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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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