Directed Tree Research Articles

Scheduling tasks of multi‐join queries in a multiprocessor

This paper deals with the problem of scheduling spawned tasks when a query is issued to a database which resides on a MIMD multiprocessor. These tasks have the property that their associated dependency scheme can be presented as a directed tree. We present a theoretical framework with extensive experimental simulations which increase the throughput of database applications. We derive a family of algorithms for scheduling tasks. Their performance is tested on several common multiprocessor configurations. For better performance the adaptation of the scheduling algorithm to the multiprocessor configuration is examined and analyzed. The scheduling algorithms are divided into two cases: (a) permitted changes in the resources connection scheme of the multiprocessor, and (b) no changes allowed. The algorithms are scalable and their complexity is computed. In particular, we present an algorithm for scheduling tasks in the case where the construction of a central storage location is permitted. One of the main tools for the construction of the above algorithms is the notion of (t, 1)-domination and k-domination sets. Copyright © 1999 John Wiley & Sons, Ltd.

Distributions of Runs and Consecutive Systems on Directed Trees

In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.

Algorithms for a Class of Isotonic Regression Problems

The isotonic regression problem has applications in statistics, operations research, and image processing. In this paper a general framework for the isotonic regression algorithm is proposed. Under this framework, we discuss the isotonic regression problem in the case where the directed graph specifying the order restriction is a directed tree with n vertices. A new algorithm is presented for this case, which can be regarded as a generalization of the PAV algorithm of Ayer et al. Using a simple tree structure such as the binomial heap, the algorithm can be implemented in O(n log n) time, improving the previously best known O(n 2 ) time algorithm. We also present linear time algorithms for special cases where the directed graph is a path or a star.

The forbidden subgraph characterization of directed vertex graphs

A graph is called a directed vertex (DV) graph if it is the intersection graph of a family of directed paths in a directed tree, i.e., a tree in which each edge is oriented, with one or more vertices of indegree zero. In this paper we present the forbidden subgraph characterization of DV graphs.

Optimal Labellings of Rooted Directed Trees

We consider an optimal labelling problem for a rooted directed tree (abbreviated as “RDT”) which is motivated by certain scheduling problem. We obtain several necessary and sufficient conditions for the optimal labellings of a RDT and give a polynomially bounded algorithm for constructing the optimal labellings of a RDT. We also generalize the problem and the corresponding results to the case of vertex weighted RDTs.

Scheduling Real-Time Jobs with Chain Tree Temporal Distance Constraints

Real-Time Job Scheduling is an important issue in the area of Real-Time systems. In many real-time systems, once a monitor job has started, another monitor job must be executed within a specific time interval (temporal distance) in order to monitor the progress of real-time processes. The problem (TDCP) of scheduling a set of jobs in which pairs of the jobs must be executed within a temporal distance constraint on a single processor is considered in this paper. For a set of jobs with arbitrary execution times, Han, Lin and Liu have already shown that TDCP is NP-hard in the ordinary sense even with just one temporal distance constraint. For a set of jobs with equal execution time, the problem will be related to the Linear Array Problem (LAP) in the sense that LAP can be reduced to TDCP. Since LAP has been shown to be NP-hard in the strong sense when the graph is of tree structure, TDCP is NP-hard for tree-structured temporal distance constraints of arbitrary values. Han, Lin and Liu have thus shown that TDCP is NP-hard in the strong sense even for a set of unit-execution-time jobs having chain-structured temporal distance constraints of arbitrary values and arbitrary number of jobs in each chain. For temporal distance constraints of the same constant value, Han, Lin and Liu introduced an O(n2) algorithm (where n is the total number of jobs), an O(m2c2) algorithm (where m is the number of chains and c is the constant temporal distance constraint) for solving a special case of TDCP where the graph is of directed chain tree and an O(n log n) algorithm for solving the special bi-level case. In this paper, another special case of the bi-level chain tree is studied. The graph of the temporal distance constraints is restricted to a bi-level directed tree with at most two sons, and present an O(n2) algorithm ND.

Enumeration of Polyhex Hydrocarbons to h = 21

An algorithm based on the boundary-edges code and the reverse search method is proposed for enumerating nonisomorphic planar simply connected polyhexes. These polyhexes are associated with vertices of a graph whose edges correspond to addition of a hexagon. A directed tree is defined on this graph. To this effect, a new father−son relationship is introduced: the father is the polyhex obtained when removing the hexagon associated with the first digit of the son's code. Then testing if a generated polyhex is a legitimate one in the enumeration can be done easily and efficiently. The resulting algorithm is used to enumerate polyhexes with h ≤ 21 hexagons, a set of over one trillion molecules, which is >600 times larger than previously done.

On avoidable and unavoidable claws

A claw of degree k is a directed tree consisting of k paths emerging from a common root. We prove that every claw of order n with degree less than 19 50 n appears in every n-vertex tournament. We also construct avoidable claws with degree approaching 11 23 n . Thus for large n, the maximum λ such that every claw with degree λn appears in every n-vertex tournament satisfies λ ⩽ 11 23 . This improves earlier bounds.

Scheduling Problems in a Practical Allocation Model

A parallel computational model is defined which addresses I/O contention,latency, and pipe-lined message passing between tasks allocated to differentprocessors. The model can be used for parallel task-allocation on either anetwork of workstations or on a multi-stage inter-connected parallel machine.To study performance bounds more closely, basic properties are developed forwhen the precedence constraints form a directed tree. It is shown that theproblem of optimally scheduling a directed one-level precedence tree on anunlimited number of identical processors in this model is NP-hard. Theproblem of scheduling a directed two-level precedence tree is also shown tobe NP-hard even when the system latency is zero.

A Digraphic Approach for Dimensional Chain Identification in Design and Manufacturing

In order to determine working dimensions and tolerances of a process plan, a tolerance chart is often employed. Dimensional chain identification is a necessary step in the tolerance charting procedures. This paper presents a digraphic representation for the tolerance chart. Two directed trees are generated from working dimensions, blueprint dimensions and stock removals of the tolerance chart. These trees are then used to identify dimensional chains in order to find the relationships among the design and manufacturing dimensions. Furthermore, the concept of a reverse dimensional chain is introduced. The reverse dimensional chains can also be identified from the directed trees. The relationship between the forward and reverse dimensional chains is also discussed. Finally, the applications of dimensional chains to the determination of the working dimensions and tolerances are briefly described.

On avoidable and unavoidable trees

A directed tree is a rooted tree if there is one vertex (the root) of in-degree 0 and every other vertex has in-degree 1. The depth of a rooted tree is the length of a longest path from the root. A directed graph G is called n-unavoidable if every tournament of order n contains it as a subgraph. M. Saks and V. Sos [“On Unavoidable Subgraphs of Tournaments,” Colloquia Mathematica Societatis Janos Bolyai 37, Finite and Infinite Sets, Eger, Hungary (1981), 663–674] constructed unavoidable rooted spanning trees of depth 3. There they wrote, “It is natural to ask how small the depth of a spanning n-unavoidable rooted tree can be.” In this paper we construct unavoidable rooted spanning trees of depth 2. Note that the depth 2 is the best we can do. For each n define λ(n) to be the largest real number such that every claw with degree d ≤ λ(n)n is n-unavoidable. The example in X. Lu [“On Claws Belonging to Every Tournament,” Combinatorica, Vol. 11 (1991), pp. 173–179] showed that λ(n) < 1/2 for sufficiently large n, but the upper bound on λ(n) given there tends to 1/2 for large n. Let λ be the lim sup of λ(n) as n tends to infinity. In this paper we show that λ is strictly less than 1/2, specifically λ ≤ 25/52. © 1996 John Wiley & Sons, Inc.

Subpath acyclic digraphs

A characterization is given for acyclic digraphs that are the acyclic intersection digraphs of subpaths of a directed tree.

Balanced allocations for tree-like inputs

We consider the following procedure for constructing a directed tree on n vertices: The underlying undirected tree is fixed in advance but the edges of the tree are presented in a random order (all orders are equally likely); each edge is oriented towards its endpoint that has the lower indegree at the time of its insertion. The question is what is E( M( n)), the expected maximum indegree? As we shall explain, this problem has connections with balanced allocations and with on-line load balancing. Previous results by Azar, Naor, and Rom imply that if the insertion order is arbitrary, for any tree, M( n) = O( log n) and that there are trees and insertion orders for which M(n) = Ω(log n). On the other hand, results by Azar, Broder, Karlin, and Upfal imply that if both the underlying tree and the insertion order are random, then E( M( n)) = Θ( log log n). Here we show an intermediate result: for any tree if the insertion order is random, then E(M(n)) = O( log n log log n ) and there are trees for which E(M(n)) = Ω( log n log log n ) .

Realizations of interlacing by tree-patterned matrics

In this note, we prove the following theorem. Given 2n real numbers x 1≥β1≥x,≥≥β2≥ [cddot] x n ≥β n ≥0 and a directed tree T on n+1 vertices with a distinguished vertex which is a source. then there exists an n+1 x n real matrix A that satisfies. 1. The directed graph of A is T (where the first row of A corresponds to the distinguished source vertex); 2. The singular values of A are α1, …, α n 3. The singular values of the n × n matrix A 0 obtained by deleting the top row of A are β1, …, β n We adapt out method to prove a similar result first found by A. L. Duarte for eigenvalue interlacing of Hermitian matrices. Some partial results of this type for invariant factor interlacing of integral matrices are given.

An efficient algorithm for generating all Kekul� patterns of a generalized benzenoid system

An algorithm for generating all Kekule patterns of a benzenoid system was given by Jiang [1]. However, for a generalized benzenoid system, this problem is more complex. In this paper, we give an efficient algorithm for generating all Kekule patterns of a generalized benzenoid system by the generalized directed tree structure [2] of the set of Kekule patterns of a generalized benzenoid system.

Separable convex programs with ratio constraints over a directed tree

This paper describes a class of separable convex programs with ratio constraints over a directed tree that has applications in production planning, regression analysis, quality improvement, and other related areas. To develop a simple procedure for solving this class of problems, we extend the procedure for solving separable convex programs with partially ordered variables (Tang, 1988). For any problem that has N variables, our simple procedure finds an optimal solution by solving at most 2 N convex subproblems with one variable.

Mathematical properties and structures of sets of sextet patterns of generalized polyhexes

Several definitions of sextet patterns and super sextets of (generalized) polyhexes have been given, first by He Wenjie and He Wenchen [1], later by Zhang Fuji and Guo Xiaofeng [2], and by Ohkami [3], respectively. The one-to-one correspondence between Kekule and sextet patterns has also been proved by the above authors using different methods. However, in a rigorous sense, their definitions of sextet patterns and super sextets are only some procedures for finding sextet patterns and super sextets, not explicit definitions. In this paper, we give for the first time such an explicit definition from properties of generalized polyhexes, and give a new proof of the Ohkami-Hosoya conjecture using the new definition. Furthermore, we investigate mathematical properties and structures of sets of generalized polyhexes, and prove that thes-sextet rotation graphRs(G) of the set of sextet patterns of a generalized polyhexG is a directed tree with a unique root corresponding to theg-root sextet pattern ofG.

Directed tree structure of the set of Kekulé patterns of generalized polyhex graphs

In this paper we define the concept of g -sextet rotation graph of a generalized polyhex graph G , and prove that the g -sextet rotation graph D ( G ) of G is a directed tree. This conclusion is a generalization of [1] and also valid for any polyhex fragment graphs. Furthermore, by our results, an error in a proof [2] of the Ohkami—Hosoya conjecture [3] about a one-to-one correspondence between Kekulé and sextet patterns of a polyhex graph is corrected.

Reliability covering problems

AbstractThis paper studies the reliability covering problem, in which given routes provides service to various stops (e.g., of a transit system). If the routes are subject to failure, it is desired to find the probability that all stops will be covered by an operating route. It is shown that this problem is NP‐hard even when routes are defined with respect to an underlying tree. Polynomially solvable cases are developed when some additional structure is imposed on the routes of a tree: e.g., when the routes are directed paths of a rooted directed tree. These cases generalize reliability computations for consecutive k‐out‐of‐n systems as well as the extensions to consecutively connected systems studied by Shanthikumar and by Hwang and Yao.

Finding minimum cost directed trees with demands and capacities

Given a directed graph, the authors consider the problem of finding a rooted directed tree (or branching) satisfying given demands at all the nodes and capacity constraints on the arcs. Various integer programming formulations are compared, including flow and multicommodity flow formulations and two partitioning-type formulations involving directed subtrees. Computational results concerning an application to the design of a low voltage electricity network are given. For the class of problems considered, one of the partitioning formulations allows one to solve problems with up to 100 nodes and several hundred arcs.