Recurrence Relation - 4

Duration: 19 min

This video lesson is available to enrolled students.

Return to /learn/UPPSC-POLYTECHNIC-LECTURER-2025-CS/paper-1-design-and-analysis-of-algorithms/time-complexity-analysis/recursion-tree-method/asset-recurrence-relation-4-20114 after enrolling

Inside: a video lesson and guided study material.

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, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, वर्णमाला, समास, संधि, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, मुहावरे और लोकोक्तियाँ, अपठित गद्यांश और अपठित पद्यांश, Multi-Topic Content (Cross-Concept Material), शब्द-भंडार, वर्ण-विचार, हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, अशुद्धि-शोधन, शब्द-रचना, शब्द-भेद, हिन्दी की रूप रचना, रूप-रचना, 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: 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
  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: Permutation and Combination, 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, 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 to determine its time complexity using the recursion tree method. The problem presented is T(n) = 1 if n <= 1, and T(n/3) + T(2n/3) + O(n) if n > 1. The instructor systematically breaks down the recurrence by identifying two distinct subproblems: T(n/3) as the 'First Subproblem' and T(2n/3) as the 'Second Subproblem'. The core of the analysis involves constructing a recursion tree to visualize how the problem size reduces at each level. The instructor demonstrates that while the subproblem sizes are uneven, the total work done at each level sums to n. By analyzing the depth of the tree and the number of leaves, he derives that the total time complexity is O(n log n).

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The lecture begins with the instructor introducing a specific recurrence relation problem asking for its time complexity. The equation is displayed on screen as T(n) = 1 if n <= 1, and T(n/3) + T(2n/3) + O(n) if n > 1. The instructor points to the equation while explaining the problem setup, specifically highlighting the recursive step involving terms like n/3 and 2n/3. He breaks down the recursive step T(n/3) + T(2n/3) + O(n) into two distinct subproblems, labeling the term T(n/3) as the 'First Subproblem' and T(2n/3) as the 'Second Subproblem'. This indicates a divide-and-conquer approach where the problem is split unevenly, setting the stage for visualizing the recursion tree.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins the core analysis by drawing a recursion tree to visualize the recurrence relation. He starts with the root node T(n) and branches out to show the two subproblems defined in the recurrence: T(n/3) and T(2n/3). The visual progression shows the expansion of these terms, demonstrating how n becomes 9n/27 in subsequent levels. The instructor explicitly writes out the expansion steps, such as (n/3)/3 = n/9 and 2n/3^2, to show the mathematical reduction of problem size. He continues to label subproblems as 'First Subproblem' and 'Second Subproblem' on the branches, ensuring students can track the distinct recursive calls at each level of the tree structure.

  3. 5:00 – 10:00 05:00-10:00

    The analysis deepens as the instructor calculates the cost at each level of the recursion tree. He demonstrates that the sum of work done at the first and second levels equals n, specifically showing (n/3) + (2n/3) = n. He generalizes this pattern to subsequent levels, indicating that the total cost is related to the number of levels multiplied by n. The instructor writes out terms like T(n/3^2) and T(2n/3^2), showing the expansion to deeper levels. He emphasizes that despite the uneven split, the work per level remains constant at n, which is a critical observation for determining the overall complexity. The tree expansion continues to deeper levels, with the instructor calculating terms like T(n/3^k) and T(2n/3^k).

  4. 10:00 – 15:00 10:00-15:00

    The instructor focuses on the pattern of problem size reduction and the generalization for level i. He writes out the inequality T(n/3^(k-1)) < n to describe the state of nodes at deeper levels. The visual progression shows the tree expanding with terms like T(n/3^i) and T(2n/3^i). He calculates the cost at specific levels, reinforcing that the sum of terms at each level consistently equals n. The instructor points to specific nodes in the tree and writes out calculations for subproblem sizes, ensuring students understand how the problem size diminishes. He identifies the pattern for general level i, showing that the work done is proportional to n at every depth of the recursion tree.

  5. 15:00 – 19:24 15:00-19:24

    The lecture concludes with the analysis of leaf nodes to determine the total cost. The instructor sets up an inequality involving the number of leaves and the problem size at the bottom level, specifically checking if (n/3^k) <= 1. Through logarithmic manipulation, he derives the relationship between k and log n, showing that 3^k >= n leads to k = log_3 n. He concludes that the total cost TC is approximately kn, which translates to O(n log n). The instructor applies logarithm properties to solve for k and relates it back to the original problem size n. The final derivation on screen shows log(3^k) <= log(n), confirming the time complexity is O(n log n).

The lecture provides a comprehensive walkthrough of solving an uneven divide-and-conquer recurrence relation using the recursion tree method. The central concept is that even when subproblems are of unequal size (n/3 and 2n/3), the total work per level can remain constant. The instructor methodically constructs the tree, starting from T(n) and expanding to T(n/3) and T(2n/3). A key insight is the calculation of work at each level, where (n/3) + (2n/3) sums to n. This pattern holds for all levels until the base case is reached. The depth of the tree, determined by solving n/3^k = 1 for k, yields log_3 n. Multiplying the work per level (n) by the number of levels (log_3 n) results in a total time complexity of O(n log n). This example illustrates that the Master Theorem's standard form does not apply directly, necessitating the recursion tree method for verification.

Loading lesson…