The document describes several algorithms related to graph searching and optimization problems:
1) Binary search and depth-first search algorithms for searching arrays and graphs.
2) Breadth-first search for graphs and finding shortest paths.
3) Topological sorting of directed acyclic graphs.
4) Backtracking algorithms for solving the n-queens problem and searching for Hamiltonian cycles in graphs.
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
It is a presentation on some Searching and Sorting Techniques for Computer Science.
It consists of the following techniques:
Sequential Search
Binary Search
Selection Sort
Bubble Sort
Insertion Sort
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
It is a presentation on some Searching and Sorting Techniques for Computer Science.
It consists of the following techniques:
Sequential Search
Binary Search
Selection Sort
Bubble Sort
Insertion Sort
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion
I am Charles B. I am an Algorithm Assignment Expert at programminghomeworkhelp.com. I hold a Ph.D. in Programming, Texas University, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Algorithms.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Algorithm assignments.
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion
I am Charles B. I am an Algorithm Assignment Expert at programminghomeworkhelp.com. I hold a Ph.D. in Programming, Texas University, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Algorithms.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Algorithm assignments.
Daniel Glazman, W3C CSS Working Group Chair and Web Tech Lead from Samsung OSG, discusses how CSS 3 and stylesheets will affect web standards in the future.
N-Queens Combinatorial Problem - Polyglot FP for Fun and Profit – Haskell and...Philip Schwarz
First see the problem solved using the List monad and a Scala for comprehension.
Then see the Scala program translated into Haskell, both using a do expressions and using a List comprehension.
Understand how the Scala for comprehension is desugared, and what role the withFilter function plays.
Also understand how the Haskell do expressions and List comprehension are desugared, and what role the guard function plays.
Scala code for Part 1: https://github.com/philipschwarz/n-queens-combinatorial-problem-scala-part-1
Errata: on slide 30, the resulting lists should be Haskell ones rather than Scala ones.
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
2. Algorithm 4.1.1 Binary Search This algorithm searches for the value key in the nondecreasing array L [ i ], ... , L[ j ]. If key is found, the algorithm returns an index k such that L [ k ] equals key . If key is not found, the algorithm returns -1, which is assumed not to be a valid index. Input Parameters: L , i , j , key Output Parameters: None bsearch ( L , i , j , key ) { while ( i = j ) { k = ( i + j )/2 if ( key == L [ k ]) // found return k if ( key < L [ k ]) // search first part j = k - 1 else // search second part i = k + 1 } return -1 // not found }
3. Algorithm 4.2.2 Depth-First Search This algorithm executes a depth-first search beginning at vertex start in a graph with vertices 1, ... , n and outputs the vertices in the order in which they are visited. The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ver , the vertex adjacent to i , and next , the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit[i ] is set to true if vertex i has been visited or to false if vertex i has not been visited.
4. Input Parameters: adj , start Output Parameters: None dfs ( adj , start ) { n = adj . last for i = 1 to n visit [ i ] = false dfs_recurs ( adj , start ) } dfs_recurs ( adj , start ) { println ( start ) visit [ start ] = true trav = adj [ start ] while ( trav != null) { v = trav . ver if (! visit [ v ]) dfs_recurs ( adj , v ) trav = trav . next } }
5. Algorithm 4.3.2 Breadth-First Search This algorithm executes a breadth-first search beginning at vertex start in a graph with vertices 1, ... , n and outputs the vertices in the order in which they are visited. The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ve r, the vertex adjacent to i , and next , a reference to the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit [ i ] is set to true if vertex i has been visited or to false if vertex i has not been visited. The algorithm uses an initially empty queue q to store pending current vertices. The expression q . enqueue ( val ) adds val to q . The expression q . front () returns the value at the front of q but does not remove it. The expression q . dequeue () removes the item at the front of q . The expression q . empty () returns true if q is empty or false if q is not empty.
6. Input Parameters: adj , start Output Parameters: None bfs ( adj , start ) { n = adj . last for i = 1 to n visit [ i ] = false visit [ start ] = true println ( start ) q . enqueue ( start ) // q is an initially empty queue while (! q . empty ()) { current = q . front () q . dequeue () trav = adj [ current ] while ( trav != null) { v = trav . ver if (! visit [ v ]) { visit [ v ] = true println ( v ) q . enqueue ( v ) } trav = trav . next } } }
7. Algorithm 4.3.4 Finding Shortest Path Lengths Using Breadth-First Search This algorithm finds the length of a shortest path from the start vertex start to every other vertex in a graph with vertices 1, ... , n . The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ve r, the vertex adjacent to i , and next , a reference to the next node in the linked list or null, for the last node in the linked list. In the array length , length [ i ] is set to the length of a shortest path from start to vertex i if this length has been computed or to ∞ if the length has not been computed. If there is no path from start to i , when the algorithm terminates, length [ i ] is ∞ .
8. Input Parameters: adj , start Output Parameters: length shortest_paths ( adj , start , length ) { n = adj . last for i = 1 to n length [ i ] = ∞ length [ start ] = 0 q . enqueue ( start ) // q is an initially empty queue while (! q . empty ()) { current = q . front () q . dequeue () trav = adj [ current ] while ( trav != null) { v = trav . ver if ( length [ v ] == ∞) { length [ v ] = 1 + length [ current ] q . enqueue ( v ) } trav = trav . next } } }
9. Algorithm 4.4.1 Topological Sort This algorithm computes a topological sort of a directed acyclic graph with vertices 1, ... , n . The vertices in the topological sort are stored in the array ts . The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ver , the vertex adjacent to i , and next , the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit[i ] is set to true if vertex i has been visited or to false if vertex i has not been visited.
10. Input Parameters: adj Output Parameters: ts top_sort ( adj , ts ) { n = adj . last // k is the index in ts where the next vertex is to be // stored in topological sort. k is assumed to be global. k = n for i = 1 to n visit [ i ] = false for i = 1 to n if (! visit [ v ]) top_sort_recurs ( adj , i , ts ) } top_sort_recurs ( adj , start , ts ) { visit [ start ] = true trav = adj [ start ] while ( trav != null) { v = trav . ver if (! visit [ v ]) top_sort_recurs ( adj , v , ts ) trav = trav . next } ts [ k ] = start k = k - 1 }
11. Algorithm n -Queens, Initial Version The n -queens problem is to place n queens on an n × n board so that no two queens are in the same row, column, or diagonal. Using backtracking, this algorithm outputs all solutions to this problem. We place queens successively in the columns beginning in the left column and working from top to bottom. When it is impossible to place a queen in a column, we return to the previous column and move its queen down.
12. n_queens ( n ) { rn_queens (1, n ) } rn_queens ( k , n ) { for row [ k ] = 1 to n if ( position_ok ( k , n )) if ( k == n ) { for i = 1 to n print ( row [ i ] + “ ”) println () } else rn_queens ( k + 1, n ) } position_ok ( k , n ) for i = 1 to k - 1 // abs is absolute value if ( row [ k ] == row [ i ] || abs ( row [ k ] - row [ i ]) == k - i ) return false return true }
13. Algorithm 4.5.2 Solving the n -Queens Problem Using Backtracking The n -queens problem is to place n queens on an n × n board so that no two queens are in the same row, column, or diagonal. Using backtracking, this algorithm outputs all solutions to this problem. We place queens successively in the columns beginning in the left column and working from top to bottom. When it is impossible to place a queen in a column, we return to the previous column and move its queen down.
14. The value of row [ k ] is the row where the queen in column k is placed. The algorithm begins when n_queens calls rn_queens (1, n ). When rn_queens (k, n) is called, queens have been properly placed in columns 1 through k - 1, and rn_queens ( k , n ) tries to place a queen in column k . If it is successful and k equals n , it prints a solution. If it is successful and k does not equal n , it calls rn_queens ( k + 1, n ). If it is not successful, it backtracks by returning to its caller rn_queens ( k - 1 , n ). The value of row_used [ r ] is true if a queen occupies row r and false otherwise. The value of ddiag_used [ d ] is true if a queen occupies ddiag diagonal d and false otherwise. According to the numbering system used, the queen in column k , row r , is in ddiag diagonal n - k + r . The value of udiag_used [ d ] is true if a queen occupies udiag diagonal d and false otherwise. According to the numbering system used, the queen in column k , row r , is in udiag k + r - 1. The function position_ok ( k , n ) assumes that queens have been placed in columns 1 through k - 1. It returns true if the queen in column k does not conflict with the queens in columns 1 through k - 1 or false if it does conflict.
15. Input Parameter: n Output Parameters: None n_queens ( n ) { for i = 1 to n row_used [ i ] = false for i = 1 to 2 * n - 1 ddiag_used [ i ] = udiag_used [ i ] = false rn_queens (1, n ) } ...
16. ... // When rn_queens (k, n ) is called, queens have been // properly placed in columns 1 through k - 1. rn_queens ( k , n ) { for row [ k ] = 1 to n if ( position_ok ( k , n )) row_used [ row [ k ]] = true ddiag_used [ n - k + row [ k ]] = true udiag_used [ k + row [ k ] - 1] = true if ( k == n ) { // Output a solution. Stop if only one // solution is desired. for i = 1 to n print ( row [ i ] + “ ”) println () } else rn_queens ( k + 1, n ) row_used [ row [ k ]] = false ddiag_used [ n - k + row [ k ]] = false udiag_used [ k + row [ k ] - 1] = false } ...
17. ... // position_ok ( k , n ) returns true if the queen in column k // does not conflict with the queens in columns 1 // through k - 1 or false if it does conflict. position_ok ( k , n ) return !( row_used [ row [ k ]] || ddiag_used [ n - k + row [ k ]] || udiag_used [ k + row [ k ] - 1 ]) }
18. Form of Backtracking Algorithm Suppose that we solve a problem using backtracking as in Algorithm 4.5.2 in which the solution is of the form x [1], ... , x [ n ]. Suppose also that the values of x [ i ] are in the set S (e.g., in Algorithm 4.5.2, S = {1, . . . , n }). We require a function bound ( k ) with the following property. Whenever x [1], ... , x [ k - 1] is a partial solution and x [ k ] has been assigned a value, then bound ( k ) has to return true if x [1], ... , x [ k ] is a partial solution and false otherwise. The key to writing a useful back- tracking algorithm is to write an efficient bound function that eliminates many potential nodes from the search tree.
19. The general form of a backtracking algorithm is backtrack(n) { rbacktrack(1,n) } rbacktrack(k,n) { for each x[k] S if (bound(k)) if (k == n) { // Output a solution. Stop if only for i = 1 to n // one solution is desired. print(x[i] + “ ”) println() } else rbacktrack(k + 1, n) }
20. Algorithm 4.5.4 Searching for a Hamiltonian Cycle This algorithm inputs a graph with vertices 1, ... , n . The graph is represented as an adjacency matrix adj ; adj [ i ][ j ] is true if ( i , j ) is an edge or false if it is not an edge. If the graph has a Hamiltonian cycle, the algorithm computes one such cycle ( x [1], ... , x [ n ], x[1]). If the graph has no Hamiltonian cycle, the algorithm returns false and the contents of the array x are not specified. In the array used , used [ i ] is true if i has been selected as one of the vertices in a potential Hamiltonian cycle or false if i has not been selected. The function path_ok ( adj , k , x ) assumes that ( x [1], ... , x [ k - 1]) is a path from x [1] to x [ k - 1] and that the vertices x [1], ... , x [ k - 1] are distinct. It then checks whether x [ k ] is different from each of x [1], ... , x [ k - 1] and whether ( x [ k - 1], x [ k ]) is an edge. If k = n , path_ok also checks whether ( x [ n ], x [1]) is an edge.
21. Input Parameter: adj Output Parameter: x hamilton ( adj , x ) { n = adj . last x [1] = 1 used [1] = true for i = 2 to n used [ i ] = false rhamilton ( adj ,2, x ) } rhamilton ( adj , k , x ) { n = adj . last for x [ k ] = 2 to n if ( path_ok ( adj , k , x )) { used [ x [ k ]] = true if ( k == n || rhamilton ( adj , k + 1, x )) return true used [ x [ k ]] = false } return false } ...
22. ... path_ok ( adj , k , x ) { n = adj . last if ( used [ x [ k ]]) return false if ( k < n ) return adj [ x [ k - 1]][ x [ k ]] else return adj [ x [ n - 1]][ x [ n ]] && adj [ x [1]][ x [ n ]] }