A kinetic minimum spanning tree is a kinetic data structure that maintains the minimum spanning tree (MST) of a graph whose edge weights are changing as a continuous function of time.
The most efficient known data structure for the general case uses a kinetic sorted list to store the edge weights, and a standard MST algorithm to compute the MST given the sorted edge weights. This data structure must process events, developing a more efficient data structure remains an open problem. [1]
Agarwal et al. developed a data structure that maintains the MST for a graph belonging to a minor closed family. It uses the idea of a "swap", calculating the amount by which the weight of the MST would increase if some edge in the tree e was replaced by an edge f outside the tree such that the circle induced by f in the tree contains e. Maintaining the tree is then equivalent to finding and swapping the next pair for which this quantity becomes negative. This data structure considers the dual view of the graph, and then divides based on Frederickson's restricted partitions [2] to make this efficient. It results in a total run time if insertions or deletions are made, or if only weight changes are allowed. These deterministic bounds are slightly improved if randomization is allowed.
In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a max heap, for any given node C, if P is a parent node of C, then the key of P is greater than or equal to the key of C. In a min heap, the key of P is less than or equal to the key of C. The node at the "top" of the heap is called the root node.
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.
Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest.
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.
In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets, and finding a representative member of a set. The last operation allows to find out efficiently if any two elements are in the same or different sets.
The Euclidean minimum spanning tree or EMST is a minimum spanning tree of a set of points in the plane or higher-dimensional Euclidean space. It connects the points by a system of line segments, so that any two points can reach each other along a path through the line segments, and it selects line segments that minimize the sum of the Euclidean distances between directly-connected pairs of points.
The distributed minimum spanning tree (MST) problem involves the construction of a minimum spanning tree by a distributed algorithm, in a network where nodes communicate by message passing. It is radically different from the classical sequential problem, although the most basic approach resembles Borůvka's algorithm. One important application of this problem is to find a tree that can be used for broadcasting. In particular, if the cost for a message to pass through an edge in a graph is significant, a MST can minimize the total cost for a source process to communicate with all the other processes in the network.
In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first published in 1993.
In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points and by an edge whenever there does not exist a third point that is closer to both and than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.
In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles.
Image segmentation strives to partition a digital image into regions of pixels with similar properties, e.g. homogeneity. The higher-level region representation simplifies image analysis tasks such as counting objects or detecting changes, because region attributes can be compared more readily than raw pixels.
A kinetic convex hull data structure is a kinetic data structure that maintains the convex hull of a set of continuously moving points. It should be distinguished from dynamic convex hull data structures, which handle points undergoing discrete changes such as insertions or deletions of points rather than continuous motion.
A kinetic closest pair data structure is a kinetic data structure that maintains the closest pair of points, given a set P of n points that are moving continuously with time in a metric space. While many efficient algorithms were known in the static case, they proved hard to kinetize, so new static algorithms were developed to solve this problem.
A kinetic sorted list is a kinetic data structure for maintaining a list of points under motion in sorted order. It is used as a kinetic predecessor data structure, and as a component in more complex kinetic data structures such as kinetic closest pair.
A kinetic diameter data structure is a kinetic data structure which maintains the diameter of a set of moving points. The diameter of a set of moving points is the maximum distance between any pair of points in the set. In the two dimensional case, the kinetic data structure for kinetic convex hull can be used to construct a kinetic data structure for the diameter of a moving point set that is responsive, compact and efficient.
A kinetic width data structure is a kinetic data structure which maintains the width of a set of moving points. In 2D, the width of a point set is the minimum distance between two parallel lines that contain the point set in the strip between them. For the two dimensional case, the kinetic data structure for kinetic convex hull can be used to construct a kinetic data structure for the width of a point set that is responsive, compact and efficient.
A kinetic Euclidean minimum spanning tree is a kinetic data structure that maintains the Euclidean minimum spanning tree (EMST) of a set P of n points that are moving continuously.
In mathematics, a minimum bottleneck spanning tree (MBST) in an undirected graph is a spanning tree in which the most expensive edge is as cheap as possible. A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree if the graph does not contain a spanning tree with a smaller bottleneck edge weight. For a directed graph, a similar problem is known as Minimum Bottleneck Spanning Arborescence (MBSA).
In computational geometry, a greedy geometric spanner is an undirected graph whose vertices represent points in a Euclidean space, and whose edges are selected by a greedy algorithm to make the shortest path distances in the graph approximate the Euclidean distances between pairs of points. The greedy spanner was first described in the PhD thesis of Gautam Das and conference paper and subsequent journal paper by Ingo Althöfer et al. These sources also credited Marshall Bern (unpublished) with the independent discovery of the same construction.
In graph theory a minimum spanning tree (MST) of a graph with and is a tree subgraph of that contains all of its vertices and is of minimum weight.
Agarwal, Pankaj; Eppstein, David; Guibas, Leonidas J.; Henzinger, Monika R. (1998). Parametric and Kinetic Minimum Spanning Trees (PDF). FOCS. Retrieved May 19, 2012.
