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Tracking the connected components of an undirected graph

Suppose we have an undirected graph and we want to efficiently make queries regarding the connected components of that graph, such as:

  • Are two vertices of the graph in the same connected component?
  • List all vertices of the graph in a particular component.
  • How many connected components are there?

If the graph is static (not changing), we can simply use breadth-first search to associate a component with each vertex. However, if we want to keep track of these components while adding additional vertices and edges to the graph, a disjoint-set data structure is much more efficient.

We assume the graph is empty initially. Each time we add a vertex, we use MakeSet to make a set containing only that vertex. Each time we add an edge, we use Union to union the sets of the two vertices incident to that edge. Now, each set will contain the vertices of a single connected component, and we can use Find to determine which connected component a particular vertex is in, or whether two vertices are in the same connected component.

This technique is used by the Boost Graph Library to implement its Incremental Connected Components functionality.

Note that this scheme doesn't allow deletion of edges — even without path compression or the rank heuristic, this is not as easy, although more complex schemes have been designed that can deal with this type of incremental update.

Computing shorelines of a terrain

When computing the contours of a 3D surface, one of the first steps is to compute the "shorelines," which surround local minima or "lake bottoms." We imagine we are sweeping a plane, which we refer to as the "water level," from below the surface upwards. We will form a series of contour lines as we move upwards, categorized by which local minima they contain. In the end, we will have a single contour containing all local minima.

Whenever the water level rises just above a new local minimum, it creates a small "lake," a new contour line that surrounds the local minimum; this is done with the MakeSet operation.

As the water level continues to rise, it may touch a saddle point, or "pass." When we reach such a pass, we follow the steepest downhill route from it on each side until we arrive a local minimum. We use Find to determine which contours surround these two local minima, then use Union to combine them. Eventually, all contours will be combined into one, and we are done.

Classifying a set of atoms into molecules or fragments

In computational chemistry, collisions involving the fragmentation of large molecules can be simulated using molecular dynamics. The result is a list of atoms and their positions. In the analysis, the union-find algorithm can be used to classify these atoms into fragments. Each atom is initially considered to be part of its own fragment. The Find step usually consists of testing the distance between pairs of atoms, though other criterion like the electronic charge between the atoms could be used. The Union merges two fragments together. In the end, the sizes and characteristics of each fragment can be analyzed.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Source:  OpenStax, Data structures and algorithms. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10765/1.1
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