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Bend Minimization
In graph drawing styles that represent the edges of a graph by polylines (sequences of line segments connected at bends), it is desirable to minimize the number of bends per edge (sometimes called the curve complexity). or the total number of bends in a drawing.. Bend minimization is the algorithmic problem of finding a drawing that minimizes these quantities. Eliminating all bends The prototypical example of bend minimization is Fáry's theorem, which states that every planar graph can be drawn with no bends, that is, with all its edges drawn as straight line segments. Drawings of a graph in which the edges are both bendless and axis-aligned are sometimes called ''rectilinear drawings'', and are one way of constructing RAC drawings in which all crossings are at right angles. However, it is NP-complete to determine whether a planar graph has a planar rectilinear drawing, and NP-complete to determine whether an arbitrary graph has a rectilinear drawing that allows crossings.. Bend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's menta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-completeness
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Programming
An integer programming problem is a mathematical optimization or Constraint satisfaction problem, feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are Linear function (calculus), linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \text && \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum-cost Flow Problem
The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. The minimum cost flow problem is one of the most fundamental among all flow and circulation problems because most other such problems can be cast as a minimum cost flow problem and also that it can be solved efficiently using the network simplex algorithm. Definition A flow network is a directed graph G=(V,E) with a source vertex s \in V and a sink vertex t \in V, where each edge (u,v) \in E has capacity c(u,v) > 0, flow f(u,v) and cost a(u,v), with most minimum-cost flow algorithms supporting edges with negative costs. The cost of sending this flow along an edge (u,v) is f(u,v)\cdot a(u,v). The problem requires an amount of flow d to be sent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Drawing
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
RAC Drawing
In graph drawing, a RAC drawing of a graph is a drawing in which the vertices are represented as points, the edges are represented as straight line segments or polylines, at most two edges cross at any point, and when two edges cross they do so at right angles to each other. In the name of this drawing style, "RAC" stands for "right angle crossing". The right-angle crossing style and the name "RAC drawing" for this style were both formulated by ,. motivated by previous user studies showing that crossings with large angles are much less harmful to the readability of drawings than shallow crossings. Even for planar graphs, allowing some right-angle crossings in a drawing of the graph can significantly improve measures of the drawing quality such as its area or angular resolution. Examples The complete graph ''K''5 has a RAC drawing with straight edges, but ''K''6 does not. Every 6-vertex RAC drawing has at most 14 edges, but ''K''6 has 15 edges, too many to have a RAC drawing. A com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (graph Theory)
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fáry's Theorem
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by , , and . Proof One way of proving Fáry's theorem is to use mathematical induction. Let be a simple plane graph with vertices; we may add edges if necessary so that is a maximally plane graph. If < 3, the result is trivial. If ≥ 3, then all faces of must be triangles, as we could add an edge into any face with more sides while preserving planarity, contradicting the assumption of maximal planarity. Choose some three vertices forming a triangular face of . We prove by induction on that there exists a straight-line combinatorially isomorphic re-embedding of in which triangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
