The document discusses graph searching algorithms breadth-first search (BFS) and depth-first search (DFS). It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. BFS explores vertices level-by-level starting from the source vertex, while DFS explores deeper into the graph first before backtracking. Both algorithms run in O(V+E) time, where V is the number of vertices and E is the number of edges.
SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time - Cancun talk 2012Jurgen Riedel
This document presents an overview of Q-balls and boson stars. It begins with introductions to solitons, topological solitons like vortices and domain walls, and non-topological solitons like Q-balls. Q-balls are described as localized rotating solutions stabilized by a conserved Noether charge. Boson stars are discussed as gravitating solutions composed of scalar fields minimally coupled to gravity. Different models of boson stars are presented, including those with self-interactions and gauge couplings. Rotating versions of both Q-balls and boson stars are also summarized.
Elementary Graph Algorithms discusses graphs and common graph algorithms. It defines graphs as G = (V, E) with vertices V and edges E. It describes the breadth-first search (BFS) and depth-first search (DFS) algorithms. BFS expands the frontier between discovered and undiscovered vertices uniformly across the breadth of the frontier. It uses a queue and runs in O(V+E) time. DFS explores edges out of the most recently discovered vertex, searching as deep as possible first before backtracking. It uses timestamps and runs in O(V+E) time. Pseudocode and examples are provided for both algorithms.
The document discusses graphs and graph algorithms. It defines graphs and their representations using adjacency lists and matrices. It then describes the Breadth-First Search (BFS) and Depth-First Search (DFS) algorithms for traversing graphs. BFS uses a queue to explore the neighbors of each vertex level-by-level, while DFS uses a stack to explore as deep as possible first before backtracking. Pseudocode and examples are provided to illustrate how the algorithms work.
Uniform cost search is similar to breadth-first search but pursues the path with the lowest accumulated cost rather than minimizing depth. It calculates the path cost g(n) as the total cost of the path from the initial node to node n, rather than just considering depth. This means uniform cost search prioritizes expanding nodes with the lowest cost paths found so far, making it optimal for problems where path costs are uniform.
This document outlines an assignment for Harshit Kumar due on March 23rd. It instructs him to use truth tables to prove 5 logical equivalence rules: (1) material implication, (2) biconditional, (3) distributive law, (4) DeMorgan's laws, and (5) idempotent law. For the idempotent law, three methods of proof are required as discussed in class.
Handout for the course Abstract Argumentation and Interfaces to Argumentative...Federico Cerutti
This document provides an overview of abstract argumentation frameworks and semantics. It begins with definitions of Dung's argumentation framework (AF), including concepts like conflict-free sets, acceptable arguments, and admissible sets. It then covers properties that argumentation semantics can satisfy, like being conflict-free or reinstating acceptable arguments. Several semantics are defined, like complete, grounded, preferred and stable extensions. The document also discusses labelling-based representations of semantics and computational properties of decision problems for different semantics. In the second half, it outlines implementations, ranking-based semantics, argumentation schemes, semantic web argumentation, and natural language interfaces for argumentation systems.
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
The document discusses the Fundamental Theorem of Calculus (FTC), which links differentiation and integration. It defines the FTC in two parts: the first part states that if a function f is continuous on an interval [a,b] and F is defined by integrating f, then F is uniformly continuous and differentiable. The second part states that if f is differentiable on [a,b] and its derivative f' is integrable, then the integral of f' from a to b is equal to f(b)-f(a). Proofs of both parts of the FTC are provided.
SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time - Cancun talk 2012Jurgen Riedel
This document presents an overview of Q-balls and boson stars. It begins with introductions to solitons, topological solitons like vortices and domain walls, and non-topological solitons like Q-balls. Q-balls are described as localized rotating solutions stabilized by a conserved Noether charge. Boson stars are discussed as gravitating solutions composed of scalar fields minimally coupled to gravity. Different models of boson stars are presented, including those with self-interactions and gauge couplings. Rotating versions of both Q-balls and boson stars are also summarized.
Elementary Graph Algorithms discusses graphs and common graph algorithms. It defines graphs as G = (V, E) with vertices V and edges E. It describes the breadth-first search (BFS) and depth-first search (DFS) algorithms. BFS expands the frontier between discovered and undiscovered vertices uniformly across the breadth of the frontier. It uses a queue and runs in O(V+E) time. DFS explores edges out of the most recently discovered vertex, searching as deep as possible first before backtracking. It uses timestamps and runs in O(V+E) time. Pseudocode and examples are provided for both algorithms.
The document discusses graphs and graph algorithms. It defines graphs and their representations using adjacency lists and matrices. It then describes the Breadth-First Search (BFS) and Depth-First Search (DFS) algorithms for traversing graphs. BFS uses a queue to explore the neighbors of each vertex level-by-level, while DFS uses a stack to explore as deep as possible first before backtracking. Pseudocode and examples are provided to illustrate how the algorithms work.
Uniform cost search is similar to breadth-first search but pursues the path with the lowest accumulated cost rather than minimizing depth. It calculates the path cost g(n) as the total cost of the path from the initial node to node n, rather than just considering depth. This means uniform cost search prioritizes expanding nodes with the lowest cost paths found so far, making it optimal for problems where path costs are uniform.
This document outlines an assignment for Harshit Kumar due on March 23rd. It instructs him to use truth tables to prove 5 logical equivalence rules: (1) material implication, (2) biconditional, (3) distributive law, (4) DeMorgan's laws, and (5) idempotent law. For the idempotent law, three methods of proof are required as discussed in class.
Handout for the course Abstract Argumentation and Interfaces to Argumentative...Federico Cerutti
This document provides an overview of abstract argumentation frameworks and semantics. It begins with definitions of Dung's argumentation framework (AF), including concepts like conflict-free sets, acceptable arguments, and admissible sets. It then covers properties that argumentation semantics can satisfy, like being conflict-free or reinstating acceptable arguments. Several semantics are defined, like complete, grounded, preferred and stable extensions. The document also discusses labelling-based representations of semantics and computational properties of decision problems for different semantics. In the second half, it outlines implementations, ranking-based semantics, argumentation schemes, semantic web argumentation, and natural language interfaces for argumentation systems.
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
The document discusses the Fundamental Theorem of Calculus (FTC), which links differentiation and integration. It defines the FTC in two parts: the first part states that if a function f is continuous on an interval [a,b] and F is defined by integrating f, then F is uniformly continuous and differentiable. The second part states that if f is differentiable on [a,b] and its derivative f' is integrable, then the integral of f' from a to b is equal to f(b)-f(a). Proofs of both parts of the FTC are provided.
This document provides instructions and examples for a geometry drill on inverse functions. Students are asked to find missing values in functions, find the inverse of given functions, use inverse sine and cosine to find angle measurements given side lengths, and provide answers to assignment questions. The document also provides example answers to practice problems involving inverse trigonometric functions and finding unknown angle measures.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
This document presents a method for finding the best polynomial approximation of degree one for a function over a given interval using the method of least parallelograms. It introduces the problem, outlines the main results which include theorems on the existence and properties of the least parallelogram and its relationship to the minimax polynomial. Examples are also provided to illustrate the method.
This document discusses how to calculate the volumes of prisms and cylinders. It provides the volume formulas for prisms (V=Bh) and cylinders (V=πr^2h), along with examples of how to use the formulas to calculate volumes of various shapes. It also includes an example problem solving for the length of prisms given their volumes.
This document discusses how to calculate the volumes of pyramids and cones. It provides the formulas for volume of a pyramid (V=1/3Bh) and volume of a cone (V=1/3πr^2h) and works through multiple examples of applying the formulas. It finds the volumes of various pyramids and cones by plugging dimensions like base area, height, radius, etc. into the appropriate volume formula.
SUSY Q-Balls and Boson Stars in Anti-de Sitter space-timeJurgen Riedel
This document discusses supersymmetric Q-balls and boson stars in anti-de Sitter spacetime. It begins with an overview of the AdS/CFT correspondence and how it relates classical gravity theories to strongly coupled quantum field theories. Next, it examines SUSY Q-balls in an AdS background through numerical analysis, finding their properties like mass and charge vary with frequency and cosmological constant. Finally, it analyzes SUSY boson stars in AdS through similar numerical methods, noting how their properties change with coupling constant and frequency.
The document discusses fields and their characteristics. It defines the characteristic of a field F as the smallest positive integer p such that 1+1+...+1 (p times) is equal to 0, if such a p exists. If no such p exists, the characteristic is 0. It states that if a field has finite characteristic p, then p must be prime. It also discusses finite fields, polynomials over fields, and field extensions.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
Nishan Nazimuddin provides his personal and contact information, including his date of birth, current location, languages spoken, and certifications. He has a Bachelor's degree in computer science and is Cisco and Microsoft certified. His career objectives are to work in a competitive environment utilizing his skills. His technical skills include experience administering Windows servers and networking equipment such as routers, switches, and firewalls. He has strengths in team management, planning, troubleshooting, and site coordination.
The document is a resume for Saied Fathy Saied applying for a beverage developer manager position. It summarizes his work experience of over 15 years in food and beverage roles at various cafes. It details his responsibilities in his current role as Beverage Developer Manager such as ensuring beverage quality standards, menu costing and profitability, developing new items, and training baristas. It also lists his education and various training courses completed to enhance his skills.
This document is the preface and introduction to a human rights resource guide for South Korea published in 2014. It provides an overview of South Korea's progress on human rights issues while acknowledging ongoing challenges. Some key points:
- South Korea has made tremendous economic and social progress but still faces challenges upholding full human rights, especially for women.
- Issues around freedom of speech and privacy online are increasingly important as South Korea is highly digitally connected. Recent court rulings and laws still threaten these rights.
- The guide aims to educate the public and track trends in human rights by surveying implementation of Universal Declaration of Human Rights provisions in Korean law and society.
- It uniquely provides details on human rights
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In just one sentence, it pitches the idea of using Haiku Deck to easily create engaging slideshows.
This document outlines the storyboard for an online course project called Project Hashtag. It includes:
- An introduction describing the course's aim to teach Microsoft Word and APA referencing skills.
- Details on the roles of four creators and an instructional design model called SAM being used.
- Storyboard sections like the main page, topic landing pages, sample PowerPoint slides, and a draft storyboard layout.
- Notes on design decisions like using levels, requiring progression, and tailoring for the target cohort.
This document describes a method for determining the phase or arrival time of individual bunches in the drive beam of a two-beam acceleration test stand using RF data from power extraction and transfer structures (PETS). The method works by modeling the relationship between the electric field in the PETS, the measured RF signals outside the PETS, and the drive beam current. This relationship can be expressed as a system of linear equations that can be solved to determine the drive beam phase for each bunch. Applying this analysis method to collected PETS RF and beam current data allows the drive beam phase and its variation along the bunch train to be characterized.
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In a single sentence, it pitches the idea of using Haiku Deck to easily create and share slideshow presentations online.
Maggie Verspoor is a Visual Arts major at SUNY New Paltz who chose her major because she enjoys transforming ideas into art. She is taking the ENG.308.01 course to further develop her critical thinking and understanding of literature, and it also fulfills a requirement for her major. She loves nature and completed her final season on the New Paltz Cross Country team last fall. Her favorite author is F. Scott Fitzgerald because she admires the 1920s time period of his writings and loves the narrative of The Great Gatsby. She is excited to read and analyze some of Fitzgerald's short stories this winter to gain a greater understanding of his work.
The document advertises sponsorship and exhibition opportunities for the 1st Saudi Clinical Pharmacy Conference taking place from February 15-17, 2015 in Qassim, Saudi Arabia. The conference will have over 600 delegates from various roles in clinical pharmacy and is supported by several partner organizations. Sponsorship offers opportunities to meet key stakeholders, decision makers, and influencers in the rapidly growing Saudi pharmaceutical industry. Various sponsorship packages are available and can be customized to suit different budgets and business needs. Interested companies are encouraged to contact the event organizers.
אמנות היצירה .משגב. הגליל המערבי.סדנאות יצירה, פסיפס mosaic,שפר רבקהRSHEFF30
אמנות היצירה. האמנית רבקה שפר יוצרת יצירות ייחודיות בפסיפס.
.סדנאות רב- דוריות . סדנאות יצירה לילדים נוער ומבוגרים.
בחצר הסדנה מקום אירוח ל 50 אורחים. ניתנת הרצאה כיבוד והתנסות בעבודת הפסיפס. בסדנה עבודות למכירה. הסדנה נמצאת בטל-אל בגליל המערבי,משגב
This document provides instructions and examples for a geometry drill on inverse functions. Students are asked to find missing values in functions, find the inverse of given functions, use inverse sine and cosine to find angle measurements given side lengths, and provide answers to assignment questions. The document also provides example answers to practice problems involving inverse trigonometric functions and finding unknown angle measures.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
This document presents a method for finding the best polynomial approximation of degree one for a function over a given interval using the method of least parallelograms. It introduces the problem, outlines the main results which include theorems on the existence and properties of the least parallelogram and its relationship to the minimax polynomial. Examples are also provided to illustrate the method.
This document discusses how to calculate the volumes of prisms and cylinders. It provides the volume formulas for prisms (V=Bh) and cylinders (V=πr^2h), along with examples of how to use the formulas to calculate volumes of various shapes. It also includes an example problem solving for the length of prisms given their volumes.
This document discusses how to calculate the volumes of pyramids and cones. It provides the formulas for volume of a pyramid (V=1/3Bh) and volume of a cone (V=1/3πr^2h) and works through multiple examples of applying the formulas. It finds the volumes of various pyramids and cones by plugging dimensions like base area, height, radius, etc. into the appropriate volume formula.
SUSY Q-Balls and Boson Stars in Anti-de Sitter space-timeJurgen Riedel
This document discusses supersymmetric Q-balls and boson stars in anti-de Sitter spacetime. It begins with an overview of the AdS/CFT correspondence and how it relates classical gravity theories to strongly coupled quantum field theories. Next, it examines SUSY Q-balls in an AdS background through numerical analysis, finding their properties like mass and charge vary with frequency and cosmological constant. Finally, it analyzes SUSY boson stars in AdS through similar numerical methods, noting how their properties change with coupling constant and frequency.
The document discusses fields and their characteristics. It defines the characteristic of a field F as the smallest positive integer p such that 1+1+...+1 (p times) is equal to 0, if such a p exists. If no such p exists, the characteristic is 0. It states that if a field has finite characteristic p, then p must be prime. It also discusses finite fields, polynomials over fields, and field extensions.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
Nishan Nazimuddin provides his personal and contact information, including his date of birth, current location, languages spoken, and certifications. He has a Bachelor's degree in computer science and is Cisco and Microsoft certified. His career objectives are to work in a competitive environment utilizing his skills. His technical skills include experience administering Windows servers and networking equipment such as routers, switches, and firewalls. He has strengths in team management, planning, troubleshooting, and site coordination.
The document is a resume for Saied Fathy Saied applying for a beverage developer manager position. It summarizes his work experience of over 15 years in food and beverage roles at various cafes. It details his responsibilities in his current role as Beverage Developer Manager such as ensuring beverage quality standards, menu costing and profitability, developing new items, and training baristas. It also lists his education and various training courses completed to enhance his skills.
This document is the preface and introduction to a human rights resource guide for South Korea published in 2014. It provides an overview of South Korea's progress on human rights issues while acknowledging ongoing challenges. Some key points:
- South Korea has made tremendous economic and social progress but still faces challenges upholding full human rights, especially for women.
- Issues around freedom of speech and privacy online are increasingly important as South Korea is highly digitally connected. Recent court rulings and laws still threaten these rights.
- The guide aims to educate the public and track trends in human rights by surveying implementation of Universal Declaration of Human Rights provisions in Korean law and society.
- It uniquely provides details on human rights
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In just one sentence, it pitches the idea of using Haiku Deck to easily create engaging slideshows.
This document outlines the storyboard for an online course project called Project Hashtag. It includes:
- An introduction describing the course's aim to teach Microsoft Word and APA referencing skills.
- Details on the roles of four creators and an instructional design model called SAM being used.
- Storyboard sections like the main page, topic landing pages, sample PowerPoint slides, and a draft storyboard layout.
- Notes on design decisions like using levels, requiring progression, and tailoring for the target cohort.
This document describes a method for determining the phase or arrival time of individual bunches in the drive beam of a two-beam acceleration test stand using RF data from power extraction and transfer structures (PETS). The method works by modeling the relationship between the electric field in the PETS, the measured RF signals outside the PETS, and the drive beam current. This relationship can be expressed as a system of linear equations that can be solved to determine the drive beam phase for each bunch. Applying this analysis method to collected PETS RF and beam current data allows the drive beam phase and its variation along the bunch train to be characterized.
This short document promotes creating presentations using Haiku Deck, a tool for making slideshows. It encourages the reader to get started making their own Haiku Deck presentation and sharing it on SlideShare. In a single sentence, it pitches the idea of using Haiku Deck to easily create and share slideshow presentations online.
Maggie Verspoor is a Visual Arts major at SUNY New Paltz who chose her major because she enjoys transforming ideas into art. She is taking the ENG.308.01 course to further develop her critical thinking and understanding of literature, and it also fulfills a requirement for her major. She loves nature and completed her final season on the New Paltz Cross Country team last fall. Her favorite author is F. Scott Fitzgerald because she admires the 1920s time period of his writings and loves the narrative of The Great Gatsby. She is excited to read and analyze some of Fitzgerald's short stories this winter to gain a greater understanding of his work.
The document advertises sponsorship and exhibition opportunities for the 1st Saudi Clinical Pharmacy Conference taking place from February 15-17, 2015 in Qassim, Saudi Arabia. The conference will have over 600 delegates from various roles in clinical pharmacy and is supported by several partner organizations. Sponsorship offers opportunities to meet key stakeholders, decision makers, and influencers in the rapidly growing Saudi pharmaceutical industry. Various sponsorship packages are available and can be customized to suit different budgets and business needs. Interested companies are encouraged to contact the event organizers.
אמנות היצירה .משגב. הגליל המערבי.סדנאות יצירה, פסיפס mosaic,שפר רבקהRSHEFF30
אמנות היצירה. האמנית רבקה שפר יוצרת יצירות ייחודיות בפסיפס.
.סדנאות רב- דוריות . סדנאות יצירה לילדים נוער ומבוגרים.
בחצר הסדנה מקום אירוח ל 50 אורחים. ניתנת הרצאה כיבוד והתנסות בעבודת הפסיפס. בסדנה עבודות למכירה. הסדנה נמצאת בטל-אל בגליל המערבי,משגב
The document discusses the algorithms of breadth-first search (BFS) and depth-first search (DFS) on graphs. It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. Key points include:
- BFS uses a queue to explore all neighbors of a vertex before moving to the next level. It finds the shortest paths from the source.
- DFS uses recursion to explore as deep as possible before backtracking. It identifies tree edges, back edges, and forward edges.
- Both BFS and DFS run in O(V+E) time on a graph with V vertices and E edges.
Breadth-first search (BFS) and depth-first search (DFS) are two graph traversal algorithms that systematically visit all vertices and edges of a graph. BFS discovers vertices in order of distance from the source vertex, allowing it to compute shortest paths. DFS recursively explores as far as possible along each branch before backtracking, imposing a tree structure on the graph. Both algorithms run in O(V+E) time, where V is the number of vertices and E is the number of edges. DFS classifies edges as tree, forward, back, or cross edges and can detect cycles based on the presence of back edges.
The document discusses graph algorithms breadth-first search (BFS) and depth-first search (DFS). It provides pseudocode for BFS and DFS algorithms and explains key aspects of each. BFS uses a queue to explore all neighbors of a node before moving to the next level, building a breadth-first tree. DFS uses recursion to explore as deep as possible along each branch before backtracking, resulting in a depth-first tree structure. Both algorithms run in O(V+E) time on a graph with V vertices and E edges.
This document discusses elementary graph algorithms, including breadth-first search (BFS) and depth-first search (DFS). It provides background on graph representations using adjacency lists and matrices. It then describes the key steps of BFS and DFS, including initializing data structures, exploring edges, and classifying edge types. BFS finds the shortest path between two nodes by exploring the closest nodes first. DFS searches deeper whenever possible and backtracks to explore other branches.
This document provides an outline and overview of a lecture on elementary graph algorithms. It begins with contact information for the lecturer, Dr. Muhammad Hanif Durad. It then outlines topics to be covered, including definition and representation of graphs, breadth-first search, depth-first search, topological sort, and strongly connected components. The document discusses the importance of graphs and examples of problems that can be modeled with graphs. It provides definitions and descriptions of basic graph terminology like vertices, edges, types of graphs. It also covers representations of graphs using adjacency lists and adjacency matrices. The document dives deeper into breadth-first search and depth-first search algorithms, providing pseudocode and examples. It discusses applications and analysis of the algorithms.
Depth-first search (DFS) is an algorithm that explores all the vertices reachable from a starting vertex by traversing edges in a depth-first manner. DFS uses a stack data structure to keep track of vertices to visit. It colors vertices white, gray, and black to indicate their status. DFS runs in O(V+E) time and can be used for applications like topological sorting and finding strongly connected components. The edges discovered during DFS can be classified as tree, back, forward, or cross edges based on the order in which vertices are discovered.
DFS explores a path fully before backtracking, coloring nodes white, gray, then black. It uses two timestamps, discover and finish times. The DFS algorithm initializes colors and timestamps, then calls DFS_VISIT on each white node to explore its subtree and color neighbors. DFS runs in O(V+E) time. It is used for topological sorting and finding strongly connected components. Edges are classified as tree, back, forward, or cross based on node colors during exploration.
The document discusses graphs and their representations. It defines a graph as a pair (V,E) where V is a set of vertices and E is a set of edges. There are two main representations of graphs: adjacency matrix and adjacency lists. The adjacency matrix represents the graph as a 2D matrix where rows and columns are vertices and entries indicate edges. The adjacency lists representation uses an array of linked lists, where each list stores the neighbors of its corresponding vertex.
This document describes graphs and basic graph algorithms. It defines what a graph is as a set of vertices and edges. It also defines common graph types like directed and undirected graphs. The document then describes two common ways to represent graphs: adjacency lists and adjacency matrices. It provides pseudocode and examples for the core graph search algorithms Breadth-first Search (BFS) and Depth-first Search (DFS). BFS explores the graph in levels, starting from the source vertex. DFS explores edges deeply first before backtracking.
This document discusses graphs and graph algorithms. It defines what a graph is - a set of vertices connected by edges. It covers different types of graphs like directed/undirected graphs and weighted graphs. It then explains two common graph search algorithms - depth-first search (DFS) and breadth-first search (BFS). DFS explores each path as deep as possible before backtracking while BFS explores all neighbors of a node before moving deeper. Both algorithms run in O(V+E) time where V is vertices and E is edges. BFS always finds the shortest path while DFS is not guaranteed to.
This document provides an overview of the Bellman-Ford algorithm for finding single-source shortest paths in a weighted graph. It begins with an outline of the lecture topics, then provides context on weighted graphs and shortest path problems. The key properties of shortest paths and edge relaxation are described. The Bellman-Ford algorithm is presented as performing relaxation of all edges |V|-1 times to find shortest paths of length up to |V|-1, to account for all possible path lengths without cycles. Examples are provided and it is shown that the algorithm runs in O(VE) time and O(V+E) space. Correctness of the algorithm is discussed based on earlier theorems regarding shortest path properties.
This document discusses graphs and graph algorithms. It defines what a graph is - a data structure containing vertices and edges. It provides examples of graphs like social networks and road maps. It explains concepts like paths, connectedness, and cycles. It then covers two graph search algorithms - depth-first search (DFS) and breadth-first search (BFS). DFS explores each path as deeply as possible before backtracking, while BFS explores all neighbors of a vertex before moving deeper. Both algorithms run in O(V+E) time where V is vertices and E is edges. BFS always finds the shortest path.
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking
Algorithm Design and Complexity - Course 7Traian Rebedea
The document discusses algorithms for graphs, including breadth-first search (BFS) and depth-first search (DFS). BFS uses a queue to traverse nodes level-by-level from a starting node, computing the shortest path. DFS uses a stack, exploring as far as possible along each branch before backtracking, and computes discovery and finish times for nodes. Both algorithms color nodes white, gray, black to track explored status and maintain predecessor pointers to reconstruct paths. Common graph representations like adjacency lists and matrices are also covered.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Film vocab for eal 3 students: Australia the movie
Graph 02
1. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
1
Graphs (Cont.)
graph (many to many)
2. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Graph Searching
Given: a graph G = (V, E), directed or undirected
Goal: methodically explore every vertex and every edge
Ultimately: build a tree on the graph
Pick a vertex as the root
Choose certain edges to produce a tree
Note: might also build a forest if graph is not connected
• There are two standard graph traversal techniques:
Breadth-First Search (BFS)
Depth-First Search (DFS)
3. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search
“Explore” a graph, turning it into a tree
One vertex at a time
Expand frontier of explored vertices across the breadth of
the frontier
Builds a tree over the graph
Pick a source vertex to be the root
Find (“discover”) its children, then their children, etc.
4. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search
Again will associate vertex “colors” to guide the
algorithm
White vertices have not been discovered
All vertices start out white
Grey vertices are discovered but not fully explored
They may be adjacent to white vertices
Black vertices are discovered and fully explored
They are adjacent only to black and grey vertices
Explore vertices by scanning adjacency list of grey
vertices
5. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search
6. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
∞
∞
∞
∞
∞
∞
∞
∞
r s t u
v w x y
7. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
∞
∞
0
∞
∞
∞
∞
∞
r s t u
v w x y
sQ:
8. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
∞
0
1
∞
∞
∞
∞
r s t u
v w x y
wQ: r
9. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
∞
0
1
2
2
∞
∞
r s t u
v w x y
rQ: t x
10. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
∞
∞
r s t u
v w x y
Q: t x v
11. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
3
∞
r s t u
v w x y
Q: x v u
12. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: v u y
13. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: u y
14. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: y
15. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Example
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: Ø
16. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
BFS - A Graphical Representation
M N O P
I J K L
E F G H
A B C D
0
M N O P
I J K L
E F G H
A B C D
0 1
M N O P
I J K L
E F G H
A C DB
0 1 2
M N O P
I J K L
E F G H
A B C D
0 1 2 3
d)c)
b)a)
17. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
BFS - A Graphical Representation
M N O P
I J K L
E F G H
A B C D
4
0 1 2 3
M N O P
I J K L
E F G H
A B C D
4
5
0 1 2 3
e) f)
18. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
BFS: The Code Again
BFS(G, s) {
initialize vertices;
Q = {s};
while (Q not empty) {
u = RemoveTop(Q);
for each v ∈ u->adj {
if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}
What will be the running time?
Touch every vertex: O(V)
u = every vertex, but only once
(Why?)
So v = every vertex
that appears in
some other vert’s
adjacency list
Total running time: O(V+E)
19. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
BFS: The Code Again
BFS(G, s) {
initialize vertices;
Q = {s};
while (Q not empty) {
u = RemoveTop(Q);
for each v ∈ u->adj {
if (v->color == WHITE)
v->color = GREY;
v->d = u->d + 1;
v->p = u;
Enqueue(Q, v);
}
u->color = BLACK;
}
}
What will be the storage cost
in addition to storing the tree?
Total space used:
O(V + Σ(degree(v))) = O(V+E)
20. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Breadth-First Search: Properties
BFS calculates the shortest-path distance to the source
node
Shortest-path distance δ(s,v) = minimum number of edges
from s to v, or ∞ if v not reachable from s
BFS builds breadth-first tree, in which paths to root
represent shortest paths in G
Thus can use BFS to calculate shortest path from one
vertex to another in O(V+E) time in an unweighted tree
21. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search
Depth-first search is another strategy for exploring a graph
Explore “deeper” in the graph whenever possible
Edges are explored out of the most recently discovered vertex v that still
has unexplored edges
When all of v’s edges have been explored, backtrack to the vertex from
which v was discovered
Vertices initially colored white
Then colored grey when discovered
Then black when finished
22. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
23. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What does u->d represent?
24. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What does u->f represent?
25. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
Will all vertices eventually be colored black?
26. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
What will be the running time?
27. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
Running time: O(n2
) because call DFS_Visit on each vertex,
and the loop over Adj[] can run as many as |V| times
28. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
BUT, there is actually a tighter bound.
How many times will DFS_Visit() actually be called?
29. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search: The Code
DFS(G)
{
for each vertex u ∈ G->V
{
u->color = WHITE;
}
time = 0;
for each vertex u ∈ G->V
{
if (u->color == WHITE)
DFS_Visit(u);
}
}
DFS_Visit(u)
{
u->color = GREY;
time = time+1;
u->d = time;
for each v ∈ u->Adj[]
{
if (v->color == WHITE)
DFS_Visit(v);
}
u->color = BLACK;
time = time+1;
u->f = time;
}
So, running time of DFS = O(V+E)
30. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Depth-First Search Analysis
This running time argument is an informal example of
amortized analysis
“Charge” the exploration of edge to the edge:
Each loop in DFS_Visit can be attributed to an edge in the
graph
Runs once/edge if directed graph, twice if undirected
Thus loop will run in O(E) time, algorithm O(V+E)
Storage requirement is O(V+E), since adj list requires
O(V+E) storage
31. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
source
vertex
32. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
|||
| |
source
vertex
d f
33. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
|||
2 | |
source
vertex
d f
34. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
||3 |
2 | |
source
vertex
d f
35. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
||3 | 4
2 | |
source
vertex
d f
36. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
|5 |3 | 4
2 | |
source
vertex
d f
37. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | | |
|5 | 63 | 4
2 | |
source
vertex
d f
38. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | 8 | |
|5 | 63 | 4
2 | 7 |
source
vertex
d f
39. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | 8 | |
|5 | 63 | 4
2 | 7 |
source
vertex
d f
40. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | 8 | |
|5 | 63 | 4
2 | 7 9 |
source
vertex
d f
41. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | 8 | |
|5 | 63 | 4
2 | 7 9 |10
source
vertex
d f
42. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 | 8 |11 |
|5 | 63 | 4
2 | 7 9 |10
source
vertex
d f
43. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 |12 8 |11 |
|5 | 63 | 4
2 | 7 9 |10
source
vertex
d f
44. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 |12 8 |11 13|
|5 | 63 | 4
2 | 7 9 |10
source
vertex
d f
45. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 |12 8 |11 13|
14|5 | 63 | 4
2 | 7 9 |10
source
vertex
d f
46. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 |12 8 |11 13|
14|155 | 63 | 4
2 | 7 9 |10
source
vertex
d f
47. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS Example
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
source
vertex
d f
48. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS introduces an important distinction among edges in
the original graph:
Tree edge: encounter new (white) vertex
The tree edges form a spanning forest
Can tree edges form cycles? Why or why not?
DFS: Kinds of edges
49. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges
DFS: Kinds of edges
50. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS: Kinds of edges
DFS introduces an important distinction among edges in
the original graph:
Tree edge: encounter new (white) vertex
Back edge: from descendent to ancestor
Encounter a grey vertex (grey to grey)
51. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Tree edges Back edges
DFS: Kinds of edges
1 | | |
||3 |
2 | |
source
vertex
d f
52. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS: Kinds of edges
DFS introduces an important distinction among edges in
the original graph:
Tree edge: encounter new (white) vertex
Back edge: from descendent to ancestor
Forward edge: from ancestor to descendent
Not a tree edge, though
From grey node to black node
53. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
Tree edges Back edges Forward edges
DFS: Kinds of edges
1 | 8 | |
|5 | 63 | 4
2 | 7 |
source
vertex
d f
54. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS: Kinds of edges
DFS introduces an important distinction among edges in
the original graph:
Tree edge: encounter new (white) vertex
Back edge: from descendent to ancestor
Forward edge: from ancestor to descendent
Cross edge: between a tree or subtrees
From a grey node to a black node
55. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
1 |12 8 |11 13|16
14|155 | 63 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges Back edges Forward edges Cross edges
DFS: Kinds of edges
56. Dr. Md. Abul Kashem Mia, Professor, CSE Dept, BUET 12/29/16
DFS: Kinds of edges
DFS introduces an important distinction among edges in
the original graph:
Tree edge: encounter new (white) vertex
Back edge: from descendent to ancestor
Forward edge: from ancestor to descendent
Cross edge: between a tree or subtrees
Note: tree and back edges are very important; some
algorithms use forward and cross edges