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return 0;

}

7.3.5. johnson’s algorithms

(From Wikipedia, the free encyclopedia)

Johnson's algorithm is a way to solve the all-pairs shortest path problem in a sparse, weighted, directed graph.

First, it adds a new node with zero weight edge from it to all other nodes, and runs the Bellman-Ford algorithm to check for negative weight cycles and find h(v), the least weight of a path from the new node to node v. Next it reweights the edges using the nodes' h(v) values. Finally for each node, it runs Dijkstra's algorithm and stores the computed least weight to other nodes, reweighted using the nodes' h(v) values, as the final weight. The time complexity is O(V2log V + VE).

7.4. union-find problem

(From Wikipedia, the free encyclopedia)

Given a set of elements, it is often useful to break them up or partition them into a number of separate, non-overlapping sets. A disjoint-set data structure is a data structure that keeps track of such a partitioning. A union-find algorithm is an algorithm that performs two useful operations on such a data structure:

  • Find: Determine which set a particular element is in. Also useful for determining if two elements are in the same set.
  • Union: Combine or merge two sets into a single set.

Because it supports these two operations, a disjoint-set data structure is sometimes called a merge-find set. The other important operation, MakeSet, which makes a set containing only a given element (a singleton), is generally trivial. With these three operations, many practical partitioning problems can be solved (see the Applications section).

In order to define these operations more precisely, we need some way of representing the sets. One common approach is to select a fixed element of each set, called its representative, to represent the set as a whole. Then, Find(x) returns the representative of the set that x belongs to, and Union takes two set representatives as its arguments.

Disjoint-set linked lists

Perhaps the simplest approach to creating a disjoint-set data structure is to create a linked list for each set. We choose the element at the head of the list as the representative.

MakeSet is obvious, creating a list of one element. Union simply appends the two lists, a constant-time operation. Unfortunately, with this implementation Find requires Ω(n) or linear time with this approach.

We can avoid this by including in each linked list node a pointer to the head of the list; then Find takes constant time. However, we've now ruined the time of Union, which has to go through the elements of the list being appended to make them point to the head of the new combined list, requiring Ω(n) time.

We can ameliorate this by always appending the smaller list to the longer, called the weighted union heuristic. This also requires keeping track of the length of each list as we perform operations to be efficient. Using this, a sequence of m MakeSet, Union, and Find operations on n elements requires O(m + nlog n) time. To make any further progress, we need to start over with a different data structure.

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
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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
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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
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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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fine, how about you?
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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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