Dijkstra's algorithm finds the shortest paths between vertices in a graph with non-negative edge weights. It works by maintaining distances from the source vertex to all other vertices, initially setting all distances to infinity except the source which is 0. It then iteratively selects the unvisited vertex with the lowest distance, marks it as visited, and updates the distances to its neighbors if a shorter path is found through the selected vertex. This continues until all vertices are visited, at which point the distances will be the shortest paths from the source vertex.
Dijkstra's algorithm was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. He received the Turing award in 1972. The Dijkstra prize is named after him which is given for outstanding papers on the principles of distributed computing. One of his famous quote is that Computer Science is no more about computers than astronomy is about telescopes.
Dijkstra's algorithm solves the shortest-path problem for any weighted graph with non-negative weights and finds shortest distance between 2 vertices. The algorithm creates the tree of the shortest paths from the starting source vertex from all other points in the graph. Works on both directed and undirected graph. It differs from the minimum spanning tree as the shortest distance between two vertices may not be included in all the vertices of the graph.
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.
Dijkstra's algorithm was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. He received the Turing award in 1972. The Dijkstra prize is named after him which is given for outstanding papers on the principles of distributed computing. One of his famous quote is that Computer Science is no more about computers than astronomy is about telescopes.
Dijkstra's algorithm solves the shortest-path problem for any weighted graph with non-negative weights and finds shortest distance between 2 vertices. The algorithm creates the tree of the shortest paths from the starting source vertex from all other points in the graph. Works on both directed and undirected graph. It differs from the minimum spanning tree as the shortest distance between two vertices may not be included in all the vertices of the graph.
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.
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.
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.
In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks.
breadth-first search (BFS) is a strategy for searching in a graph when search is limited to essentially two operations
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
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.
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.
In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks.
breadth-first search (BFS) is a strategy for searching in a graph when search is limited to essentially two operations
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
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.
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An Activity Selection Problem
The activity selection problem is a mathematical optimization problem. Our first illustration is the problem of scheduling a resource among several challenge activities. We find a greedy algorithm provides a well designed and simple method for selecting a maximum- size set of manually compatible activities.
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage.
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When we have to display a large portion of the picture, then not only scaling & translation is necessary, the visible part of picture is also identified. This process is not easy. Certain parts of the image are inside, while others are partially inside. The lines or elements which are partially visible will be omitted.
For deciding the visible and invisible portion, a particular process called clipping is used. Clipping determines each element into the visible and invisible portion. Visible portion is selected. An invisible portion is discarded.
Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California. It is the world's largest semiconductor chip manufacturer by revenue, and is the developer of the x86 series of microprocessors, the processors found in most personal computers
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A motherboard is the main printed circuit board in general-purpose computers and other expandable systems. It holds and allows communication between many of the crucial electronic components of a system, such as the central processing unit and memory, and provides connectors for other peripherals.
In graph theory, the shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized
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.
Python is an interpreted high-level general-purpose programming language. Its design philosophy emphasizes code readability with its use of significant indentation. Its language constructs as well as its object-oriented approach aim to help programmers write clear, logical code for small and large-scale projects
import in python is similar to #include header_file in C/C++. Python modules can get access to code from another module by importing the file/function using import. The import statement is the most common way of invoking the import machinery, but it is not the only way. import module_name .When the import is used, it searches for the module initially in the local scope by calling __import__() function. The value returned by the function is then reflected in the output of the initial code.
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:
An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems.
A grammar is said to be regular, if the production is in the form -
A → αB,
A -> a,
A → ε,
for A, B ∈ N, a ∈ Σ, and ε the empty string
A regular grammar is a 4 tuple -
G = (V, Σ, P, S)
V - It is non-empty, finite set of non-terminal symbols,
Σ - finite set of terminal symbols, (Σ ∈ V),
P - a finite set of productions or rules,
S - start symbol, S ∈ (V - Σ)
Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, removing those solutions that fail to satisfy the constraints of the problem at any point of time
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
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2. Dijkstra's algorithm
• Dijkstra's algorithm: finds shortest (minimum weight) path between a
particular pair of vertices in a weighted directed graph with nonnegative
edge weights
– solves the "one vertex, shortest path" problem
– basic algorithm concept: create a table of information about the
currently known best way to reach each vertex (distance, previous
vertex) and improve it until it reaches the best solution
• in a graph where:
– vertices represent cities,
– edge weights represent driving distances between pairs of cities
connected by a direct road,
Dijkstra's algorithm can be used to find the shortest route between
one city and any other
2
3. Dijkstra pseudocode
Dijkstra(v1, v2):
for each vertex v: // Initialization
v's distance := infinity.
v's previous := none.
v1's distance := 0.
List := {all vertices}.
while List is not empty:
v := remove List vertex with minimum distance.
mark v as known.
for each unknown neighbor n of v:
dist := v's distance + edge (v, n)'s weight.
if dist is smaller than n's distance:
n's distance := dist.
n's previous := v.
reconstruct path from v2 back to v1,
following previous pointers.
3
4. 4
Example: Initialization
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0
Pick vertex in List with minimum distance.
Distance(source) =
0
Distance (all vertices
but source) =
6. 6
Example: Remove vertex with
minimum distance
Pick vertex in List with minimum distance, i.e., D
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
1
7. 7
Example: Update neighbors
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
3 3
1
9 5
Distance(C) = 1 + 2 = 3
Distance(E) = 1 + 2 = 3
Distance(F) = 1 + 8 = 9
Distance(G) = 1 + 4 = 5
8. 8
Example: Continued...
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
3 3
1
Pick vertex in List with minimum distance (B) and update neighbors
9 5
Note : distance(D) not
updated since D is
already known and
distance(E) not updated
since it is larger than
previously computed
10. 10
Example: Continued...
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
3 3
1
8 5
Pick vertex List with minimum distance (C) and update neighbors
Distance(F) = 3 + 5 = 8
11. 11
Example: Continued...
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
3 3
1
6 5
Distance(F) = min (8, 5+1) = 6
Previous distance
Pick vertex List with minimum distance (G) and update neighbors
12. 12
Example (end)
A
G
F
B
E
C D
4 1
2
10
3
6
4
2
2
8
5
1
0 2
3 3
1
Pick vertex not in S with lowest cost (F) and update neighbors
6 5
13. 13
Correctness
• Dijkstra’s algorithm is a greedy algorithm
– make choices that currently seem the best
– locally optimal does not always mean globally optimal
• Correct because maintains following two properties:
– for every known vertex, recorded distance is shortest
distance to that vertex from source vertex
– for every unknown vertex v, its recorded distance is shortest
path distance to v from source vertex, considering only
currently known vertices and v
14. 14
THE KNOWN
CLOUD
v
Next shortest path from
inside the known cloud
v'
“Cloudy” Proof: The Idea
• If the path to v is the next shortest path, the path to v' must be at
least as long. Therefore, any path through v' to v cannot be shorter!
Source
Least cost node
competitor
15. Dijkstra pseudocode
Dijkstra(v1, v2):
for each vertex v: // Initialization
v's distance := infinity.
v's previous := none.
v1's distance := 0.
List := {all vertices}.
while List is not empty:
v := remove List vertex with minimum distance.
mark v as known.
for each unknown neighbor n of v:
dist := v's distance + edge (v, n)'s weight.
if dist is smaller than n's distance:
n's distance := dist.
n's previous := v.
reconstruct path from v2 back to v1,
following previous pointers.
15
16. 16
Time Complexity: Using List
The simplest implementation of the Dijkstra's algorithm
stores vertices in an ordinary linked list or array
– Good for dense graphs (many edges)
• |V| vertices and |E| edges
• Initialization O(|V|)
• While loop O(|V|)
– Find and remove min distance vertices O(|V|)
– Potentially |E| updates
• Update costs O(1)
• Reconstruct path O(|E|)
Total time O(|V2| + |E|) = O(|V2| )
17. 17
Time Complexity: Priority Queue
For sparse graphs, (i.e. graphs with much less than |V2| edges)
Dijkstra's implemented more efficiently by priority queue
• Initialization O(|V|) using O(|V|) buildHeap
• While loop O(|V|)
– Find and remove min distance vertices O(log |V|) using O(log |V|)
deleteMin
– Potentially |E| updates
• Update costs O(log |V|) using decreaseKey
• Reconstruct path O(|E|)
Total time O(|V|log|V| + |E|log|V|) = O(|E|log|V|)
• |V| = O(|E|) assuming a connected graph
19. Minimum spanning tree
• tree: a connected, directed acyclic graph
• spanning tree: a subgraph of a graph, which meets
the constraints to be a tree (connected, acyclic) and
connects every vertex of the original graph
• minimum spanning tree: a spanning tree with
weight less than or equal to any other spanning tree
for the given graph
19
20. Min. span. tree applications
• Consider a cable TV company laying cable to a new
neighborhood...
– If it is constrained to bury the cable only along certain paths, then
there would be a graph representing which points are connected by
those paths.
– Some of those paths might be more expensive, because they are
longer, or require the cable to be buried deeper.
• These paths would be represented by edges with larger weights.
– A spanning tree for that graph would be a subset of those paths that
has no cycles but still connects to every house.
• There might be several spanning trees possible. A minimum spanning tree
would be one with the lowest total cost.
20