Prim's algorithm finds a minimum spanning tree of a graph using a greedy approach. It treats the nodes as a single tree and adds new nodes from the graph one by one, choosing the connection with the lowest weight at each step. This grows the tree until all nodes are included. Kruskal's algorithm also uses greedy approach but treats the graph as separate trees that merge as it adds edges from lowest to highest weight while ensuring no cycles are formed.
A presentation on prim's and kruskal's algorithmGaurav Kolekar
This slides are for a presentation on Prim's and Kruskal's algorithm. Where I have tried to explain how both the algorithms work, their similarities and their differences.
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.
Greedy with Task Scheduling Algorithm.pptRuchika Sinha
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
A presentation on prim's and kruskal's algorithmGaurav Kolekar
This slides are for a presentation on Prim's and Kruskal's algorithm. Where I have tried to explain how both the algorithms work, their similarities and their differences.
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.
Greedy with Task Scheduling Algorithm.pptRuchika Sinha
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
Mastering Greedy Algorithms: Optimizing Solutions for Efficiency"22bcs058
Greedy algorithms are fundamental techniques used in computer science and optimization problems. They belong to a class of algorithms that make decisions based on the current best option without considering the overall future consequences. Despite their simplicity and intuitive appeal, greedy algorithms can provide efficient solutions to a wide range of problems across various domains.
At the core of greedy algorithms lies a simple principle: at each step, choose the locally optimal solution that seems best at the moment, with the hope that it will lead to a globally optimal solution. This principle makes greedy algorithms easy to understand and implement, as they typically involve iterating through a set of choices and making decisions based on some criteria.
One of the key characteristics of greedy algorithms is their greedy choice property, which states that at each step, the locally optimal choice leads to an optimal solution overall. This property allows greedy algorithms to make decisions without needing to backtrack or reconsider previous choices, resulting in efficient solutions for many problems.
Greedy algorithms are commonly used in problems involving optimization, scheduling, and combinatorial optimization. Examples include finding the minimum spanning tree in a graph (Prim's and Kruskal's algorithms), finding the shortest path in a weighted graph (Dijkstra's algorithm), and scheduling tasks to minimize completion time (interval scheduling).
Despite their effectiveness in many situations, greedy algorithms may not always produce the optimal solution for a given problem. In some cases, a greedy approach can lead to suboptimal solutions that are not globally optimal. This occurs when the greedy choice property does not guarantee an optimal solution at each step, or when there are conflicting objectives that cannot be resolved by a greedy strategy alone.
To mitigate these limitations, it is essential to carefully analyze the problem at hand and determine whether a greedy approach is appropriate. In some cases, greedy algorithms can be augmented with additional techniques or heuristics to improve their performance or guarantee optimality. Alternatively, other algorithmic paradigms such as dynamic programming or divide and conquer may be better suited for certain problems.
Overall, greedy algorithms offer a powerful and versatile tool for solving optimization problems efficiently. By understanding their principles and characteristics, programmers and researchers can leverage greedy algorithms to tackle a wide range of computational challenges and design elegant solutions that balance simplicity and effectiveness.
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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2. Prim's algorithm is a minimum spanning
tree algorithm that takes a graph as input and finds
the subset of the edges of that graph which
•form a tree that includes every vertex
•has the minimum sum of weights among all the
trees that can be formed from the graph
Prim's Algorithm
3. Prim's Spanning Tree
Algorithm
Prim's algorithm to find minimum cost
spanning tree (as Kruskal's algorithm) uses
the greedy approach. Prim's algorithm
shares a similarity with the shortest path
first algorithms.
Prim's algorithm, in contrast with Kruskal's
algorithm, treats the nodes as a single tree
and keeps on adding new nodes to the
spanning tree from the given graph.
6. Remove all loops and parallel edges from the
given graph. In case of parallel edges, keep the
one which has the least cost associated and
remove all others.
7. Step 2 - Choose any arbitrary node as
root node
In this case, we choose S node as the
root node of Prim's spanning tree.
This node is arbitrarily chosen, so any
node can be the root node. One may
wonder why any video can be a root
node. So the answer is, in the
spanning tree all the nodes of a graph
are included and because it is
connected then there must be at least
one edge, which will join it to the rest
of the tree
8. Step 3 - Check outgoing edges and select the one with less cost
After choosing the root node S, we see that S,A and S,C are two edges
with weight 7 and 8, respectively. We choose the edge S,A as it is lesser
than the other.
9. Now, the tree S-7-A is treated as
one node and we check for all
edges going out from it. We select
the one which has the lowest cost
and include it in the tree.
10. After this step, S-7-A-3-C tree is formed. Now we'll again
treat it as a node and will check all the edges again.
However, we will choose only the least cost edge. In this
case, C-3-D is the new edge, which is less than other
edges' cost 8, 6, 4, etc.
11. After adding node D to the spanning tree, we
now have two edges going out of it having the
same cost, i.e. D-2-T and D-2-B. Thus, we can
add either one. But the next step will again yield
edge 2 as the least cost. Hence, we are showing
a spanning tree with both edges included.
12. Kruskal's Spanning Tree Algorithm
Kruskal's algorithm to find the minimum
cost spanning tree uses the greedy
approach. This algorithm treats the
graph as a forest and every node it has
as an individual tree. A tree connects to
another only and only if, it has the least
cost among all available options and
does not violate MST properties.
13. How Kruskal's algorithmworks
It falls under a class of algorithms called greedy
algorithms that find the local optimum in the
hopes of finding a global optimum.
We start from the edges with the lowest weight
and keep adding edges until we reach our goal.
The steps for implementing Kruskal's algorithm
are as follows:
1. Sort all the edges from low weight to high
Take the edge with the lowest weight and add it
to the spanning tree.
2.If adding the edge created a cycle, then reject
this edge.
3.Keep adding edges until we reach all vertices.
15. Step 1 - Remove all loops graph.
and Parallel Edges
Remove all loops and parallel edges
from the given
16. In case of parallel edges, keep
the one which has the least cost
associated and remove all
others.
17. Step 2 - Arrange all edges in their increasing order of
weight
The next step is to create a set of edges and weight, and
arrange them in an ascending order of weightage (cost).
18. Step 3 - Add the edge which has the least
weightage
Now we start adding edges to the graph
beginning from the one which has the least
weight. Throughout, we shall keep checking that
the spanning properties remain intact. In case, by
adding one edge, the spanning tree property does
not hold then we shall consider not to include the
edge in the graph.
19. The least cost is 2 and edges involved are
B,D and D,T. We add them. Adding them
does not violate spanning tree properties, so
we continue to our next edge selection.
20. Next cost is 3, and associated
edges are A,C and C,D. We add
them again −
21. Next cost in the table is 4, and we observe that
adding it will create a circuit in the graph. −
22. the process we shall ignore/avoid all
edges that create We ignore it. In a
circuit.
23. We observe that edges with cost 5
and 6 also create circuits. We ignore
them and move on.
24. Now we are left with only one node to be
added. Between the two least cost edges
available 7 and 8, we shall add the edge
with cost 7.
25. PRIM’S ALGORITHM KRUSKAL’S ALGORITHM
It starts to build the Minimum
Spanning Tree from any vertex
in the graph.
It starts to build the Minimum
Spanning Tree from the vertex
carrying minimum weight in
the graph.
It traverses one node more
than one time to get the
minimum distance.
It traverses one node only
once.
Prim’s algorithm has a time
complexity of O(V2), V being
the number of vertices and can
be improved up to O(E + log V)
using Fibonacci heaps.
Kruskal’s algorithm’s time
complexity is O(E log V), V
being the number of vertices.
Prim’s algorithm gives
connected component as well
as it works only on connected
graph.
Kruskal’s algorithm can
generate forest(disconnected
components) at any instant as
well as it can work on
disconnected components
Prim’s algorithm runs faster in
dense graphs.
Kruskal’s algorithm runs faster
in sparse graphs.