The document discusses algorithms for finding shortest paths and minimal spanning trees in graphs. It describes Dijkstra's algorithm for finding the shortest path between two nodes in a weighted graph. The algorithm works by gradually building up a set of nodes whose shortest paths from the source node have been determined. It also describes Prim's algorithm for finding a minimal spanning tree, which proceeds similarly to Dijkstra's algorithm but finds the minimum weighted connecting tree rather than shortest path between two nodes. Both algorithms run in O(n2) time in the worst case.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
Kriging and spatial design accelerated by orders of magnitudeAlexander Litvinenko
We suggest new method, which combine a low-rank tensor technique and Fast Fourier Transformation to speed up kriging, estimation of the variance and geostatistical optimal design. We do 3D Kriging with O(10e+13) estimation points, A-criterion for optimal design with O(10e+12) points in 30 sec.
Introduction to complexity theory that solves your assignment problem it contains about complexity class,deterministic class,big- O notation ,proof by mathematical induction, L-Space ,N-Space and characteristics functions of set and so on
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Download the latest issue of the mobileSQUARED newsletter. Includes
Conversational advertising
Google targets smartphones
The Weather Channel going mobile
UK smartphone usage
and more
Kriging and spatial design accelerated by orders of magnitudeAlexander Litvinenko
We suggest new method, which combine a low-rank tensor technique and Fast Fourier Transformation to speed up kriging, estimation of the variance and geostatistical optimal design. We do 3D Kriging with O(10e+13) estimation points, A-criterion for optimal design with O(10e+12) points in 30 sec.
Introduction to complexity theory that solves your assignment problem it contains about complexity class,deterministic class,big- O notation ,proof by mathematical induction, L-Space ,N-Space and characteristics functions of set and so on
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Download the latest issue of the mobileSQUARED newsletter. Includes
Conversational advertising
Google targets smartphones
The Weather Channel going mobile
UK smartphone usage
and more
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Internet of Things (IoT) two-factor authentication using blockchainDavid Wood
Presented at the Ethereum Engineering Group Meetup in Brisbane, Australia, on 13 Nov 2019. We report on research to use an Ethereum blockchain as an MFA and/or MPA device to secure command channels on IoT networks, even when the underlying network may be compromised.
Methods for Securing Spacecraft Tasking and Control via an Enterprise Ethereu...David Wood
Presentation at ICSSC 2019 (see http://www.kaconf.org) associated with the following academic paper:
David Hyland-Wood, Peter Robinson, Roberto Saltini, Sandra Johnson, Christopher Hare. Method for Securing Spacecraft Tasking and Control via an Enterprise Ethereum Blockchain. Proc. 37th International Communications Satellite Systems Conference (ICSSC), 29 October - 1 November 2019.
Implementing the Verifiable Claims data modelDavid Wood
The W3C Verifiable Claims data model arguably requires a decentralised, distributed database controllable by three types of parties; issuers, inspectors, and holders. This presentation explores the benefits of implementing the Verifiable Claims data model using the RDF and Linked Data technology stack.
Metaphors define civilized life. They are all around us in the stories that we teach our children and tell each other to justify our actions. But social metaphors have a dark side. They can cause entire civilizations to self destruct. Metaphors can kill. This presentation explores the power, and danger, of metaphors as social memes.
These slides are from a talk given to the Fredericksburg Secular Humanists (FSH) in Fredericksburg, Virginia, on 8 November 2015. FSH is sub-chapter of the United Coalition of Reason (unitedcor.org). The talk compared the secular societies of the United States and Australia.
Building a writer's platform with social mediaDavid Wood
This presentation reports on my progress in trying to build my writer's platform using social media. It focuses on Twitter, but the advice is generally applicable. Kudos to my mentors @DanCitizen and @RayneHall.
A summary of the Hero's Journey, Joseph Campbell's formulation of the "monomyth" in mythology and literature. Originally presented to the Fredericksburg Writing as a Business Meetup, 24 January 2015.
Open Data is the idea that "certain data should be freely available to everyone to use and republish as they wish, without restrictions from copyright, patents or other mechanisms of control”. Open Data follows similar “open” concepts that have proven to be valuable in the information economy such as Open Standards, Open Source Software, Open Content and has been followed more recently by variations on the theme such as Open Science and Open Government.
Open Data allows information of common value to be reused without needing to be recreated. The economic benefits of Open Data include cost reduction, organizational efficiencies and the facilitation of commonly held understanding. The costs of implementing Open Data deployment strategies tend to be iterative on top of existing information infrastructure.
This presentation will describe Open Data and its place in the ecosystem of economic and governmental discourse.
2. Shortest Path Problem
● Assume that we have a simple, weighted, connected
graph, where the weights are positive. Then a path
exists between any two nodes x and y.
● How do we find a path with minimum weight?
● For example, cities connected by roads, with the
weight being the distance between them.
● The shortest-path algorithm is known as Dijkstra’s
algorithm.
Section 6.3 Shortest Path and Minimal Spanning Tree 1
3. Shortest-Path Algorithm
● To compute the shortest path from x to y using
Dijkstra’s algorithm, we build a set (called IN ) that
initially contains only x but grows as the algorithm
proceeds.
● IN contains every node whose shortest path from x,
using only nodes in IN, has so far been determined.
● For every node z outside IN, we keep track of the
shortest distance d[z] from x to that node, using a path
whose only non-IN node is z.
● We also keep track of the node adjacent to z on this
path, s[z].
Section 6.3 Shortest Path and Minimal Spanning Tree 2
4. Shortest-Path Algorithm
● Pick the non-IN node p with the smallest distance d.
● Add p to IN, then recompute d for all the remaining
non-IN nodes, because there may be a shorter path
from x going through p than there was before p
belonged to IN.
● If there is a shorter path, update s[z] so that p is now
shown to be the node adjacent to z on the current
shortest path.
● As soon as y is moved into IN, IN stops growing. The
current value of d[y] is the distance for the shortest
path, and its nodes are found by looking at y, s[y], s[s
[y]], and so forth, until x is reached.
Section 6.3 Shortest Path and Minimal Spanning Tree 3
5. Shortest-Path Algorithm
● ALGORITHM ShortestPath
ShortestPath (n × n matrix A; nodes x, y)
//Dijkstra’s algorithm. A is a modified adjacency matrix for a
//simple, connected graph with positive weights; x and y are
//nodes in the graph; writes out nodes in the shortest path from x
// to y, and the distance for that path
Local variables:
set of nodes IN //set of nodes whose shortest path from x
//is known
nodes z, p //temporary nodes
array of integers d //for each node, the distance from x using
//nodes in IN
array of nodes s //for each node, the previous node in the
//shortest path
integer OldDistance //distance to compare against
//initialize set IN and arrays d and s
IN = {x}
d[x] = 0
Section 6.3 Shortest Path and Minimal Spanning Tree 4
6. Shortest-Path Algorithm
for all nodes z not in IN do
d[z] = A[x, z]
s [z] = x
end for//process nodes into IN
while y not in IN do
//add minimum-distance node not in IN
p = node z not in IN with minimum d[z]
IN = IN ∪ {p}
//recompute d for non-IN nodes, adjust s if necessary
for all nodes z not in IN do
OldDistance = d[z]
d[z] = min(d[z], d[ p] + A[ p, z])
if d[z] != OldDistance then
s [z] = p
end if
end for
end while
Section 6.3 Shortest Path and Minimal Spanning Tree 5
7. Shortest-Path Algorithm
//write out path nodes
write(“In reverse order, the path is”)
write (y)
z = y
repeat
write (s [z])
z = s [z]
until z = x
// write out path distance
write(“The path distance is,” d[y])
end ShortestPath
Section 6.3 Shortest Path and Minimal Spanning Tree 6
8. Shortest-Path Algorithm Example
● Trace the algorithm using the following graph and
adjacency matrix:
● At the end of the initialization phase:
Section 6.3 Shortest Path and Minimal Spanning Tree 7
9. Shortest-Path Algorithm Example
● The circled nodes are those in set IN. heavy lines show the
current shortest paths, and the d-value for each node is
written along with the node label.
Section 6.3 Shortest Path and Minimal Spanning Tree 8
10. Shortest-Path Algorithm Example
● We now enter the while loop and search through the d-values
for the node of minimum distance that is not in IN. Node 1 is
found, with d[1] = 3.
● Node 1 is added to IN. We recompute all the d-values for the
remaining nodes, 2, 3, 4, and y.
p = 1
IN = {x,1}
d[2] = min(8, 3 + A[1, 2]) = min(8, ∞) = 8
d[3] = min(4, 3 + A[1, 3]) = min(4, 9) = 4
d[4] = min(∞, 3 + A[1, 4]) = min(∞, ∞) = ∞
d[y] = min(10, 3 + A[1, y]) = min(10, ∞) = 10
● This process is repeated until y is added to IN.
Section 6.3 Shortest Path and Minimal Spanning Tree 9
11. Shortest-Path Algorithm Example
● The second pass through the while loop:
■ p = 3 (3 has the smallest d-value, namely 4, of 2, 3, 4, or y)
■ IN = {x, 1, 3}
■ d[2] = min(8, 4 + A[3, 2]) = min(8, 4 + ∞) = 8
■ d[4] = min(∞, 4 + A[3, 4]) = min(∞, 4 + 1) = 5 (a change, so
update s[4] to 3)
■ d[y] = min(10, 4 + A[3, y]) = min(10, 4 + 3) = 7 (a change, so
update s[y] to 3)
Section 6.3 Shortest Path and Minimal Spanning Tree 10
12. Shortest-Path Algorithm Example
● The third pass through the while loop:
■ p = 4 (d-value 5)
■ IN = {x, 1, 3, 4}
■ d[2] = min(8, 5 + 7) = 8
■ d[y] = min(7, 5 + 1) = 6 (a change, update s[y])
Section 6.3 Shortest Path and Minimal Spanning Tree 11
13. Shortest-Path Algorithm Example
● The third pass through the while loop:
■ p = y
■ IN = {x, 1, 3, 4, y}
■ d[2] = min(8, 6 ) = 8
● y is now part of IN, so the while loop terminates. The
(reversed) path goes through y, s[y]= 4, s[4]= 3, and s
[3] = x.
Section 6.3 Shortest Path and Minimal Spanning Tree 12
14. Shortest-Path Algorithm Analysis
● ShortestPath is a greedy algorithm - it does what
seems best based on its limited immediate knowledge.
● The for loop requires Θ(n) operations.
● The while loop takes Θ(n) operations.
● In the worst case, y is the last node brought into IN,
and the while loop will be executed n - 1 times.
● The total number of operations involved in the while
loop is Θ(n(n - 1)) = Θ(n2 ).
● Initialization and writing the output together take Θ(n)
operations.
● So the algorithm requires Θ(n + n2) = Θ(n2 )
operations in the worst case.
Section 6.3 Shortest Path and Minimal Spanning Tree 13
15. Minimal Spanning Tree Problem
● DEFINITION: SPANNING TREE
A spanning tree for a connected graph is a nonrooted tree
whose set of nodes coincides with the set of nodes for the
graph and whose arcs are (some of) the arcs of the graph.
● A spanning tree connects all the nodes of a graph with no
excess arcs (no cycles). There are algorithms for
constructing a minimal spanning tree, a spanning tree
with minimal weight, for a given simple, weighted,
connected graph.
● Prim’s Algorithm proceeds very much like the shortest-
path algorithm, resulting in a minimal spanning tree.
Section 6.3 Shortest Path and Minimal Spanning Tree 14
16. Prim’s Algorithm
● There is a set IN, which initially contains one arbitrary node.
● For every node z not in IN, we keep track of the shortest
distance d[z] between z and any node in IN.
● We successively add nodes to IN, where the next node added is
one that is not in IN and whose distance d[z] is minimal.
● The arc having this minimal distance is then made part of the
spanning tree.
● The minimal spanning tree of a graph may not be unique.
● The algorithm terminates when all nodes of the graph are in IN.
● The difference between Prim’s and Dijkstra’s algorithm is how
d[z] (new distances) are calculated.
Dijkstra’s: d[z] = min(d[z], d[p] + A[ p, z])
Prim’s: d[z] = min(d[z], A[ p, z]))
Section 6.3 Shortest Path and Minimal Spanning Tree 15