Discrete Mathematics ( Dijkstra Algorithm & Prims Algorithm)

Feb 11, 2025Download as PPTX, PDF

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The document discusses Dijkstra's and Prim's algorithms, which are used for finding the shortest path in graphs and minimum spanning trees, respectively. Dijkstra's algorithm finds the shortest path from a source vertex to all other vertices, while Prim's algorithm connects all vertices with the minimum sum of edge weights. It provides real-world examples such as GPS navigation and network design, illustrating the practical applications of these algorithms.

Education

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Dijkstra’s
Algorithm & Prim’s
Algorithm
Understanding the Fundamentals of Shortest Path and
Minimum Spanning Tree

Our Team
• Seerat Fatima [2024-DS-32]
• Aleena Ahmad [2024-DS-9]
• Minahil Samreen [2024-DS-11]
• Amna Naveed [2024-DS-42]
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Today’s Agenda
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Dijestra’s
Algorithm
01
Prim’s
Algorithm
03
Real
World
Example
02
Real
World
Example
04

What is Dijkstra's Algorithm?
• Dijkstra's algorithm finds the shortest path from one
vertex to all other vertices.
• It does so by repeatedly selecting the nearest unvisited
vertex and calculating the distance to all the unvisited
neighboring vertices.

How it works?
• Start from the source
node.
• Assign initial distances
(infinity except for the
source = 0).
• Visit neighbors of the
current node, update
shortest paths.
• Repeat until all nodes
are visited.

Answer
A = 0
B = 1+2=3 (A -> D -> B)
C = 1+1+5=7 (A -> D -> E -> C)
D = 1 (A -> D)
E = 1+1=2 (A -> D -> E)

• GPS navigation systems
• Network routing protocols
• Game AI pathfinding
Where is it used?
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Finding the shortest route from Lahore to all other
cities
Lahor
e
Karach
i
Islamaba
d
Multa
n
Peshawa
r
Quett
a

Solution
1.A = 0
2.B = 4 (A -> B)
3.C = 5 (A -> C)
4.D = 4 + 9 = 13 (A -> B -> D)
5.E = 5 + 3 = 8 (A -> C -> E)
6.F = 5 + 3 + 6 = 14 (A -> C -> E -> F)

What is Prim's Algorithm?
• Prim's algorithm finds the Minimum Spanning Tree
(MST) in a connected and undirected graph.
• The MST found by Prim's algorithm is the collection of
edges in a graph, that connects all vertices,with a
minimum sum of edge weights.

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0 Fundamental Concepts
A spanning tree is a subgraph of a connected, undirected
graph that includes all the vertices of the graph and is a tree
(i.e., it has no cycles).
Spanning tree
A spanning tree is a subgraph of a connected, undirected
graph that includes all the vertices of the graph and is a tree
(i.e., it has no cycles).
Minimum Spanning
tree

How it works?
• Start with any node.
• Add the smallest edge that
connects an unvisited node.
• Repeat until all nodes are
connected.

Cost of MST = 4 + 2 + 1 + 3 = 10 units.vv
Answer
Cost of MST = 4 + 2 + 1 + 3 = 10 units.

• Network design
• Clustering analysis
• Circuit Design
Where is it used?
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Designing a minimum-cost communications
network

Its Quiz Time!!!!
Let's see how much you've learned so far!
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The document details Dijkstra's and Prim's algorithms, which are used for finding the shortest path in graphs and constructing minimum spanning trees, respectively. Dijkstra's algorithm guarantees optimal paths in weighted graphs while Prim's algorithm focuses on minimizing the total edge weight in a tree structure. Both algorithms have their merits and demerits, with applications spanning networking, robotics, and transportation.

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This document presents an implementation of Dijkstra's algorithm to find the shortest path between computers connected by routers on a network. It introduces Dijkstra's algorithm and how it works, provides an example graph and step-by-step workings, then applies it to a problem of finding the lowest cost route between computers on an internet connection. A simulation models the network as a graph and runs Dijkstra's algorithm to determine the shortest path from computer A to G has a total cost of 70. Screenshots confirm the algorithm was implemented and found the lowest cost route.

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Kruskal's and Prim's algorithms can both be used to find a minimum spanning tree for a graph. Kruskal's algorithm considers edges and selects the lowest weight edge that does not form a cycle at each step. Prim's algorithm considers vertices and selects the lowest weight edge connecting a vertex already in the tree to one not yet in the tree. Dijkstra's algorithm can find the shortest path between two vertices in a graph by iteratively assigning final labels to vertices starting from the source vertex.

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This document provides an overview of greedy algorithms, including examples and analysis. It discusses how greedy algorithms work by making locally optimal choices at each step in hopes of finding a global optimum. Examples covered include counting change using coins, scheduling jobs on processors, minimum spanning trees, and Dijkstra's shortest path algorithm. The document notes that greedy algorithms may not always find the optimal solution and provides examples where they fail. It also analyzes the time complexity of various greedy algorithms.

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1. Dijkstra's algorithm is a greedy algorithm that finds the shortest path between nodes in a graph. It maintains a priority queue of all nodes and iteratively selects the node with the smallest distance. 2. It initializes all distances as infinity except the source node which is 0. On each iteration, it selects the node with the smallest distance and updates neighboring nodes' distances if a shorter path is found. 3. The example shows running Dijkstra's on a graph from node A. It iteratively selects the closest node and updates distances until all nodes are visited and the shortest paths from A are found.

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Dijkstra's algorithm is a widely used method for finding the shortest path from a source vertex to other vertices in a graph with non-negative edge weights. This document covers its functionality, implementation, and comparisons with other algorithms like Bellman-Ford, Floyd-Warshall, and A*, highlighting its applications in routing, optimization, and navigation. Additionally, it discusses the algorithm's complexity, its suitability for directed and undirected graphs, and concludes with its significance in various fields.

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This PowerPoint presentation is prepared for Unit 10 – Non-Communicable Diseases and National Health Programs, as per the 5th Semester B.Sc Nursing syllabus outlined by the Indian Nursing Council (INC) under the subject Community Health Nursing – I. This unit focuses on equipping students with knowledge of the causes, prevention, and control of non-communicable diseases (NCDs), which are a major public health challenge in India. The presentation emphasizes the nurse’s role in early detection, screening, management, and referral services under national-level programs. 🔹 Key Topics Included: Definition, burden, and impact of NCDs in India Epidemiology, risk factors, signs/symptoms, prevention, and management of: Diabetes Mellitus Hypertension Cardiovascular Diseases Stroke & Obesity Thyroid Disorders Blindness Deafness Injuries and Accidents (incl. road traffic injuries and trauma guidelines) NCD-2 Cancers: Breast Cancer Cervical Cancer Oral Cancer Risk factors, screening, diagnosis, early signs, referral & palliative care Role of nurse in screening, referral, counseling, and continuum of care National Programs: National Program for Prevention and Control of Cancer, Diabetes, Cardiovascular Diseases and Stroke (NPCDCS) National Program for Control of Blindness National Program for Prevention and Control of Deafness National Tobacco Control Program (NTCP) Introduction to Universal Health Coverage and Ayushman Bharat Use of standard treatment protocols and referral flowcharts This presentation is ideal for: Classroom lectures, field assignments, health education planning, and student projects Preparing for university exams, class tests, and community field postings

Community Health Nursing Approaches, Concepts, Roles & Responsibilities – Uni...RAKESH SAJJAN

This PowerPoint presentation is based on Unit 6 – Community Health Nursing Approaches, Concepts, Roles & Responsibilities of Community Health Nursing Personnel, designed for B.Sc Nursing 5th Semester students under the subject Community Health Nursing – I, following the syllabus of the Indian Nursing Council (INC). This unit focuses on the various approaches in community health, the organizational framework, and the responsibilities of different levels of nursing staff in the healthcare system. It emphasizes the real-world application of nursing principles to provide comprehensive and preventive care to the community. 📘 Key Areas Covered in this Presentation: Introduction to the concept of community health nursing Approaches to community health: Nursing Process Approach Epidemiological Approach Evidence-Based Approach Problem-Solving Approach Nursing Theories in Community Health Practice Explanation of teamwork and intersectoral coordination Concept of primary health care and its application in community nursing Levels of health care delivery – primary, secondary, and tertiary care Home visit process: principles, planning, implementation, and follow-up Use of community bag and record maintenance Roles and responsibilities of: Auxiliary Nurse Midwives (ANMs) Community Health Officers (CHOs) Staff Nurses ASHA workers Public Health Nurses (PHNs) Documentation and reporting in community settings Promotion of health education, immunization, maternal and child health, and nutritional support Role of nurse in disease surveillance, outbreak control, and health promotion Ethical principles in community nursing Coordination with health team members and village health committees This presentation is useful for: Nursing students preparing for university theory exams, class tests, or viva Nursing educators conducting lectures or field discussions Interns and trainees working in PHCs, sub-centers, or community settings Community nurses and health educators involved in rural and urban outreach The content is simplified, clear, and enhanced with point-wise explanations, flowcharts, and field-related examples for better retention and application.

YSPH VMOC Special Report - Measles Outbreak Southwest US 6-14-2025.pptxYale School of Public Health - The Virtual Medical Operations Center (VMOC)

BLUF: The Texas outbreak has slowed down, but sporadic cases continue to emerge in Kansas, Oklahoma, and New Mexico. Elsewhere in the US, we continue to see signs of acceleration due to outbreaks outside the Southwest (North Dakota, Montana, and Colorado) and travel-related cases. Measles exposures due to travel are expected to pose a significant challenge throughout the summer. The U.S. is on track to exceed its 30-year high for measles cases (1,274) within the next two weeks. Here is the latest update: CURRENT CASE COUNT: 919 •Texas: 744 (+2) (55% of cases are in Gaines County). •New Mexico: 81 (83% of cases are from Lea County). •Oklahoma: 20 (+2) •Kansas: 74 (+5) (38.89% of the cases are from Gray County). HOSPITALIZATIONS: 104 • Texas: 96 (+2) – This accounts for 13% of all cases in Texas. • New Mexico: 7 – This accounts for 9.47% of all cases in New Mexico. • Kansas: 3 – This accounts for 5.08% of all cases in the state of Kansas. DEATHS: 3 •Texas: 2 – This is 0.27% of all cases in Texas. •New Mexico: 1 – This is 1.23% of all cases in New Mexico. US NATIONAL CASE COUNT: 1,197 INTERNATIONAL SPREAD •Mexico: 2337 (+257), 5 fatalities ‒Chihuahua, Mexico: 2,179 (+239) cases, 4 fatalities, 7 currently hospitalized. •Canada: 3,207 (+208), 1 fatality ‒Ontario Outbreak, Canada: 2,115 (+74) cases, 158 hospitalizations, 1 fatality. ‒Alberta, Canada: 879(+118) cases, 5 currently hospitalized.

IIT KGP Quiz Week 2024 Sports Quiz (Prelims + Finals)IIT Kharagpur Quiz Club

LAZY SUNDAY QUIZ "A GENERAL QUIZ" JUNE 2025 SMC QUIZ CLUB, SILCHAR MEDICAL CO...Ultimatewinner0342

🧠 Lazy Sunday Quiz | General Knowledge Trivia by SMC Quiz Club – Silchar Medical College Presenting the Lazy Sunday Quiz, a fun and thought-provoking general knowledge quiz created by the SMC Quiz Club of Silchar Medical College & Hospital (SMCH). This quiz is designed for casual learners, quiz enthusiasts, and competitive teams looking for a diverse, engaging set of questions with clean visuals and smart clues. 🎯 What is the Lazy Sunday Quiz? The Lazy Sunday Quiz is a light-hearted yet intellectually rewarding quiz session held under the SMC Quiz Club banner. It’s a general quiz covering a mix of current affairs, pop culture, history, India, sports, medicine, science, and more. Whether you’re hosting a quiz event, preparing a session for students, or just looking for quality trivia to enjoy with friends, this PowerPoint deck is perfect for you. 📋 Quiz Format & Structure Total Questions: ~50 Types: MCQs, one-liners, image-based, visual connects, lateral thinking Rounds: Warm-up, Main Quiz, Visual Round, Connects (optional bonus) Design: Simple, clear slides with answer explanations included Tools Needed: Just a projector or screen – ready to use! 🧠 Who Is It For? College quiz clubs School or medical students Teachers or faculty for classroom engagement Event organizers needing quiz content Quizzers preparing for competitions Freelancers building quiz portfolios 💡 Why Use This Quiz? Ready-made, high-quality content Curated with lateral thinking and storytelling in mind Covers both academic and pop culture topics Designed by a quizzer with real event experience Usable in inter-college fests, informal quizzes, or Sunday brain workouts 📚 About the Creators This quiz has been created by Rana Mayank Pratap, an MBBS student and quizmaster at SMC Quiz Club, Silchar Medical College. The club aims to promote a culture of curiosity and smart thinking through weekly and monthly quiz events. 🔍 SEO Tags: quiz, general knowledge quiz, trivia quiz, SlideShare quiz, college quiz, fun quiz, medical college quiz, India quiz, pop culture quiz, visual quiz, MCQ quiz, connect quiz, science quiz, current affairs quiz, SMC Quiz Club, Silchar Medical College 📣 Reuse & Credit You’re free to use or adapt this quiz for your own events or sessions with credit to: SMC Quiz Club – Silchar Medical College & Hospital Curated by: Rana Mayank Pratap

LDMMIA Practitioner Student Reiki Yoga S2 Video PDF Without Yogi GoddessLDM & Mia eStudios

A bonus dept update. Happy Summer 25 almost. Do Welcome or Welcome back. Our 10th Free workshop will be released the end of this week, June 20th Weekend. All Materials/updates/Workshops are timeless for future students. ♥ Your Attendance is valued. We hit over 5k views for Spring Workshops and Updates-TY. ♥ Coming to our Shop This Weekend. Timeless for Future Grad Level Students. Practitioner Student. Level/Session 2 Packages. * ♥The Review & Topics: * All virtual, adult, education students must be over 18 years to attend LDMMIA eClasses and vStudio Thx. * Please refer to our Free Workshops anytime for review/notes. * Orientation Counts as S1 on introduction. Sold Separately as a PDF. Our S2 includes 2 Videos within 2 Mp4s. Sold Separately for Uploading. * Reiki Is Japanese Energy Healing used Globally. * Yoga is over 5k years old from India. It hosts many styles, teacher versions, and it’s Mainstream now vs decades ago. * Teaching Vod, 720 Res, Mp4: Yoga Therapy is Reviewed as a Hatha, Classical, Med Yoga (ND) Base. Take practice notes as needed or repeat videos. * Fused Teaching Vod, 720 Res, Mp4: Yoga Therapy Meets Reiki Review. Take Practice notes as needed or repeat videos. * Video, 720 Res, Mp4: Practitioner Congrats and Workshop Visual Review with Suggestions. ♥ Bonus Studio Video, 720 Res, Mp4: Our 1st Reiki Video. Produced under Yogi Goddess, LDM Recording. As a Reiki, Kundalini, ASMR Spa, Music Visual. For Our Remastered, Beatz Single for Goddess Vevo Watchers. https://www.reverbnation.com/yogigoddess * ♥ Our Videos are Vevo TV and promoted within the LDMMIA Profiles. * Scheduled upload for or by Weekend Friday June 13th. * LDMMIA Digital & Merch Shop: https://ldm-mia.creator-spring.com * ♥ As a student, make sure you have high speed connections/wifi for attendance. This sounds basic, I know lol. But, for our video section. The High Speed and Tech is necessary. Otherwise, any device can be used. Our Zip drive files should serve MAC/PC as well. * ♥ On TECH Emergency: I have had some rare, rough, horrid timed situations as a Remote Student. Pros and Cons to being on campus. So Any Starbucks (coffee shop) or library can be used for wifi hot spots. You can work at your own speed and pace. * ♥ We will not be hosting deadlines, tests/exams. * ♥Any homework will be session practice and business planning. Nothing stressful or assignment submissions.

How to Customize Quotation Layouts in Odoo 18Celine George

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Battle of Bookworms 2025 - U25 Literature Quiz by Pragya Pragya - UEM Kolkata Quiz Club

Battle of Bookworms is a literature quiz organized by Pragya, UEM Kolkata, as part of their cultural fest Ecstasia. Curated by quizmasters Drisana Bhattacharyya, Argha Saha, and Aniket Adhikari, the quiz was a dynamic mix of classical literature, modern writing, mythology, regional texts, and experimental literary forms. It began with a 20-question prelim round where ‘star questions’ played a key tie-breaking role. The top 8 teams moved into advanced rounds, where they faced audio-visual challenges, pounce/bounce formats, immunity tokens, and theme-based risk-reward questions. From Orwell and Hemingway to Tagore and Sarala Das, the quiz traversed a global and Indian literary landscape. Unique rounds explored slipstream fiction, constrained writing, adaptations, and true crime literature. It included signature IDs, character identifications, and open-pounce selections. Questions were crafted to test contextual understanding, narrative knowledge, and authorial intent, making the quiz both intellectually rewarding and culturally rich. Battle of Bookworms proved literature quizzes can be insightful, creative, and deeply enjoyable for all.

Assisting Individuals and Families to Promote and Maintain Health – Unit 7 | ...RAKESH SAJJAN

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Communicable Diseases and National Health Programs – Unit 9 | B.Sc Nursing 5t...RAKESH SAJJAN

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IIT KGP Quiz Week 2024 Sports Quiz (Prelims + Finals)IIT Kharagpur Quiz Club

LAZY SUNDAY QUIZ "A GENERAL QUIZ" JUNE 2025 SMC QUIZ CLUB, SILCHAR MEDICAL CO...Ultimatewinner0342

LDMMIA Practitioner Student Reiki Yoga S2 Video PDF Without Yogi GoddessLDM & Mia eStudios

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Discrete Mathematics ( Dijkstra Algorithm & Prims Algorithm)

1. 1 4 9 6 3 7 5 2 0 8 Dijkstra’s Algorithm & Prim’s Algorithm Understanding the Fundamentals of Shortest Path and Minimum Spanning Tree

2. Our Team • Seerat Fatima [2024-DS-32] • Aleena Ahmad [2024-DS-9] • Minahil Samreen [2024-DS-11] • Amna Naveed [2024-DS-42] 1 3 4 9 7 5 2 0 6 8

3. Today’s Agenda 1 3 8 9 7 5 0 6 2 Dijestra’s Algorithm 01 Prim’s Algorithm 03 Real World Example 02 Real World Example 04

4. 1 4 9 6 3 7 5 2 0 8 Dijkstra Algorithm

5. What is Dijkstra's Algorithm? • Dijkstra's algorithm finds the shortest path from one vertex to all other vertices. • It does so by repeatedly selecting the nearest unvisited vertex and calculating the distance to all the unvisited neighboring vertices.

6. How it works? • Start from the source node. • Assign initial distances (infinity except for the source = 0). • Visit neighbors of the current node, update shortest paths. • Repeat until all nodes are visited.

7. Example Problem A D B C E 6 1 5 5 1 2 2

8. Answer A = 0 B = 1+2=3 (A -> D -> B) C = 1+1+5=7 (A -> D -> E -> C) D = 1 (A -> D) E = 1+1=2 (A -> D -> E)

9. • GPS navigation systems • Network routing protocols • Game AI pathfinding Where is it used? 1 3 8 9 7 5 0 6 2

10. 1 4 9 6 3 7 5 2 0 8 Real World Example

11. Finding the shortest route from Lahore to all other cities Lahor e Karach i Islamaba d Multa n Peshawa r Quett a

12. Solution 1.A = 0 2.B = 4 (A -> B) 3.C = 5 (A -> C) 4.D = 4 + 9 = 13 (A -> B -> D) 5.E = 5 + 3 = 8 (A -> C -> E) 6.F = 5 + 3 + 6 = 14 (A -> C -> E -> F)

13. 1 4 9 6 3 7 5 2 0 8 Prim’s Algorithm

14. What is Prim's Algorithm? • Prim's algorithm finds the Minimum Spanning Tree (MST) in a connected and undirected graph. • The MST found by Prim's algorithm is the collection of edges in a graph, that connects all vertices,with a minimum sum of edge weights.

15. 1 3 6 8 9 7 5 2 0 Fundamental Concepts A spanning tree is a subgraph of a connected, undirected graph that includes all the vertices of the graph and is a tree (i.e., it has no cycles). Spanning tree A spanning tree is a subgraph of a connected, undirected graph that includes all the vertices of the graph and is a tree (i.e., it has no cycles). Minimum Spanning tree

16. How it works? • Start with any node. • Add the smallest edge that connects an unvisited node. • Repeat until all nodes are connected.

17. Example Problem

18. Cost of MST = 4 + 2 + 1 + 3 = 10 units.vv Answer Cost of MST = 4 + 2 + 1 + 3 = 10 units.

19. • Network design • Clustering analysis • Circuit Design Where is it used? 1 3 8 9 7 5 0 6 2

20. 1 4 9 6 3 7 5 2 0 8 Real World Example

21. Designing a minimum-cost communications network

22. Solution

23. Solution

24. Its Quiz Time!!!! Let's see how much you've learned so far! 1 3 4 9 7 5 2 0 6 8

25. 8 1 4 9 6 3 7 5 2 0 Thank you!

Discrete Mathematics ( Dijkstra Algorithm & Prims Algorithm)

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