lecture 18
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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.
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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.
topological_sort_strongly Connected Components
The document describes algorithms for depth-first search (DFS) and how it can be used to find strongly connected components (SCCs) in a directed graph. DFS assigns discovery and finish times to vertices and classifies graph edges. It can determine if a graph is acyclic by checking for back edges. To find SCCs, DFS is run on the original graph and its transpose to group vertices reachable from each other into components. The result is a directed acyclic graph of components. Lemmas are presented about paths between components based on discovery/finish times.
Graps 2
The document discusses breadth-first search (BFS) algorithms for graphs. It explains that BFS runs in O(n^2) time on adjacency matrices due to checking all elements in each row, but in O(n+m) time on adjacency lists where m is the number of edges. It then describes how to modify BFS to record the shortest path between nodes using a predecessor array. The rest of the document provides examples of how BFS works and applications such as finding connected components and directed acyclic graphs.
graphin-c1.pnggraphin-c1.txt1 22 3 83 44 5.docx
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
&"Helvetica,Regular"&12&K000000&P
Sheet2
&"Helvetica,Regular"&12&K000000&P
Sheet3
&"Helvetica,Regular"&12&K000000&P
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 ...
Algorithm Design and Complexity - Course 8
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.
Algorithm Design and Complexity - Course 7
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.
Shortest Path in Graph
The document discusses algorithms for finding shortest paths in graphs. It describes Dijkstra's algorithm and Bellman-Ford algorithm for solving the single-source shortest paths problem and Floyd-Warshall algorithm for solving the all-pairs shortest paths problem. Dijkstra's algorithm uses a priority queue to efficiently find shortest paths from a single source node to all others, assuming non-negative edge weights. Bellman-Ford handles graphs with negative edge weights but is slower. Floyd-Warshall finds shortest paths between all pairs of nodes in a graph.
Graph Traversal Algorithm
This document describes graph search algorithms like breadth-first search (BFS) and depth-first search (DFS). It provides details on how BFS works, including that it maintains distances from the source vertex and uses a queue to search levels outwards. BFS runs in O(V+E) time, visiting each vertex and edge once. It outputs the shortest path distances and predecessor graph. The document proves BFS is correct by showing the distances computed are less than or equal to the actual shortest path distances.
lecture24.ppt
The document discusses depth-first search (DFS) algorithms for exploring graphs. It begins by reviewing DFS, noting that it explores deeper in the graph whenever possible by exploring edges out of the most recently discovered vertex. It then shows pseudocode for performing DFS on a graph. The document proceeds to provide an example run of DFS on a graph, demonstrating the discovery order of vertices and labeling of tree, back, forward, and cross edges. It concludes by discussing properties of DFS, such as only producing tree and back edges in undirected graphs and using DFS to detect cycles in O(V+E) time.
19-graph1 (1).ppt
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.
kumattt).pptx
This document provides an overview of depth-first search (DFS) and breadth-first search (BFS) algorithms. It introduces the presenter, Kumar Ayush Singh Deo, and their subject on analyzing and digitizing algorithms. Key topics covered include definitions of graphs, how DFS and BFS work by exploring adjacent vertices, analysis of time complexity when using adjacency lists or matrices, examples of DFS on directed and undirected graphs illustrated with pseudocode, and applications of DFS and BFS.
Graph Representation, DFS and BFS Presentation.pptx
The document describes the breadth-first search (BFS) algorithm for traversing graphs. BFS uses a queue to visit all neighboring nodes at the current level before moving to the next level out. It marks nodes as discovered, processes their neighbors, and marks them finished after processing. The algorithm initializes all nodes as undiscovered, enqueues the source node, and dequeues nodes to process their neighbors until the queue is empty.
Chapter 23 aoa
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.
Lecture13
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.
FADML 06 PPC Graphs and Traversals.pdf
The document provides an introduction to graphs and graph algorithms. It defines graphs as consisting of vertices/nodes and edges connecting the nodes. It discusses different graph types including undirected graphs, directed graphs, and weighted graphs. It then describes various graph traversal algorithms like depth-first search, breadth-first search, and their applications to problems like finding paths, cycles, and connected components. Finally, it discusses algorithms for shortest paths in weighted graphs.
Graph Traversal Algorithms - Depth First Search Traversal
This document discusses graph traversal techniques, specifically depth-first search (DFS) and breadth-first search (BFS). It provides pseudocode for DFS and explains key properties like edge classification, time complexity of O(V+E), and applications such as finding connected components and articulation points.
Ppt 1
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
Problem Solving with Algorithms and Data Structure - Graphs
Study slides for http://interactivepython.org/runestone/static/pythonds/index.html
Focus on Graphs and Graph Algorithms
lec 09-graphs-bfs-dfs.ppt
The document discusses different types of graphs and graph terminology. It begins by defining what a graph is, including vertices and edges. It then describes different types of graphs such as undirected vs directed graphs, weighted graphs, and multigraphs. The document defines common graph terminology like paths, cycles, connectedness, trees, and more. It also discusses ways of representing graphs using adjacency lists and matrices. Finally, it briefly mentions some common graph algorithms and questions like traversal, shortest paths, minimum spanning trees, and connectivity.
Similar to lecture 18 (20)
Graph Representation, DFS and BFS Presentation.pptx
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Graph Traversal Algorithms - Depth First Search Traversal
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Problem Solving with Algorithms and Data Structure - Graphs
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lecture 25
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lecture 21
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lecture 20
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lecture 16
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lecture 13
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lecture 12
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lecture 10
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2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
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lecture 18
- 1. CS 332: Algorithms Graph Algorithms
- 9. Breadth-First Search: Example r s t u v w x y
- 10. Breadth-First Search: Example 0 r s t u v w x y s Q:
- 11. Breadth-First Search: Example 1 0 1 r s t u v w x y w Q: r
- 12. Breadth-First Search: Example 1 0 1 2 2 r s t u v w x y r Q: t x
- 13. Breadth-First Search: Example 1 2 0 1 2 2 r s t u v w x y Q: t x v
- 14. Breadth-First Search: Example 1 2 0 1 2 2 3 r s t u v w x y Q: x v u
- 15. Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: v u y
- 16. Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: u y
- 17. Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: y
- 18. Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: Ø
- 33. DFS Example source vertex
- 34. DFS Example 1 | | | | | | | | source vertex d f
- 35. DFS Example 1 | | | | | | 2 | | source vertex d f
- 36. DFS Example 1 | | | | | 3 | 2 | | source vertex d f
- 37. DFS Example 1 | | | | | 3 | 4 2 | | source vertex d f
- 38. DFS Example 1 | | | | 5 | 3 | 4 2 | | source vertex d f
- 39. DFS Example 1 | | | | 5 | 6 3 | 4 2 | | source vertex d f
- 40. DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 | source vertex d f
- 41. DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 | source vertex d f
- 42. DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 9 | source vertex d f What is the structure of the grey vertices? What do they represent?
- 43. DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 44. DFS Example 1 | 8 |11 | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 45. DFS Example 1 |12 8 |11 | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 46. DFS Example 1 |12 8 |11 13| | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 47. DFS Example 1 |12 8 |11 13| 14| 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 48. DFS Example 1 |12 8 |11 13| 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 49. DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
- 51. DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges
- 53. DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges
- 55. DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges
- 57. DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges Cross edges