Graphs are data structures consisting of nodes and edges connecting nodes. They can be directed or undirected. Trees are special types of graphs. Common graph algorithms include depth-first search (DFS) and breadth-first search (BFS). DFS prioritizes exploring nodes along each branch as deeply as possible before backtracking, using a stack. BFS explores all nodes at the current depth before moving to the next depth, using a queue.
A graph consists of vertices and edges, where vertices represent entities and edges represent relationships between vertices. Graphs can be represented sequentially using matrices like adjacency and incidence matrices, or linked using data structures like adjacency lists. Adjacency matrices allow fast addition/removal of edges but use more memory, while adjacency lists use less memory but are slower to modify. The best representation depends on whether the graph is dense or sparse.
A graph G is composed of a set of vertices V connected by edges E. It can be represented using an adjacency matrix, with a 1 or 0 in position (i,j) indicating whether vertices i and j are connected, or an adjacency list storing the neighbors of each vertex. Graph search algorithms like depth-first search (DFS) and breadth-first search (BFS) are used to traverse the graph and find paths between vertices, with DFS prioritizing depth and BFS prioritizing breadth of exploration. DFS uses recursion to implicitly store paths while BFS uses queues and must store paths separately.
The document discusses various graph theory concepts including:
- Types of graphs such as simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs which differ based on allowed edge connections.
- Graph terminology including vertices, edges, degrees, adjacency, incidence, paths, cycles, and representations using adjacency lists and matrices.
- Weighted graphs and algorithms for finding shortest paths such as Dijkstra's algorithm.
- Euler and Hamilton paths/circuits and conditions for their existence.
- The traveling salesman problem of finding the shortest circuit visiting all vertices.
Graph terminologies & special type graphsNabeel Ahsen
The document discusses various graph terminologies and special types of graphs. It defines undirected and directed graphs, and describes degrees, handshaking theorem, and other properties. Special graph types covered include complete graphs, cycles, wheels, n-cubes, and bipartite graphs. It provides examples of constructing new graphs from existing ones and an application using a bipartite graph model for employee skills and job assignments.
This document discusses adjacency lists, a method for representing graphs. An adjacency list stores a list of adjacent vertices for each vertex. Each vertex object stores a reference to other adjacent nodes. Common graph terms are defined such as vertices, edges, paths, and weights. Methods for an adjacency list implementation include adding and removing vertices and edges. Pros are that it is easy to retrieve adjacent nodes and check for edges, while a con is that checking for a specific edge is more complex. Graph traversal algorithms and applications of adjacency lists are also discussed.
A graph G consists of a non empty set V called the set of nodes (points, vertices) of the graph, a set E, which is the set of edges of the graph and a mapping from the set of edges E to a pair of elements of V.
Any two nodes, which are connected by an edge in a graph are called "adjacent nodes".
In a graph G(V,E) an edge which is directed from one node to another is called a "directed edge", while an edge which has no specific direction is called an "undirected edge". A graph in which every edge is directed is called a "directed graph" or a "digraph". A graph in which every edge is undirected is called an "undirected graph".
If some of edges are directed and some are undirected in a graph then the graph is called a "mixed graph".
Any graph which contains some parallel edges is called a "multigraph".
If there is no more than one edge but a pair of nodes then, such a graph is called "simple graph."
A graph in which weights are assigned to every edge is called a "weighted graph".
In a graph, a node which is not adjacent to any other node is called "isolated node".
A graph containing only isolated nodes is called a "null graph". In a directed graph for any node v the number of edges which have v as initial node is called the "outdegree" of the node v. The number of edges to have v as their terminal node is called the "Indegree" of v and Sum of outdegree and indegree of a node v is called its total degree.
Graphs are data structures consisting of nodes and edges connecting nodes. They can be directed or undirected. Trees are special types of graphs. Common graph algorithms include depth-first search (DFS) and breadth-first search (BFS). DFS prioritizes exploring nodes along each branch as deeply as possible before backtracking, using a stack. BFS explores all nodes at the current depth before moving to the next depth, using a queue.
A graph consists of vertices and edges, where vertices represent entities and edges represent relationships between vertices. Graphs can be represented sequentially using matrices like adjacency and incidence matrices, or linked using data structures like adjacency lists. Adjacency matrices allow fast addition/removal of edges but use more memory, while adjacency lists use less memory but are slower to modify. The best representation depends on whether the graph is dense or sparse.
A graph G is composed of a set of vertices V connected by edges E. It can be represented using an adjacency matrix, with a 1 or 0 in position (i,j) indicating whether vertices i and j are connected, or an adjacency list storing the neighbors of each vertex. Graph search algorithms like depth-first search (DFS) and breadth-first search (BFS) are used to traverse the graph and find paths between vertices, with DFS prioritizing depth and BFS prioritizing breadth of exploration. DFS uses recursion to implicitly store paths while BFS uses queues and must store paths separately.
The document discusses various graph theory concepts including:
- Types of graphs such as simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs which differ based on allowed edge connections.
- Graph terminology including vertices, edges, degrees, adjacency, incidence, paths, cycles, and representations using adjacency lists and matrices.
- Weighted graphs and algorithms for finding shortest paths such as Dijkstra's algorithm.
- Euler and Hamilton paths/circuits and conditions for their existence.
- The traveling salesman problem of finding the shortest circuit visiting all vertices.
Graph terminologies & special type graphsNabeel Ahsen
The document discusses various graph terminologies and special types of graphs. It defines undirected and directed graphs, and describes degrees, handshaking theorem, and other properties. Special graph types covered include complete graphs, cycles, wheels, n-cubes, and bipartite graphs. It provides examples of constructing new graphs from existing ones and an application using a bipartite graph model for employee skills and job assignments.
This document discusses adjacency lists, a method for representing graphs. An adjacency list stores a list of adjacent vertices for each vertex. Each vertex object stores a reference to other adjacent nodes. Common graph terms are defined such as vertices, edges, paths, and weights. Methods for an adjacency list implementation include adding and removing vertices and edges. Pros are that it is easy to retrieve adjacent nodes and check for edges, while a con is that checking for a specific edge is more complex. Graph traversal algorithms and applications of adjacency lists are also discussed.
A graph G consists of a non empty set V called the set of nodes (points, vertices) of the graph, a set E, which is the set of edges of the graph and a mapping from the set of edges E to a pair of elements of V.
Any two nodes, which are connected by an edge in a graph are called "adjacent nodes".
In a graph G(V,E) an edge which is directed from one node to another is called a "directed edge", while an edge which has no specific direction is called an "undirected edge". A graph in which every edge is directed is called a "directed graph" or a "digraph". A graph in which every edge is undirected is called an "undirected graph".
If some of edges are directed and some are undirected in a graph then the graph is called a "mixed graph".
Any graph which contains some parallel edges is called a "multigraph".
If there is no more than one edge but a pair of nodes then, such a graph is called "simple graph."
A graph in which weights are assigned to every edge is called a "weighted graph".
In a graph, a node which is not adjacent to any other node is called "isolated node".
A graph containing only isolated nodes is called a "null graph". In a directed graph for any node v the number of edges which have v as initial node is called the "outdegree" of the node v. The number of edges to have v as their terminal node is called the "Indegree" of v and Sum of outdegree and indegree of a node v is called its total degree.
This document defines and provides examples of graphs and their representations. It discusses:
- Graphs are data structures consisting of nodes and edges connecting nodes.
- Examples of directed and undirected graphs are given.
- Graphs can be represented using adjacency matrices or adjacency lists. Adjacency matrices store connections in a grid and adjacency lists store connections as linked lists.
- Key graph terms are defined such as vertices, edges, paths, and degrees. Properties like connectivity and completeness are also discussed.
This Presentation Explains a very Important topic of Data Structure in Programming which is Graphs and Methods of Graph Traversing.
I think this presentation will help you in understanding the Data Structure Graph and its Traversing Methods.
A graph is a data structure consisting of vertices and edges connecting the vertices. Graphs can be directed or undirected. Depth-first search (DFS) and breadth-first search (BFS) are algorithms for traversing graphs by exploring neighboring vertices. DFS uses a stack and explores as far as possible along each branch before backtracking, while BFS uses a queue and explores all neighbors at each depth level first before moving to the next level.
This document defines key graph concepts like paths, cycles, degrees of vertices, and different types of graphs like trees, forests, and directed acyclic graphs. It also describes common graph representations like adjacency matrices and lists. Finally, it covers graph traversal algorithms like breadth-first search and depth-first search, outlining their time complexities and providing examples of their process.
This document discusses planar graphs, including:
- Definitions of planar graphs and examples.
- Theorems like Kuratowski's and Euler's theorems about planar graphs.
- Algorithms for planarity testing and drawing planar graphs.
- Applications of planar graphs in VLSI circuits, computer vision, and electrical circuit design.
Attributed Graph Matching of Planar GraphsRaül Arlàndez
Many fields such as computer vision, scene analysis, chemistry and molecular biology have
applications in which images have to be processed and some regions have to be searched for
and identified. When this processing is to be performed by a computer automatically without
the assistance of a human expert, a useful way of representing the knowledge is by using
attributed graphs. Attributed graphs have been proved as an effective way of representing
objects. When using graphs to represent objects or images, vertices usually represent regions
(or features) of the object or images, and edges between them represent the relations
between regions. Nonetheless planar graphs are graphs which can be drawn in the plane
without intersecting any edge between them. Most applications use planar graphs to
represent an image.
Graph matching (with attributes or not) represents an NP-complete problem, nevertheless
when we use planar graphs without attributes we can solve this problem in polynomial time
[1]. No algorithms have been presented that solve the attributed graph-matching problem and
use the planar-graphs properties. In this master thesis, we research about Attributed-Planar-
Graph matching. The aim is to find a fast algorithm through studying in depth the properties
and restrictions imposed by planar graphs.
Graph in data structure it gives you the information of the graph application. How to represent the Graph and also Graph Travesal is also there many terms are there related to garph
The document discusses graphs and their applications. It defines graphs as consisting of vertices and edges. Graphs can be undirected or directed. It also defines concepts like paths, cycles, and connectivity in graphs. Common applications of graphs mentioned are maps, computer networks, and CAD designs. The document then discusses different ways to represent graphs using adjacency matrices and adjacency lists. It also discusses using graphs to model and solve problems like traffic light timing by coloring the vertices.
This document introduces some basic concepts in graph theory, including:
- A graph G is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Edges connect pairs of vertices and can be directed or undirected. Special types of edges include parallel edges and loops.
- Special graphs include simple graphs without parallel edges/loops, weighted graphs with numerical edge weights, and complete graphs where all vertex pairs are connected.
- Graphs can be represented by adjacency matrices and incidence matrices showing vertex-edge connections.
- Paths and cycles traverse vertices and edges, with Euler cycles passing through every edge once.
The document defines and describes various graph concepts and data structures used to represent graphs. It defines a graph as a collection of nodes and edges, and distinguishes between directed and undirected graphs. It then describes common graph terminology like adjacent/incident nodes, subgraphs, paths, cycles, connected/strongly connected components, trees, and degrees. Finally, it discusses two common ways to represent graphs - the adjacency matrix and adjacency list representations, noting their storage requirements and ability to add/remove nodes.
The document discusses different types of graphs including directed and undirected graphs. It describes graph implementations using adjacency matrices and adjacency lists. It also covers graph traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). DFS prioritizes exploring paths as deeply as possible while BFS explores all paths at the same depth level before moving deeper. Real-world examples of graphs include social networks and transportation networks.
This document provides definitions and theorems related to graph theory. It begins with definitions of simple graphs, vertices, edges, degree, and the handshaking lemma. It then covers definitions and properties of paths, cycles, adjacency matrices, connectedness, Euler paths and circuits. The document also discusses Hamilton paths, planar graphs, trees, and other special types of graphs like complete graphs and bipartite graphs. It provides examples and proofs of many graph theory concepts and results.
A graph G is composed of vertices V connected by edges E. It can be represented using an adjacency matrix or adjacency lists. Graph search algorithms like depth-first search (DFS) and breadth-first search (BFS) are used to traverse the graph and find paths between vertices. DFS recursively explores edges until reaching the end of a branch before backtracking, while BFS explores all neighbors at each level before moving to the next.
Graph theory is the study of points and lines, and how sets of points called vertices can be connected by lines called edges. It involves types of graphs like regular graphs where each vertex has the same number of neighbors, and bipartite graphs where the vertices can be partitioned into two sets with no edges within each set. Graphs can be represented using adjacency matrices and adjacency lists. Basic graph algorithms include depth-first search, breadth-first search, and finding shortest paths between vertices. Graph coloring assigns colors to vertices so that no adjacent vertices have the same color.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
This document provides an overview of graph theory and applications. It begins with a brief history of graph theory and examples of early applications. It then covers basic graph theory concepts like paths, trees, connectivity, and graph representations. The document discusses representing graphs with adjacency matrices and incidence matrices. It also covers algorithms for determining connectivity in graphs and searching graphs using depth-first search. The document aims to provide an introduction to fundamental graph theory topics and applications in large graphs.
This document provides an overview of graph algorithms and graph theory concepts. It defines different types of graphs including simple graphs, directed graphs, multi graphs, pseudo graphs, and weighted graphs. It also defines graph concepts such as paths, circuits, cycles, degrees of vertices, adjacency matrices, incidence matrices, and graph traversals. Finally, it provides an explanation of the depth-first search algorithm for traversing graphs.
The document discusses various topics related to graphs:
- It defines directed and undirected graphs, paths, connected graphs, trees, degree, isomorphic graphs, cut sets, and labeled graphs.
- Key aspects include paths being sequences of vertices with edges connecting them, connected graphs having paths between all vertex pairs, trees being connected and acyclic graphs, and isomorphic graphs having the same structure.
- It also covers graph concepts such as degrees measuring incident edges, cut sets separating graphs, and labeling providing additional data to graphs' vertices or edges.
Problem Solving with Algorithms and Data Structure - ListsYi-Lung Tsai
This document discusses lists and their time complexities. It describes unordered and ordered lists, with unordered lists having O(1) complexity for operations like add and search but O(n) for others like remove. Ordered lists have higher complexities of O(n) for all operations. The document provides examples of list implementations and discusses linked lists and arrays.
This document defines and provides examples of graphs and their representations. It discusses:
- Graphs are data structures consisting of nodes and edges connecting nodes.
- Examples of directed and undirected graphs are given.
- Graphs can be represented using adjacency matrices or adjacency lists. Adjacency matrices store connections in a grid and adjacency lists store connections as linked lists.
- Key graph terms are defined such as vertices, edges, paths, and degrees. Properties like connectivity and completeness are also discussed.
This Presentation Explains a very Important topic of Data Structure in Programming which is Graphs and Methods of Graph Traversing.
I think this presentation will help you in understanding the Data Structure Graph and its Traversing Methods.
A graph is a data structure consisting of vertices and edges connecting the vertices. Graphs can be directed or undirected. Depth-first search (DFS) and breadth-first search (BFS) are algorithms for traversing graphs by exploring neighboring vertices. DFS uses a stack and explores as far as possible along each branch before backtracking, while BFS uses a queue and explores all neighbors at each depth level first before moving to the next level.
This document defines key graph concepts like paths, cycles, degrees of vertices, and different types of graphs like trees, forests, and directed acyclic graphs. It also describes common graph representations like adjacency matrices and lists. Finally, it covers graph traversal algorithms like breadth-first search and depth-first search, outlining their time complexities and providing examples of their process.
This document discusses planar graphs, including:
- Definitions of planar graphs and examples.
- Theorems like Kuratowski's and Euler's theorems about planar graphs.
- Algorithms for planarity testing and drawing planar graphs.
- Applications of planar graphs in VLSI circuits, computer vision, and electrical circuit design.
Attributed Graph Matching of Planar GraphsRaül Arlàndez
Many fields such as computer vision, scene analysis, chemistry and molecular biology have
applications in which images have to be processed and some regions have to be searched for
and identified. When this processing is to be performed by a computer automatically without
the assistance of a human expert, a useful way of representing the knowledge is by using
attributed graphs. Attributed graphs have been proved as an effective way of representing
objects. When using graphs to represent objects or images, vertices usually represent regions
(or features) of the object or images, and edges between them represent the relations
between regions. Nonetheless planar graphs are graphs which can be drawn in the plane
without intersecting any edge between them. Most applications use planar graphs to
represent an image.
Graph matching (with attributes or not) represents an NP-complete problem, nevertheless
when we use planar graphs without attributes we can solve this problem in polynomial time
[1]. No algorithms have been presented that solve the attributed graph-matching problem and
use the planar-graphs properties. In this master thesis, we research about Attributed-Planar-
Graph matching. The aim is to find a fast algorithm through studying in depth the properties
and restrictions imposed by planar graphs.
Graph in data structure it gives you the information of the graph application. How to represent the Graph and also Graph Travesal is also there many terms are there related to garph
The document discusses graphs and their applications. It defines graphs as consisting of vertices and edges. Graphs can be undirected or directed. It also defines concepts like paths, cycles, and connectivity in graphs. Common applications of graphs mentioned are maps, computer networks, and CAD designs. The document then discusses different ways to represent graphs using adjacency matrices and adjacency lists. It also discusses using graphs to model and solve problems like traffic light timing by coloring the vertices.
This document introduces some basic concepts in graph theory, including:
- A graph G is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Edges connect pairs of vertices and can be directed or undirected. Special types of edges include parallel edges and loops.
- Special graphs include simple graphs without parallel edges/loops, weighted graphs with numerical edge weights, and complete graphs where all vertex pairs are connected.
- Graphs can be represented by adjacency matrices and incidence matrices showing vertex-edge connections.
- Paths and cycles traverse vertices and edges, with Euler cycles passing through every edge once.
The document defines and describes various graph concepts and data structures used to represent graphs. It defines a graph as a collection of nodes and edges, and distinguishes between directed and undirected graphs. It then describes common graph terminology like adjacent/incident nodes, subgraphs, paths, cycles, connected/strongly connected components, trees, and degrees. Finally, it discusses two common ways to represent graphs - the adjacency matrix and adjacency list representations, noting their storage requirements and ability to add/remove nodes.
The document discusses different types of graphs including directed and undirected graphs. It describes graph implementations using adjacency matrices and adjacency lists. It also covers graph traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). DFS prioritizes exploring paths as deeply as possible while BFS explores all paths at the same depth level before moving deeper. Real-world examples of graphs include social networks and transportation networks.
This document provides definitions and theorems related to graph theory. It begins with definitions of simple graphs, vertices, edges, degree, and the handshaking lemma. It then covers definitions and properties of paths, cycles, adjacency matrices, connectedness, Euler paths and circuits. The document also discusses Hamilton paths, planar graphs, trees, and other special types of graphs like complete graphs and bipartite graphs. It provides examples and proofs of many graph theory concepts and results.
A graph G is composed of vertices V connected by edges E. It can be represented using an adjacency matrix or adjacency lists. Graph search algorithms like depth-first search (DFS) and breadth-first search (BFS) are used to traverse the graph and find paths between vertices. DFS recursively explores edges until reaching the end of a branch before backtracking, while BFS explores all neighbors at each level before moving to the next.
Graph theory is the study of points and lines, and how sets of points called vertices can be connected by lines called edges. It involves types of graphs like regular graphs where each vertex has the same number of neighbors, and bipartite graphs where the vertices can be partitioned into two sets with no edges within each set. Graphs can be represented using adjacency matrices and adjacency lists. Basic graph algorithms include depth-first search, breadth-first search, and finding shortest paths between vertices. Graph coloring assigns colors to vertices so that no adjacent vertices have the same color.
This document provides an introduction to fundamental concepts in graph theory. It defines what a graph is composed of and different graph types including simple graphs, directed graphs, bipartite graphs, and complete graphs. It discusses graph terminology such as vertices, edges, paths, cycles, components, and subgraphs. It also covers graph properties like connectivity, degrees, isomorphism, and graph coloring. Examples are provided to illustrate key graph concepts and theorems are stated about properties of graphs like the Petersen graph and graph components.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
This document provides an overview of graph theory and applications. It begins with a brief history of graph theory and examples of early applications. It then covers basic graph theory concepts like paths, trees, connectivity, and graph representations. The document discusses representing graphs with adjacency matrices and incidence matrices. It also covers algorithms for determining connectivity in graphs and searching graphs using depth-first search. The document aims to provide an introduction to fundamental graph theory topics and applications in large graphs.
This document provides an overview of graph algorithms and graph theory concepts. It defines different types of graphs including simple graphs, directed graphs, multi graphs, pseudo graphs, and weighted graphs. It also defines graph concepts such as paths, circuits, cycles, degrees of vertices, adjacency matrices, incidence matrices, and graph traversals. Finally, it provides an explanation of the depth-first search algorithm for traversing graphs.
The document discusses various topics related to graphs:
- It defines directed and undirected graphs, paths, connected graphs, trees, degree, isomorphic graphs, cut sets, and labeled graphs.
- Key aspects include paths being sequences of vertices with edges connecting them, connected graphs having paths between all vertex pairs, trees being connected and acyclic graphs, and isomorphic graphs having the same structure.
- It also covers graph concepts such as degrees measuring incident edges, cut sets separating graphs, and labeling providing additional data to graphs' vertices or edges.
Problem Solving with Algorithms and Data Structure - ListsYi-Lung Tsai
This document discusses lists and their time complexities. It describes unordered and ordered lists, with unordered lists having O(1) complexity for operations like add and search but O(n) for others like remove. Ordered lists have higher complexities of O(n) for all operations. The document provides examples of list implementations and discusses linked lists and arrays.
We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.
Prepares the students for (and is a prerequisite for) the more advanced material students will encounter in later courses. Data structures organize data Þ more efficient programs.
Analysis and design of algorithms part 3Deepak John
This document discusses graph representation and traversal techniques. It begins by defining graph terminology like vertices, edges, adjacency, and paths. It then covers different ways to represent graphs, including adjacency lists and matrices. It describes the pros and cons of each representation. The document also explains depth-first search and breadth-first search traversal algorithms in detail, including pseudocode. It analyzes the time complexity of these algorithms. Finally, it briefly discusses other graph topics like strongly connected components and biconnected components.
The document discusses how to create and process two-dimensional arrays in Java. It explains how to initialize a 2D array, print its values, input data into the array, find the largest element in each row, and also covers n-dimensional arrays. Code examples are provided to demonstrate initializing a 2D array to all 10s, printing the values, getting input, and finding the largest element in each row. Multidimensional arrays of any number of dimensions are also described.
The document discusses identifiers, variables, and constants in Java. It defines identifiers as names given to variables, constants, classes and methods. Variables are names for locations in memory that store values, and must be declared before use. Constants hold fixed values. The document outlines Java's rules for identifiers, describes how to declare and initialize variables, and lists common Java keywords.
Basics in algorithms and data structure Eman magdy
The document discusses data structures and algorithms. It notes that good programmers focus on data structures and their relationships, while bad programmers focus on code. It then provides examples of different data structures like trees and binary search trees, and algorithms for searching, inserting, deleting, and traversing tree structures. Key aspects covered include the time complexity of different searching algorithms like sequential search and binary search, as well as how to implement operations like insertion and deletion on binary trees.
Webpage Visual Design and Online Prototypeamoore155
This document lists taglines from the websites of 5 companies: Glass/google with "Go on take it outside", Starburst with "Unexplainably Juicy", Snapple with "Bring home the best stuff on earth", Harleydavidson with "American by birth, Rebel by choice", and Addidas with "Impossible is Nothing". It then provides a link to an online prototype of a homepage visual design for an assignment on webpage design.
This document discusses data structures and their organization in computer memory. It defines a data structure as a way of storing and organizing data in memory for efficient use. There are two main types of data structures - linear and non-linear. Linear data structures like arrays and linked lists represent sequential relationships, while non-linear structures like trees and graphs represent hierarchical relationships. Memory can be allocated contiguously, linked, or indexed to implement different data structure formats. Common examples discussed are arrays, linked lists, stacks, queues, trees and graphs.
A Beginner’s Guide to Programming Logic, Introductory
Chapter 2
Working with Data, Creating Modules, and Designing High-Quality Programs
Objectives
In this chapter, you will learn about:
- Declaring and using variables and constants
- Assigning values to variables
- The advantages of modularization
- Modularizing a program
- The most common configuration for mainline logic
COURSE TECHNOLOGY
CENGAGE Learning
This document discusses constants, variables, and data types in C programming. It defines tokens and keywords as the basic units, and explains constants and variables as values that either remain fixed or can change. Various data types are covered, including primary, user-defined, and derived types, along with modifiers that alter storage capacity. Rules for variable names and declarations are provided. The document also discusses assigning values to variables through constants, multiple assignments, and initialization.
The electric force between two charged particles is:
- Inversely proportional to the square of the distance between them
- Directed along the line joining the particles
- Attractive if charges have opposite signs, repulsive if the same
The electric field E at a point is defined as the electric force on a positive test charge at that point divided by the magnitude of the test charge. Electric field lines are drawn tangent to the field, with a higher density of lines indicating a greater field magnitude.
A fluid enters a steady flow apparatus with a specific volume of 0.40 m3/kg, pressure of 550 kPa, and velocity of 20 m/s, and exits with a specific volume of 0.82 m3/kg, pressure of 100 kPa, and velocity of 280 m/s. The apparatus produces 140 kJ of work per kg and experiences 12 kJ/kg of heat loss. The change in internal energy is determined to be -58.32 kJ/kg.
Design And Implementation Of A Bangla CompilerMJ Ferdous
It is evident that the familiar programming environment accelerates the efficiency of the
programmer. If a programmer has a chance to write a program in his/her native language
then undoubtedly it will enhance the overall performance of programming. Keeping in
this mind, this project tries to design and implement a Bangla compiler. This compiler
has been named as Ekushe. The whole project may be divided into two phases. The first
phase is the designing of a high-level Bangla programming language. The implemented
language follows the top down approach of popular structured programming languages.
Many basic features of imperative programming languages such as: loop, block structure,
conditional branching etc have been included in Ekushe.
In the second phase the compiler has been designed and implemented. This part actually
includes lexical, syntax and semantics issues of the language. Finally, it deals with
generating assembly code. The final job is to execute the file by creating executable file.
However, as this is the first step of our venture, many complexities of the language have
been avoided. But the major features of any contemporary languages have been included
in our implemented compiler Ekushe.
This document provides information about cryptography and encryption techniques. It discusses that cryptography is the science of information security and includes techniques to hide information. It then describes three main goals of cryptography: confidentiality, integrity, and authentication. It proceeds to define encryption as converting a message into cipher text and decryption as converting cipher text back to the original message. Finally, it explains the differences between symmetric and asymmetric encryption methods.
The document discusses various algorithms for query processing operations like selection, sorting, and join. It provides cost estimates for each algorithm based on factors like the number of block transfers and seeks. The most efficient algorithms depend on characteristics of the relations and whether indices are available. Nested loop and block nested loop joins have high costs, while merge join and hash join may have lower costs depending on the situation.
graphin-c1.png
graphin-c1.txt
1: 2
2: 3 8
3: 4
4: 5
5: 3
6: 7
7: 3 6 8
8: 1 9
9: 1
graphin-c2.jpg
graphin-c2.txt
1: 2 9
2: 3 8
3: 4
4: 5 9
5: 3
6: 7
7: 3 6 8
8: 1
9:
graphin-DAG.png
graphin-DAG.txt
1: 2
2: 3 8
3: 4
4: 5
5: 9
6: 4 7
7: 3 8
8: 9
9:
CS 340 Programming Assignment III:
Topological Sort
Description: You are to implement the Depth-First Search (DFS) based algorithm for (i)
testing whether or not the input directed graph G is acyclic (a DAG), and (ii) if G is a DAG,
topologically sorting the vertices of G and outputting the topologically sorted order.
I/O Specifications: You will prompt the user from the console to select an input graph
filename, including the sample file graphin.txt as an option. The graph input files must be of
the following adjacency list representation where each xij is the j'th neighbor of vertex i (vertex
labels are 1 through n):
1: x11 x12 x13 ...
2: x21 x22 x23 ...
.
.
n: xn1 xn2 xn3 ...
Your output will be to the console. You will first output whether or not the graph is acyclic. If
the graph is NOT acyclic, then you will output the set of back edges you have detected during
DFS. Otherwise, if the graph is acyclic, then you will output the vertices in topologically
sorted order.
Algorithmic specifications:
Your algorithm must use DFS appropriately and run in O(E + V) time on any input graph. You will
need to keep track of edge types and finish times so that you can use DFS for detecting
cyclicity/acyclicity and topologically sorting if the graph is a DAG. You may implement your graph
class as you wish so long as your overall algorithm runs correctly and efficiently.
What to Turn in: You must turn in a single zipped file containing your source code, a Makefile
if your language must be compiled, appropriate input and output files, and a README file
indicating how to execute your program (especially if not written in C++ or Java). Refer to
proglag.pdf for further specifications.
This assignment is due by MIDNIGHT of Monday, February 19. Late submissions
carry a minus 40% per-day late penalty.
Sheet1Name:Possible:Score:Comments:10Graph structure with adjacency list representationDFS16Correct and O(V+E) time10Detecting cycles, is graph DAG?Topological Sort16Correctness of Topo-Sort algorithm and output18No problems in compilation and execution? Non-compiling projects receive max total 10 points, and code that compiles but crashes during execution receives max total 18 points.700Total
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Sheet2
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Sheet3
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DFS and topological sort
CS340
Depth first search
breadth
depth
Search "deeper" whenever possible
*example shows discovery times
Depth first search
Input: G = (V,E), directed or undirected.
No source vertex is given!
Output: 2 timestamps on each vertex:
v.d discovery time
v.f finishing time
These will be useful ...
This document discusses graph algorithms using depth-first search (DFS). It begins by defining graph terminology such as vertices, edges, directed and undirected graphs. It then explains DFS, including classifying edges based on the DFS spanning tree. It describes applications of DFS such as finding biconnected components, articulation points, and strongly connected components in directed graphs. The document provides pseudocode for DFS and algorithms for these applications that run in linear time with respect to the number of vertices and edges.
The document discusses using graphs to model real-world problems and algorithms for solving graph problems. It introduces basic graph terminology like vertices, edges, adjacency matrix/list representations. It then summarizes algorithms for finding shortest paths like Dijkstra's and Bellman-Ford. It also discusses using depth-first search to solve problems like detecting cycles, topological sorting of vertices, and calculating longest paths in directed acyclic graphs.
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.
This document discusses graph algorithms and graph search techniques. It begins with an introduction to graphs and their representations as adjacency matrices and adjacency lists. It then covers graph terminology like vertices, edges, paths, cycles, and weighted graphs. Common graph search algorithms like breadth-first search and depth-first search are explained. Variations of these algorithms like recursive depth-first search and Dijkstra's algorithm for finding shortest paths in weighted graphs are also covered. Examples are provided throughout to illustrate the concepts and algorithms.
Algorithm Design and Complexity - Course 8Traian Rebedea
The document discusses algorithms for finding strongly connected components (SCCs) in directed graphs. It describes Kosaraju's algorithm, which uses two depth-first searches (DFS) to find the SCCs. The first DFS computes a finishing time ordering, while the second DFS uses the transpose graph and the finishing time ordering to find the SCCs, outputting each SCC as a separate DFS tree. The algorithm runs in O(V+E) time and uses the property that the first DFS provides a topological sorting of the SCCs graph.
The document describes an algorithm for finding a (p,q)-partition of a graph G. A (p,q)-partition partitions the vertices of G into clusters such that each cluster has size at most p and cut size at most q. The algorithm works by first showing that finding a (p,q)-partition can be reduced to finding a (p,q)-cluster for each vertex. It then gives a randomized algorithm for finding a (p,q)-cluster for a given vertex v in time 2O(q)nO(1) by reducing the problem to an instance of the satellite problem, which can be solved in polynomial time. The algorithm works by sampling important cuts in the graph to construct an instance of
The document discusses algorithms for solving single-source shortest path problems on directed graphs. It begins by defining the problem and describing Bellman-Ford's algorithm, which can handle graphs with negative edge weights but runs in O(VE) time. It then describes how Dijkstra's algorithm improves this to O(ElogV) time using a priority queue, but requires non-negative edge weights. It also discusses how shortest paths can be found in O(V+E) time on directed acyclic graphs by relaxing edges topologically. Finally, it discusses how difference constraints can be modeled as shortest path problems on a corresponding constraint graph.
This document discusses several graph algorithms:
1) Topological sort is an ordering of the vertices of a directed acyclic graph (DAG) such that for every edge from vertex u to v, u comes before v in the ordering. It can be used to find a valid schedule respecting dependencies.
2) Strongly connected components are maximal subsets of vertices in a directed graph such that there is a path between every pair of vertices. An algorithm uses depth-first search to find SCCs in linear time.
3) Minimum spanning trees find a subset of edges that connects all vertices at minimum total cost. Prim's and Kruskal's algorithms find minimum spanning trees using greedy strategies in O(E
This document discusses several graph algorithms:
1) Topological sort is an ordering of the vertices of a directed acyclic graph (DAG) such that for every edge from vertex u to v, u comes before v in the ordering. It can be used to find a valid schedule respecting dependencies.
2) Strongly connected components are maximal subsets of vertices in a directed graph such that there is a path between every pair of vertices. An algorithm uses depth-first search to find SCCs in linear time.
3) Minimum spanning trees find a subset of edges that connects all vertices at minimum total cost. Prim's and Kruskal's algorithms find minimum spanning trees using greedy strategies in O(E
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.
The document describes the single-source shortest paths algorithm for directed acyclic graphs (DAGs). It involves topologically sorting the vertices of the DAG, initializing distances from the source vertex s, and then relaxing edges in topologically sorted order. This guarantees that when a vertex u is relaxed, the shortest path distances from s to its neighbors will be accurate. The algorithm runs in O(V+E) time. It is used to find critical paths in PERT charts by finding the longest path after negating or reversing edge weights.
The document discusses parallel graph algorithms. It describes Dijkstra's algorithm for finding single-source shortest paths and its parallel formulations. It also describes Floyd's algorithm for finding all-pairs shortest paths and its parallel formulation using a 2D block mapping. Additionally, it discusses Johnson's algorithm, a modification of Dijkstra's algorithm to efficiently handle sparse graphs, and its parallel formulation.
Bellman Ford's algorithm finds the shortest paths from a source vertex to all other vertices in a weighted graph, even if edge weights are negative. It works by repeatedly relaxing all edges to update the distance and previous vertex for each vertex. After iterating through all edges |V|-1 times, if an edge can still be relaxed, then a negative cycle exists in the graph.
This document discusses various greedy algorithms, including Dijkstra's algorithm for finding shortest paths in graphs. It provides details on Dijkstra's algorithm, including its greedy approach, proof of correctness, and efficient implementations using priority queues. It also discusses applications of shortest path algorithms and how the techniques extend to related problems involving closed semirings rather than just numbers.
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.
This document contains a term end examination for a higher mathematics course. It includes two parts - Part A contains 8 multiple choice or short answer questions, and Part B contains 3 longer answer questions requiring proofs or calculations. Some of the mathematical topics covered include graphs, lattices, modular arithmetic, logic, and probability. The exam is worth a total of 100 marks and students must answer 5 questions in Part A and 3 in Part B.
This document discusses graph kernels, which are positive definite kernels defined on graphs that allow applying machine learning algorithms to graph-structured data like molecules. It covers different types of graph kernels like subgraph kernels, path kernels, and walk kernels. Walk kernels count the number of walks between two graphs and can be computed efficiently in polynomial time, unlike subgraph and path kernels. The document also discusses using product graphs to compute walk kernels and presents results on classifying mutagenicity using random walk kernels. It concludes by proposing using graph kernels and product graphs to define data depth measures for labeled graph ensembles.
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3. Graph
A set of objects where some pairs of objects are
connected by links
road map
airline flights
internet connection
skill tree
Tree is special kind of graph
Vertex
(node)
Edge
(arc)
3
8. Path and Cycle
Path
Sequence of vertices that are connected by edges
[v_1, v_5, v_4, v_6]
Cycle
A path that starts and ends at same vertex
[v_2, v_3, v_4, v_5, v_2]
8
28. Choose Better Next Vertex
Visit hard-to-reach corners early
Use the middle square to hop across board
only when necessary
28
29.
30. General Depth First Search (DFS)
Knight’s Tour is special case of depth first
search
create the depth first tree
General depth first search is to search as
deeply as possible
30
34. P versus NP problem
P: questions for which some algorithms can
provide an answer in polynomial time
NP: questions for which an answer can be
verified in polynomial time
Subset sum problem
{-7, -3, -2, 5, 8} -> {-3, -2, 5}
34
36. NP-complete
Reduction: algorithm for transforming one
problem into another problem
NP-complete: a problem p in NP is NP-
Complete if every other problem in NP can
be transformed into p in polynomial time
hardest problems in NP
36
37. NP-hard
Decision problem: any arbitrary yes-or-no
question on an infinite set of inputs
NP-hard: a decision problem H in NP-hard
when for any problem L in NP, there is
polynomial-time reduction from L to H
at least as hard as the hardest problems in
NP
37
38. Topological Sorting
Linear ordering of all vertices of a directed
graph
(u, v) is an edge of directed graph and u is
before v in ordering
38
42. Lemma
A directed graph G is acyclic if and only if a
depth-first search of G yields no back edges.
Back edge: edge (u, v) connecting vertex u to
an ancestor v in depth-first tree (v ↝ u → v)
=>: back edge exists then cycle exists
<=: cycle exists then back edge exists
42
43. Proof of Implementation
For any pair of distinct vertices u, v ∈ V, if there is an
edge in G from u to v, then f[v] < f[u].
Consider edge (u, v) explored by DFS, v cannot be
gray since then that edge would be back edge.
Therefore v must be white or black.
If v is white, v is descendant of u, and so f[v] < f[u]
If v is black, it has already been finished, so that
f[v] < f[u]
43
44. Strongly Connected Components
G = (V, E) and C ⊂ V
such that ∀ (v_i, v_j) ∈ C, path v_i to v_j
and path v_j to v_i both exist
C is strongly connected components (SCC)
44
45.
46. Implementation
1. Call dfs(g)
2. Compute transpose of g as g_t
3. Call dfs(g_t) but explore each vertex in
decreasing order of finish time
4. Each tree of step 3 is a SCC
original transpose
46
49. Lemma
Let C and C’ be distinct SCCs in directed
graph G = (V, E), let u, v ∈ C, let u’, v’ ∈ C’,
and suppose that there is a path u ↝ u’ in G.
Then there cannot also be a path v’ ↝ v in G.
if v’ ↝ v exists then both u ↝ u’ ↝ v’ and v’
↝ v ↝ u exist
C and C’ are not distinct SCCs
49
50. Lemma
Let C and C’ be distinct SCCs in directed
graph G = (V, E). Suppose that there is an
edge (u, v) ∈ E, where u ∈ C and v ∈ C’. Then
f(C) > f(C’)
from x ∈ C to w ∈ C’
from y ∈ C’ cannot reach any vertex in C
50
51. Corollary
Let C and C’ be distinct SCCs in directed
graph G = (V, E). Suppose that there is an
edge (u, v) ∈ ET, where u ∈ C and v ∈ C’.
Then f(C) < f(C’).
(v, u) ∈ E then f(C’) < f(C)
51
53. Shortest Path Problem
Find the path with the smallest total weight
along which to route any given message
53
54. Dijkstra’s Algorithm
Iterative algorithm providing the shortest
path from one particular starting node to all
other nodes in graph
Path distance, priority queue
54