λ are the same, then Gw is said to have property (SR). A characterization of the class of nonsingular, weighted trees Tw with at least 8 vertices that have property (R) was given in [13] under some restriction on the weights. It is natural to ask for such a characterization for the whole of G, possibly with some weaker restrictions on the weights. We supply such a characterization. In particular, for trees it settles an open problem raised in [13].

The inverse convex ordered 1-median problem on trees under Chebyshev norm and Hamming distance

We investigate the inverse convex ordered 1-median problem on unweighted trees under the cost functions related to the Chebyshev norm and the Hamming distance. By the special structure of the problem under Chebyshev norm, we deduce the so-called maximum modification to modify the edge lengths of the tree. Additionally, the cost function of the problem receives only finite values under the bottleneck Hamming distance. Therefore, we can find the optimal cost of the problem by applying binary search. It is shown that both of the problems, under Chebyshev norm and under the bottleneck Hamming distance, can be solved in O(n2log n) time in all situations, with or without essential topology changes. Here, n is the number of vertices of the tree. Finally, we prove that the problem under weighted sum Hamming distance is NP-hard.