Recurrence Relation - 13

Duration: 8 min

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Module outline

  1. About Course: UPPSC Course Overview
  2. Interview Preparation & Guidance: Interview Preparation
  3. Paper 1 | General Hindi: भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, हिन्दी की रूप रचना, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, वर्णमाला, समास, संधि, Multi-Topic Content (Cross-Concept Material), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी शब्द रचना, हिन्दी साहित्येतिहास दर्शन, भक्तिकालीन काव्यधाराएँ, भारतीय काव्यशास्त्र की परंपरा, भारतीय नवजागरण और स्वाधीनता आंदोलन, अन्य गद्य विधाएँ एवं प्रवासी साहित्य, निर्धारित उपन्यासों का तुलनात्मक अध्ययन, गोदान, हिन्दी कहानी का विकास, स्वातंत्र्योत्तर और नयी कहानी, हिन्दी निबंध का विकास, शुक्लोत्तर निबंध, द्विवेदी और शुक्ल युग के निबंध, हिन्दी की संवैधानिक स्थिति, आधुनिक भारतीय आर्यभाषाएँ और हिन्दी का उद्भव, हिन्दी भाषा प्रयोग के विविध रूप, रीतिकाल, जीवनी, द्विवेदी युग और छायावाद, रीति, वक्रोक्ति और औचित्य सिद्धान्त, शब्दशक्ति और ध्वनि सिद्धान्त, रस सिद्धान्त और साधारणीकरण, उत्तर आधुनिकतावाद, लोहिया दर्शन, हिन्दी उपन्यास का विकास, हिन्दी नवजागरण, अंबेडकर दर्शन, आदिकाल, प्रयोगवाद और नयी कविता, छायावादी काव्य: प्रसाद और निराला, प्रगतिवाद से समकालीन कविता, निर्गुण भक्ति काव्य, द्विवेदीयुगीन काव्य, रीतिकालीन काव्य, आरंभिक हिन्दी उपन्यास, स्वातंत्र्योत्तर उपन्यास, आत्मकथा, स्वाधीनता-पूर्व प्रेमचंदोत्तर उपन्यास, निर्धारित कहानियों का तुलनात्मक अध्ययन, साठोत्तरी नाटक, प्रसाद और प्रसादोत्तर युग के नाटक, भारतेन्दु युगीन नाटक, स्वातंत्र्योत्तर नाटक, भारतेन्दु युगीन निबंध, संस्मरण और रेखाचित्र, व्यंग्य, एक साहित्यिक की डायरी, संस्कृति के चार अध्याय, सगुण भक्ति काव्य, हिन्दी नाटक, एकांकी और रंगमंच, आधुनिक गद्य और हिन्दी नवजागरण, अलंकार सिद्धान्त, हिन्दी आलोचना का विकास, भक्ति आंदोलन, कल्पना, बिम्ब, प्रतीक, मिथक और फंतासी, नयी कविता का विस्तार और साठोत्तरी कविता, उत्तर-छायावादी और प्रगतिवादी काव्य, काव्यगुण और काव्यदोष, प्लेटो, अरस्तू और पाश्चात्य परंपरा, मार्क्सवाद, स्वच्छन्दतावादी काव्यशास्त्र, आधुनिक पाश्चात्य आलोचना, भाषा और भाषा परिवार, देवनागरी लिपि
  4. Paper 2 | General Studies: State Wise GK, Polity, Geography, History, GK- Current Affairs
  5. Paper 1 | Programming Language: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++
  6. Paper 1 | Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, ORDBMS, NoSQL
  7. Paper 1 | Cyber Security: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, प्रत्यक्ष बोध / Literal Comprehension, कंप्यूटर सॉफ्टवेयर / Computer Software, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, साइबर सुरक्षा / Cyber Security, अपशिष्ट प्रबंधन / Waste Management, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, डिजिटल एवं नव माध्यम / Digital and New Media, शोध नैतिकता / Research Ethics, उभरती प्रौद्योगिकियाँ / Emerging Technologies, शिक्षा और शासन में ICT / ICT in Education and Governance, Society, Law & Ethics, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Electronic Data Interchange and E-Commerce, Physical Security and Cryptography Basics, Intrusion Detection Systems (IDS), Cyber Safety and Security, Ethical Hacking, भौगोलिक संकेतों का संरक्षण / Protection of Geographical Indications
  8. Paper 1 | Computer Organization and Architecture: Basics of COA, Programming the Basic Computer, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design
  9. Paper 1 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Security
  10. Paper 1 | Design and Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  11. Paper 1 | Data Structures: Introduction to DS, Array, Linked List, Stack, Queue, Tree, Graphs, Hashing
  12. Paper 1 | Discrete Structures and Theory of Logic: क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
  13. Paper 1 | Software Project Management: Software Project Management
  14. Paper 2 | Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Security
  15. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment, Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Pushdown Automata & CFG, Complexity Theory
  16. Paper 2 | Artificial Intelligence & High Performance Computing: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Natural Language Processing, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, Emerging Tech Trends
  17. Paper 2 | Image Processing: Digital Image Fundamentals, Image Enhancement, Image Restoration, Color Image Processing, Image Compression
  18. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  19. Paper 2 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  20. Paper 2 | Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, HTML Basics, JavaScript-Basics, Java
  21. Paper 2 | Soft Computing: Fuzzy Sets, Artificial Neural Networks, Genetic Algorithms
  22. Paper 2 | Distributed System: Distributed Systems
  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
  24. Previous Year Papers:
  25. Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture segment focuses on solving a specific recurrence relation problem to determine its asymptotic time complexity. The instructor begins by presenting the problem statement, which defines a function T(n) with a base case of 1 when n equals 1 and a recursive step defined as T(n-2) + n for all n greater than 1. The core of the lesson involves applying the iterative substitution method to expand this recurrence relation into a summation series. By repeatedly substituting the recursive term, the instructor identifies a pattern that transforms the problem into an arithmetic progression of odd numbers. The final derivation utilizes standard summation formulas to simplify the series, ultimately concluding that the time complexity of the recurrence is O(n^2).

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the problem by displaying a question asking for the time complexity of a recurrence relation. On-screen text clearly defines the function as T(n) = 1 for n=1 and T(n-2) + n for n > 1. The instructor writes the recursive step T(n) = T(n-2) + n on the screen and begins the solution process by substituting the term for T(n-2). This initial expansion shows the substitution of (n-2) into the recursive formula, resulting in T(n-4) + (n-2), demonstrating the first step of the iterative substitution method.

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues the iterative expansion of the recurrence relation to identify a general pattern. He writes out successive substitutions, showing T(n-2) = T(n-4) + (n-2), followed by T(n-4) = T(n-6) + (n-4). The instructor generalizes this process for k steps, writing the term T(n) = T(n-2k) + ... and explicitly noting the sequence of added terms as (n-2k+1). This section emphasizes recognizing the arithmetic progression within the recursive steps and setting up the equation for a general k-step expansion.

  3. 5:00 – 7:55 05:00-07:55

    The instructor completes the derivation by expanding the recurrence into a full summation series. He identifies the resulting sequence as an arithmetic progression of odd numbers: 1 + 3 + 5 + 7 + ... + (n-2) + n. Using the summation formula for odd numbers, he applies the expression n(n+1)/4 to simplify the series. The final calculation shows (n^2+n)/4, leading to the conclusion that the asymptotic time complexity is O(n^2). The instructor also notes the value of k as (n-1)/2 to determine when the base case T(1) is reached.

The lecture demonstrates a systematic approach to solving recurrence relations using the iterative substitution method. The problem T(n) = T(n-2) + n is solved by expanding the recursive term repeatedly until a pattern emerges. The key insight is recognizing that the expansion results in an arithmetic series of odd numbers, which can be summed using standard formulas. The derivation shows that the sum is proportional to n squared, confirming a quadratic time complexity of O(n^2). This method highlights the importance of pattern recognition in algorithm analysis.

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