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Frankl, P. Gera, R. This one builds on top of what we learned. Eight time complexities that every programmer should know. Trees Data Structures for Beginners. Self-balanced Binary Search Trees. Appendix I: Analysis of Recursive Algorithms.
A graph is a data structure where a node can have zero or more adjacent elements. The connection between two nodes is called edge. Nodes can also be called vertices. The degree is the number of edges connected to a vertex. If the edges are bi-directional, then we have an undirected graph. But, if the edges have a direction, then we have a directed graph or di-graph for short. You can think of it as a one-way street directed or two-way street undirected.
Vertex can have edges that go to itself e. A graph can have cycles which means that if you traverse through the node, you could get the same node more than once. The graph without cycles is called acyclic graph. Also, acyclic undirected graphs are called tree. We are going to cover trees in depth in the next post. Not all vertices have to be connected in the graph. You might have isolated nodes or even separated subgraphs.
If all nodes have at least one edge, then we have a connected graph. When all nodes are connected to all other nodes, then we have a complete graph.
In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the. Graph Structure Theory cover image. Contemporary Mathematics Volume: ; ; pp; Softcover MSC: Primary 05; Secondary 68; 06;.
For a complete graph, each node should have nodes - 1 edges. In the previous example, we have seven vertices, so each node has six edges. Weighted graphs have many applications depending on the domain where you need to solve a problem. To name a few:.
We just learned the basics of graphs and some applications. We digraph with 4 nodes. When a vertex has a link to itself e. The adjacency matrix is one way of representing a graph using a two-dimensional array NxN matrix. In the intersection of nodes, we add 1 or other weight if they are connected and 0 or - if they are not connected. As you can see, the matrix list all nodes horizontally and vertically.
If there a few connections we called sparse graph if there are many connections close to the max number of links we called it dense graph. If all possible connections are reached, then we have a complete graph. Querying if two nodes are connected in an adjacency matrix takes a constant time or O 1.
Storing a graph as an adjacency matrix has a space complexity of O n 2 , where n is the number of vertices. Also, represented as O V 2. The vertices are stored as a V x V matrix. Since the matrix has a VxV matrix, to get all the adjacent nodes to a given vertex, we would have to go to the node row and get all its edges with the other nodes. This method is particularly useful to reveal hierarchical structures in a planar network.
The organization of nodes and links in a graph conveys a structure that can be described and labeled.
The basic structural properties of a graph are:. Symmetry and Asymmetry. A graph is symmetrical if each pair of nodes linked in one direction is also linked in the other. By convention, a line without an arrow represents a link where it is possible to move in both directions. However, both directions have to be defined in the graph. Most transport systems are symmetrical but asymmetry can often occur as it is the case for maritime pendulum and air services. Asymmetry is rare on road transportation networks, unless one-way streets are considered. Assortativity and disassortativity. Assortative networks are those characterized by relations among similar nodes, while disassortative networks are found when structurally different nodes are often connected.
Transport or technological networks are often disassortative when they are non-planar, due to the higher probability for the network to be centralized into a few large hubs. A graph is complete if two nodes are linked in at least one direction. A complete graph has no sub-graph and all its nodes are interconnected.
A complete graph is described as connected if for all its distinct pairs of nodes there is a linking chain. Direction does not have importance for a graph to be connected, but may be a factor for the level of connectivity. There are various levels of connectivity, depending on the degree at which each pair of nodes is connected. Two sub graphs are complementary if their union results in a complete graph. Multimodal transportation networks are complementary as each sub-graph modal network benefits from the connectivity of other sub-graphs.
A node r where every other node is the extremity of a path coming from r is a root.
Direction has an importance. A root is generally the starting point of a distribution system, such as a factory or a warehouse. A connected graph without a cycle is a tree. A tree has the same number of links than nodes plus one. If a link is removed, the graph ceases to be connected. If a new link between two nodes is provided, a cycle is created. A branch of root r is a tree where no links are connecting any node more than once. River basins are typical examples of tree-like networks based on multiple sources connecting towards a single estuary. This structure strongly influences river transport systems.
Articulation Node. In a connected graph, a node is an articulation node if the sub-graph obtained by removing this node is no longer connected. An articulation node is generally a port or an airport, or an important hub of a transportation network, which serves as a bottleneck. It is also called a bridge node. In a connected graph, an isthmus is a link that is creating, when removed, two sub-graphs having at least one connection.
Most central links in a complex network are often isthmuses, which removal by reiteration helps revealing dense communities clusters. Authors: Dr. Jean-Paul Rodrigue and Dr. Cesar Ducruet Graph theory is a branch of mathematics concerned about how networks can be encoded and their properties measured.