Aggregate nearest neighbor query, which returns an optimal target point that minimizes the aggregate distance for a given query point set, is one of the most important operations in spatial databases and their application domains. This paper addresses the problem of finding the aggregate nearest neighbor for a merged set that consists of the given query point set and multiple points needed to be selected from a candidate set, which we name as merged aggregate nearest neighbor(MANN) query. This paper proposes two algorithms to process MANN query on road networks when aggregate function is max. Then, we extend the algorithms to support other aggregate functions (e.g., sum). Extensive experiments are conducted to examine the behaviors of the solutions in terms of five parameters affecting the performance. The overall experiments show that our strategies to minimize the response time are effective.
The world of perfect graphs has grown to include over 200 special graph classes. The Venn diagrams that are used to show some of the inclusions between classes in the First Generation—for example, Hasse diagrams for the Second Generation. Perhaps the most important new development in the theory of perfect graphs is the recent proof of the strong perfect graph conjecture by Chudnovsky, Robertson, Seymour, and Thomas announced in May 2002. On the algorithmic side, many of the problems that were open in 1980 have subsequently been settled, and algorithms on new classes of perfect graphs have been studied. For example, tolerance graphs generalize both interval graphs and permutation graphs, and coloring tolerance graphs in polynomial time is important in solving scheduling problems where a measure of flexibility or tolerance is allowed for sharing or relinquishing resources when total exclusivity prevents a solution.
This chapter discusses the characterization of split graphs by presenting various theorems. An undirected graph G + (V, E) is defined to be split if there is a partition V + S + K of its vertex set into a stable set S and a complete set K. There is no restriction on edges between the vertices of S and the vertices of K. An undirected graph G is a split graph if its complement G is a split graph. The chapter discusses the condition for a sequence to be graphic. A simple necessary condition for a sequence to be graphic comes from Euler's theorem: The sum Σdi must be even. However, an even sum is not sufficient to insure graphicness. The chapter presents two classical theorems characterizing graphic sequences, which are work of Havel and Hakimi, Erdös and Gallai. It also discusses a third classical theorem on graphic sequences.
Th perfect graph satifies the following properties: (P1): ω(GA) + χ(GA) (for all A⊆V) and (P2): α(GA ) + k(GA) (for all A⊆V). It is clear by duality that a graph G satisfies (P1) if its complement satisfies (P2). A much stronger result was conjectured by Berge, cultivated by Fulkerson, and finally proven by Lovâsz, namely, that (P1 and (P 2 ) are equivalent. This has become known as the perfect graph theorem. This chapter presents the proof of this theorem.
We extend a result of D. J. Rose [9] on perfect Gaussian elimination for symmetric matrices. It is proved that the restriction that all pivots are to be chosen along the main diagonal can be removed without loss of generality.
This chapter presents basic definitions and notations of graph theory. A function that is both injective and surjective is called a bijection. A permutation is simply a bijection from a set to itself. A binary relation R on X may satisfy one or more properties. Such a relation is said to be equivalence if it is reflexive, symmetric, and transitive. A binary relation is called a strict partial order if it is irreflexive and transitive. It is a simple exercise to show that a strict partial order will also be antisymmetric. A graph is defined as a set and a certain relation on that set. It is often convenient to draw a “picture” of the graph. This may be done in many ways. Usually one draws a circle for each vertex and connects vertex x and vertex y with a directed arrow whenever xy is an edge. If both xy and yx are edges, then sometimes a single line joins x and y without arrows.
Publisher Summary This chapter presents a class of perfect graphs, which has a large number of applications. An undirected graph G[Π] from Π can be constructed in the following manner: G[Π] has vertices numbered from 1 to n; two vertices are joined by an edge if the larger of their corresponding numbers is to the left of the smaller in Π (that is, they occur out of their proper order reading left to right). The graph G[Π] is sometimes called the inversion graph of Π. Permutation graphs have many interesting properties. When one reverse the sequence K. Each pair of numbers that occurred in the correct order in Π is now in the wrong order, and vice versa. Thus, the permutation graph one obtains is the complement of G[Π]. The chapter shows that “the complement of a permutation graph is also a permutation graph.” Another property of the graph G[Π] is that it is transitively orientable.
This chapter discusses the notion of perfection in weighted graphs—superperfection. In the process, a more general type of coloring the vertices of a graph are introduced, suggesting many interesting applications. The concept of superperfection is by Alan Hoffman and Ellis Johnson. They were motivated by the shipbuilding problem and most of the early results are theirs. There are two basic methods for demonstrating superperfection: providing a suitable coloring or giving a suitable acyclic orientation. It has been reported that superperfect graphs properly contain the comparability graphs. This leads one to ask under what conditions these two classes coincide. Therefore, the chapter provides one answer to this question and discusses some open problems. The chapter also relates the concept of superperfection to some ideas of linear programming.
One of the first classes of graphs to be recognized as being perfect was the class of triangulated graphs. Triangulated graphs satisfy the perfect property P2 (α-perfection), and satisfy P1 (χ-perfection). These two results, in large measure, inspired the conjecture that P1 and P2 were equivalent. The study of triangulated graphs can well be thought of as the beginning of the theory of perfect graphs. An undirected graph G is called triangulated if every cycle of length strictly greater than 3 possesses a chord—that is, an edge joining two nonconsecutive vertices of the cycle. Equivalently, G does not contain an induced subgraph isomorphic to Cn for n > 3. Being triangulated is a hereditary property inherited by all the induced subgraphs of G. Triangulated graphs are also called as chordal, rigid-circuit, monotone transitive, and perfect elimination graphs.
This chapter discusses a particularly simple technique for distinguishing between stable and nonstable subsets of vertices in a special class of graphs. The graphs that admit this technique, which involves assigning certain weights to the vertices, are called threshold graphs. The chapter introduces the notion of threshold dimension. An induced subgraph of a threshold graph is a threshold graph. Therefore, any graph that contains an induced subgraph isomorphic to one of those is not threshold. The threshold dimension θ (G) of an arbitrary graph G can be defined in an alternate but equivalent manner using threshold graphs. Take θ (G) to be the minimum number of threshold graphs needed to cover the edges of G—that is, partial subgraphs of G, which are themselves threshold, and include every edge at least once.
This chapter discusses the class of perfect graphs known as comparability graphs or transitively orientable graphs; these graphs are discussed in connection with interval graphs. It also discusses the interaction between implication classes. In the process, the formula for the number t(G) of transitive orientations of a comparability graph G and a procedure is obtained for constructing them. The treatment follows Golumbic, in which most of the theory was developed. An alternate method for calculating t(G) appears in Shevrin and Filippov. The chapter describes an algorithm for calculating transitive orientations and for determining whether or not a graph is a comparability graph. This technique is a modification of one first presented by Pnueli, Lempel, and Even. A transitive orientation F could be constructed for a comparability graph G in 0(δe + n) steps, where δ is the maximum degree of a vertex, e is the number of edges, and n is the number of vertices. From the transitive orientation F, one can assign a minimum coloring of G using the height function in 0(n + e) additional steps. At the same time, a maximum clique could also be calculated.
Publisher Summary This chapter discusses the properties, characterization, and theorems of interval graphs. The earliest characterization of interval graphs was obtained by Lekerkerker and Boland. Their result embodies the notion that an interval graph neither can branch into more than two directions nor can circle back onto itself. Theorem by Gilmore and Hoffman establishes where the class of interval graphs belongs in the world of perfect graphs. A theorem by Fulkerson and Gross states that an undirected graph G is an interval graph if its clique matrix M has the consecutive 1's property for columns. Interval graphs are characterized as those graphs whose clique matrices satisfy the consecutive 1's property for columns. One may apply this characterization to a recognition algorithm for interval graphs G + (V, E) in a two-step process. Booth and Leuker have shown that this step can also be executed in linear time. The chapter discusses the main ideas behind their algorithm and its implementation.
In designing algorithms, one frequently requires a mechanism for exploring the vertices and edges of a graph. Two criteria of priority prove to be especially useful in exploring a graph. They are depth-first search (DFS) and breadth-first search (BFS). In both methods, each edge is traversed exactly once in the forward and reverse directions and each vertex is visited. By examining a graph in such a structured way, some algorithms become easier to understand and faster to execute. The choice of which method to use will often affect the efficiency of the algorithm. Thus, simply selecting a clever data structure is not sufficient to insure a good implementation. A carefully chosen search technique is also needed. One search technique may be preferable over another, that is, it may give one a more efficient implementation. This chapter presents the list of some instances of problems for which DFS and BFS are most effective.
This chapter discusses the equivalence and characterization of the stack sorting graphs, the circle graphs, and the overlap graphs. There are a number of open problems concerning this class of graphs: (1) find an algorithm that recognizes circle graphs and constructs a representation for the graph as intersecting chords of a circle, (2) are the coloring and clique-cover problems NP-complete for circle graphs, And (3) is the strong perfect graph conjecture true for circle graphs. The stable set problem and the clique problem are tractable when restricted to the not-so-perfect graphs. In the context of open problem (1), it is essential that one be given, a priori, a representation of the graph as overlapping intervals, intersecting chords, or a permutation to be sorted. The chapter discusses the use of the overlap graph model.
This chapter discusses graphs and related notions. Graphs should be visualized as points joined by lines with or without arrows rather than be thought of as formal objects. A graph G is a triple (V(G), E(G), iG) where V(G) i s a finite set of vertices, E(G) is a finite set of edges, and iG is an incidence function, which associates with each edge a pair of vertices, not necessarily distinct, called its end points or end. Vertices are also called “nodes” or “junctions” while edges are also called “arcs” or “branches.” An edge may have a single end point; such edges are called “selfloops.” A vertex may have no edges incident on it; such vertices are isolated. A connected graph with each vertex having degree two is called a “circuit graph” or a “polygon graph.”
This chapter provides an overview of sets. A set (or collection) is specified by the elements (or members) that belong to it. If element x belongs to the set X, and is written as x ∈ X (x ∉ X). Two sets are equal if they have the same members. A set is finite if it has a finite number of elements. Otherwise, it is infinite. A set is often specified by actually listing its members, for example, {e1, e2, e3}, is the set with members e1, e2, e3. The chapter also describes vectors, matrices, and related notions. The set of all vectors linearly dependent on a collection C of vectors can be shown to form a vector space, which is generated by or spanned by C. Generally, maximal and minimal members of a collection of sets may not be largest and smallest in terms of size.
Publisher Summary This chapter discusses a method of network decomposition that is a topological generalization of hybrid analysis (analysis where unknowns involve both voltages and currents). The result states that solving a network is equivalent to solving two derived subnetworks matching certain current and voltage boundary conditions. To state the result precisely, a formal description of a general electrical network is presented. Usually an electrical network has a device characteristic in which the constraints on sets of branches can be specified independently of each other. The chapter discusses three results; two of these viz. “v-shift” and “i-shift” are well known basic results. Issues concerned with optimal application of network equations are discussed. The application of the method to the case of linear electrical networks is described.
This chapter discusses the Dilworth truncation (truncation for short) operation on submodular functions and the related notion of principal lattice of partitions. The theory of Dilworth truncation bears a strong resemblance to that of convolution. The chapter presents the properties of the truncation operation and presents a number of examples, including Dilworth's own, relevant to the operation. The principal lattice of partitions (PLP) of a submodular function is discussed and the similarity in its properties with those of the principal partition (PP) and the characteristic properties are highlighted. The chapter also describes a technique for building approximation algorithms for optimum partitioning problems (including vertex partitioning for a graph, minimizing number of cut edges). It also discusses the relation of the PLP of a given submodular function to the PLP of naturally derived functions.
Matroids are important combinatorial structures both from the point of view of theory and from that of applications. One of the subjects to which applications were found early was electrical network theory. This chapter explains the theory with electrical networks. A matroid can be defined in several equivalent ways. Each of these is based on an axiom system. The primitive objects of each axiom system can be identified with either the primitive or some derived objects of every other axiom system. The words “maximal” and “minimal” are used in the statements of the axioms. Independent sets of a matroid correspond to subtrees (or, dually, subcotrees) of graphs and to independent sets of columns of matrices. The chapter generalizes the notion of minors of graphs and vector spaces to matroids.
