Recurrence Relation - 5

Duration: 13 min

This video lesson is available to enrolled students.

Return to /learn/RSSB-SENIOR-COMPUTER-INSTRUCTOR/algorithms/time-complexity-analysis/recursion-tree-method/asset-recurrence-relation-5-20115 after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About the Course: About the Course
  2. Paper-1 & 2 | Basic Numeracy: Ratio and Proportion (Ratios), Divisibility Rules, Mixture and Alligation, Permutation and Combination, Logarithm, Number System, HCF LCM, Sequence and Series (Series), Speed Time and Distance, Percentage, Average, Mensuration and Geometry, Age Problems, Arithmetic, Statistics, Probability, Profit and Loss, Simple Interest and Compound Interest, Partnership, Speed Mathematics and Basic Mathematics, Algebra, Powers and Exponents (Surds and Indices), Time and Work, Pipe and Cistern, RSSB Specific Videos - Basic Numeracy
  3. Paper-1 & 2 | Data Interpretation: Data Interpretation
  4. Paper-1 & 2 | General Mental Ability: Mental Ability, RSSB Specific Videos - Mental Ability
  5. Paper-1 & 2 | Logical Reasoning and Analytical Ability: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Syllogisms, Seating Arrangements, Blood Relations, Directions (Direction Test), Analytical Reasoning (Counting Figures Reasoning), Puzzle Solving (Puzzles), Calendar, Clock, Cubes & Dices, Venn Diagram, Analogy, Ranking, Order and Sequence, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Data Arrangement, Logical Sequence Of Words, Crypt Arithmetic
  6. Paper-1 & 2 | Decision Making and Problem Solving: Selection Decision Table (Decision Making)
  7. Paper-1 | General Science: Physics, Biology, Chemistry, RSSB Specific Videos
  8. Paper-1 | General Awareness: State Wise GK, GK- Current Affairs
  9. Paper-2 | Fundamentals of Computer: Basics of COA, Number System, Floating Point Rep, Number Representation
  10. Paper - 2 | Digital Logic Systems: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits
  11. Paper-2 | Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Linux OS, Distributed Systems
  12. Paper-2 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, Relational Calculus, File Organization & Indexing, Transaction Management, Concurrency Control
  13. Paper-2 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  14. Paper-2 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  15. Paper 2 | Oops, C++, Java: Language Design, OOPS with C++, Java, .Net Programming, Java Fundamentals
  16. Paper 2 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  17. Paper 2 | Emerging Trends and AI: Emerging Tech Trends, Artificial Intelligence
  18. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Advanced Algorithms
  19. Paper 2 | Data Communication and Network Security: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, 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, Ethical Hacking
  20. Paper-2 I Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  21. Paper-2 | Computer Organization and Architecture: Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Control Unit Design, Register Transfer and Microoperations
  22. Paper-2 | Web Development: JavaScript-Basics, HTML Basics, CSS, XML
  23. Pedagogy: Child Development, Concept of Pedagogy, Psychology, Theories, Inclusive Education, Experiential Pedagogy, Curriculum Framework & Reforms, National Education Policy (NEP) 2020
  24. Live Sessions: RSSB Live Sessions
  25. RSSB Paper 1 - Full Mock Tests:
  26. RSSB Paper 2 - Full Mock Tests:
  27. SCI Previous Year Papers:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture segment demonstrates the application of the recursion tree method to determine the time complexity of a specific recurrence relation. The instructor systematically breaks down the problem T(n) = T(n/5) + T(3n/4) + O(n) by visualizing the recursive calls as a tree structure. The analysis focuses on calculating the work performed at each level of the tree, identifying a geometric series pattern in the summation of costs, and applying standard formulas to derive the final asymptotic complexity. The core pedagogical goal is to show how recursive structures can be flattened into summations to reveal their growth rates.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The session begins with the instructor presenting a specific recurrence relation problem: T(n) = 1 if n <= 1, and T(n/5) + T(3n/4) + O(n) if n > 1. The instructor explicitly points to the recursive terms T(n/5) and T(3n/4), underlining them to emphasize the branching nature of the problem. He then initiates a visual representation by drawing a recursion tree, placing T(n) at the root node. This initial phase establishes the problem structure and sets up the visual framework for analyzing how input size decreases at each recursive step.

  2. 2:00 – 5:00 02:00-05:00

    The instructor proceeds to expand the recursion tree, branching T(n) into its two subproblems: T(n/5) on the left and T(3n/4) on the right. He methodically calculates the problem size for subsequent levels, showing how T(n/5) further splits into T(n/25) and T(3n/20). The visual focus shifts to tracking the input size reduction, specifically simplifying fractions like (3n/4)/5 to 3n/20. This section demonstrates the mechanical process of expanding recursive calls and deriving the arguments for child nodes based on the parent node's size.

  3. 5:00 – 10:00 05:00-10:00

    The analysis shifts from structural expansion to cost calculation at each level of the tree. The instructor sums the non-recursive work O(n) contributed by nodes at each depth, simplifying terms like n/5 + 3n/4 into a single fraction (19n)/20. He continues this process for deeper levels, deriving expressions such as n(19/20)^k. The visual evidence shows the instructor writing out summation formulas and identifying a geometric progression pattern where the total work per level decreases by a factor of 19/20. This step is crucial for transitioning from the tree structure to a mathematical series.

  4. 10:00 – 12:43 10:00-12:43

    In the final phase, the instructor consolidates the level-wise costs into a summation series representing the total work done. He identifies the terms as a geometric progression with a common ratio r = 19/20, which is less than 1. Using the formula for a decreasing geometric series sum, he factors out 'n' and applies the standard summation notation to conclude that the total complexity is O(n). The board displays the final derivation, confirming that the dominant term comes from the root level due to the decreasing nature of the series.

The lecture provides a clear, step-by-step walkthrough of solving recurrence relations using the recursion tree method. The instructor effectively bridges the gap between abstract recursive definitions and concrete mathematical analysis by visualizing the problem as a tree. Key takeaways include the method for calculating subproblem sizes at each level, the technique for summing costs per depth to identify geometric series patterns, and the application of summation formulas to determine asymptotic bounds. The specific example T(n) = T(n/5) + T(3n/4) + O(n) serves as a practical demonstration of how unequal splits in recursion can still result in linear time complexity if the sum of fractions is less than one.

Loading lesson…