a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree. This algorithm is often used in routing and as a subroutine in other graph algorithms.
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.
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.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
This presentation gives a conceptual idea of a graphical algorithm that is Dijkstra's Algorithm. Includes general introduction , the pseudocode, code in form of QR Card, graphical. Also discussing the algorithm in form of graphical images and nodes. Also it include the complexity and application of algorithm in various ranges of fields. This is a fun, eye-catching, conceptual presentation, best suited for students into engineering.
It is related to Analysis and Design Of Algorithms Subject.Basically it describe basic of topological sorting, it's algorithm and step by step process to solve the example of topological sort.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
This presentation gives a conceptual idea of a graphical algorithm that is Dijkstra's Algorithm. Includes general introduction , the pseudocode, code in form of QR Card, graphical. Also discussing the algorithm in form of graphical images and nodes. Also it include the complexity and application of algorithm in various ranges of fields. This is a fun, eye-catching, conceptual presentation, best suited for students into engineering.
It is related to Analysis and Design Of Algorithms Subject.Basically it describe basic of topological sorting, it's algorithm and step by step process to solve the example of topological sort.
Huffman coding is a coding technique for lossless compression of data base based upon the frequency of occurance of a symbol in that file.
In huffman coding every Data is based upon 0’s and 1’s which reduces the size of file.
Using binary representation, the number of bits required to represent each character depends upon the number of characters that have to be represented. Using one bit we can represent two characters, i.e., 0 represents the first character and 1 represents the second character.
a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees.
This is the most important concept in advance java. Why java is so much popular than other? answer is its implicit objects. It provides many implicit object in the library. So you don't need to declare object to use it. You just have to use whenever you need it.
This Presentation contains all the topics in design concept of software engineering. This is much more helpful in designing new product. You have to consider some of the design concepts that are given in the ppt
A Complete Presentation about how to Find a LCS of a given Strings Using LCS algorithm. This PPT contains easiest way to solve LCS problem With example.
The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers.
Graphs: Finding shortest paths
Topics:
Definitions
Floyd-Warshall algorithm
Dijkstra algorithm
Bellman-Ford algorithm
Teaching material for the course of "Tecniche di Programmazione" at Politecnico di Torino in year 2012/2013. More information: http://bit.ly/tecn-progr
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. C#ODE STUDIO
Algorithms L17.2
Paths in graphs
Consider a digraph G = (V, E) with edge-weight
function w : E R. The weight of path p = v1
v2 L vk is defined to be
1
1
1),()(
k
i
ii vvwpw .
3. C#ODE STUDIO
Algorithms L17.3
Paths in graphs
Consider a digraph G = (V, E) with edge-weight
function w : E R. The weight of path p = v1
v2 L vk is defined to be
1
1
1),()(
k
i
ii vvwpw .
v
1 v
2
v
3 v
4
v
5
4 –2 –5 1
Example:
w(p) = –2
4. C#ODE STUDIO
Algorithms L17.4
Shortest paths
A shortest path from u to v is a path of
minimum weight from u to v. The shortest-
path weight from u to v is defined as
(u, v) = min{w(p) : p is a path from u to v}.
Note: (u, v) = if no path from u to v exists.
12. C#ODE STUDIO
Algorithms L17.12
Single-source shortest paths
Problem. From a given source vertex s V, find
the shortest-path weights (s, v) for all v V.
If all edge weights w(u, v) are nonnegative, all
shortest-path weights must exist.
IDEA: Greedy.
1. Maintain a set S of vertices whose shortest-
path distances from s are known.
2. At each step add to S the vertex v V – S
whose distance estimate from s is minimal.
3. Update the distance estimates of vertices
adjacent to v.
14. C#ODE STUDIO
Algorithms L17.14
Dijkstra’s algorithm
d[s] 0
for each v V – {s}
do d[v]
S
Q V ⊳ Q is a priority queue maintaining V – S
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
15. C#ODE STUDIO
Algorithms L17.15
Dijkstra’s algorithm
d[s] 0
for each v V – {s}
do d[v]
S
Q V ⊳ Q is a priority queue maintaining V – S
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
relaxation
step
Implicit DECREASE-KEY
21. C#ODE STUDIO
Algorithms L17.21
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C }
0
7
3 5
11
10
7 11 5
Relax all edges leaving C:
22. C#ODE STUDIO
Algorithms L17.22
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C, E }
0
7
3 5
11
10
7 11 5
“E” EXTRACT-MIN(Q):
23. C#ODE STUDIO
Algorithms L17.23
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C, E }
0
7
3 5
11
10
7 11 5
7 11
Relax all edges leaving E:
24. C#ODE STUDIO
Algorithms L17.24
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C, E, B }
0
7
3 5
11
10
7 11 5
7 11
“B” EXTRACT-MIN(Q):
25. C#ODE STUDIO
Algorithms L17.25
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C, E, B }
0
7
3 5
9
10
7 11 5
7 11
Relax all edges leaving B:
9
26. C#ODE STUDIO
Algorithms L17.26
Example of Dijkstra’s
algorithm
A
B D
C E
10
3
1 4 7 9
8
2
2A B C D EQ:
0
S: { A, C, E, B, D }
0
7
3 5
9
10
7 11 5
7 11
9
“D” EXTRACT-MIN(Q):
27. C#ODE STUDIO
Algorithms L17.27
Correctness — Part I
Lemma. Initializing d[s] 0 and d[v] for all
v V – {s} establishes d[v] (s, v) for all v V,
and this invariant is maintained over any sequence
of relaxation steps.
28. C#ODE STUDIO
Algorithms L17.28
Correctness — Part I
Lemma. Initializing d[s] 0 and d[v] for all
v V – {s} establishes d[v] (s, v) for all v V,
and this invariant is maintained over any sequence
of relaxation steps.
Proof. Suppose not. Let v be the first vertex for
which d[v] < (s, v), and let u be the vertex that
caused d[v] to change: d[v] = d[u] + w(u, v). Then,
d[v] < (s, v) supposition
(s, u) + (u, v) triangle inequality
(s,u) + w(u, v) sh. path specific path
d[u] + w(u, v) v is first violation
Contradiction.
29. C#ODE STUDIO
Algorithms L17.29
Correctness — Part II
Lemma. Let u be v’s predecessor on a shortest
path from s to v. Then, if d[u] = (s, u) and edge
(u, v) is relaxed, we have d[v] (s, v) after the
relaxation.
30. C#ODE STUDIO
Algorithms L17.30
Correctness — Part II
Lemma. Let u be v’s predecessor on a shortest
path from s to v. Then, if d[u] = (s, u) and edge
(u, v) is relaxed, we have d[v] (s, v) after the
relaxation.
Proof. Observe that (s, v) = (s, u) + w(u, v).
Suppose that d[v] > (s, v) before the relaxation.
(Otherwise, we’re done.) Then, the test d[v] >
d[u] + w(u, v) succeeds, because d[v] > (s, v) =
(s, u) + w(u, v) = d[u] + w(u, v), and the
algorithm sets d[v] = d[u] + w(u, v) = (s, v).
32. C#ODE STUDIO
Algorithms L17.32
Correctness — Part III
Theorem. Dijkstra’s algorithm terminates with
d[v] = (s, v) for all v V.
Proof. It suffices to show that d[v] = (s, v) for every v
V when v is added to S. Suppose u is the first vertex
added to S for which d[u] (s, u). Let y be the first
vertex in V – S along a shortest path from s to u, and
let x be its predecessor:
s
x y
u
S, just before
adding u.
33. C#ODE STUDIO
Algorithms L17.33
Correctness — Part III
(continued)
Since u is the first vertex violating the claimed
invariant, we have d[x] = (s, x). When x was
added to S, the edge (x, y) was relaxed, which
implies that d[y] = (s, y) (s, u) d[u]. But,
d[u] d[y] by our choice of u. Contradiction.
s
x y
uS
34. C#ODE STUDIO
Algorithms L17.34
Analysis of Dijkstra
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
35. C#ODE STUDIO
Algorithms L17.35
Analysis of Dijkstra
|V|
times
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
36. C#ODE STUDIO
Algorithms L17.36
Analysis of Dijkstra
degree(u)
times
|V|
times
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
37. C#ODE STUDIO
Algorithms L17.37
Analysis of Dijkstra
degree(u)
times
|V|
times
Handshaking Lemma (E) implicit DECREASE-KEY’s.
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
38. C#ODE STUDIO
Algorithms L17.38
Analysis of Dijkstra
degree(u)
times
|V|
times
Handshaking Lemma (E) implicit DECREASE-KEY’s.
Time = (V·TEXTRACT-MIN + E·TDECREASE-KEY)
Note: Same formula as in the analysis of Prim’s
minimum spanning tree algorithm.
while Q
do u EXTRACT-MIN(Q)
S S {u}
for each v Adj[u]
do if d[v] > d[u] + w(u, v)
then d[v] d[u] + w(u, v)
44. C#ODE STUDIO
Algorithms L17.44
Unweighted graphs
• Use a simple FIFO queue instead of a priority
queue.
Suppose that w(u, v) = 1 for all (u, v) E.
Can Dijkstra’s algorithm be improved?
45. C#ODE STUDIO
Algorithms L17.45
Unweighted graphs
while Q
do u DEQUEUE(Q)
for each v Adj[u]
do if d[v] =
then d[v] d[u] + 1
ENQUEUE(Q, v)
• Use a simple FIFO queue instead of a priority
queue.
Breadth-first search
Suppose that w(u, v) = 1 for all (u, v) E.
Can Dijkstra’s algorithm be improved?
46. C#ODE STUDIO
Algorithms L17.46
Unweighted graphs
while Q
do u DEQUEUE(Q)
for each v Adj[u]
do if d[v] =
then d[v] d[u] + 1
ENQUEUE(Q, v)
• Use a simple FIFO queue instead of a priority
queue.
Analysis: Time = O(V + E).
Breadth-first search
Suppose that w(u, v) = 1 for all (u, v) E.
Can Dijkstra’s algorithm be improved?
59. C#ODE STUDIO
Algorithms L17.59
Correctness of BFS
Key idea:
The FIFO Q in breadth-first search mimics
the priority queue Q in Dijkstra.
• Invariant: v comes after u in Q implies that
d[v] = d[u] or d[v] = d[u] + 1.
while Q
do u DEQUEUE(Q)
for each v Adj[u]
do if d[v] =
then d[v] d[u] + 1
ENQUEUE(Q, v)