Recurrence Relation - 5

Duration: 15 min

This video lesson is available to enrolled students.

Return to /learn/RSSB-SENIOR-COMPUTER-INSTRUCTOR/algorithms/time-complexity-analysis/substitution-method/asset-recurrence-relation-5 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 focuses on analyzing the time and space complexity of a specific recursive algorithm using recurrence relations. The instructor begins by presenting an algorithm that returns 1 if n equals 1, otherwise it recursively calls itself with n/2 and adds n to the result. The core of the lesson involves deriving mathematical recurrence relations for both time complexity T(n) and space complexity V(n). The instructor employs the substitution method to expand these relations, identifying patterns across iterations. Key derivations include establishing base cases where n=1 takes constant time and space, followed by recursive steps involving division of the input size. The analysis culminates in determining that while space complexity is logarithmic, time complexity follows a linearithmic pattern O(n log n) due to the accumulation of work at each level of recursion.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces a recursive algorithm problem asking for its time and space complexity. He begins to set up the recurrence relation by writing T(n) on the board, indicating the start of a mathematical analysis for the algorithm's efficiency. The visible code shows Algorithm rec(n) with a base case if(n == 1) return 1; and an else block returning (2*rec(n/2)+n). The instructor explicitly writes T(n) = { 1 if n=1, T(n/2)+1 if n>1 } to formalize the time complexity analysis.

  2. 2:00 – 5:00 02:00-05:00

    The instructor shifts focus to space complexity, writing V(n) on the board. He establishes a base case of 1 for n=1 and begins writing the recursive step, indicating a space cost of 2 for the else block. The board displays V(n) = { 1 if m=1, ... } alongside the time complexity recurrence. The instructor points to specific parts of the code, specifically the recursive call and the addition operation, to explain how they contribute to the overall space usage. The visible text includes V(n) = 2V(n/2) + n, suggesting a focus on stack depth or local variable storage.

  3. 5:00 – 10:00 05:00-10:00

    The instructor solves the recurrence relation for time complexity using the substitution method. He expands T(n/2) repeatedly to identify a pattern, showing how the constant '1' accumulates with each iteration. The process involves substituting T(n/2) into the original equation to derive terms like T(n/4), T(n/8), and so on, eventually generalizing the pattern to k iterations. The visible derivation shows T(n) = T(n/2^k) + k, leading to the conclusion that log base 2 of n is involved in the final complexity calculation.

  4. 10:00 – 14:32 10:00-14:32

    The instructor derives the total cost by summing up the work done at each level of the recursion tree, specifically focusing on the term 2^k V(n/2^k) + kn. He determines the value of k by setting n/2^k = 1, leading to k = log_2 n. Finally, he substitutes this value back into the equation to find the final time complexity of O(n log_2 n). The board shows steps like = 2^3 V(n/2^3) + n+n+n and concludes with the final complexity result.

The lecture provides a structured approach to analyzing recursive algorithms by separating time and space complexity. The instructor uses the substitution method to expand recurrence relations, demonstrating how constants accumulate over iterations. Key evidence includes the derivation of T(n) = T(n/2^k) + k and V(n) = 2V(n/2) + n. The final conclusion establishes that the algorithm has a time complexity of O(n log n), derived from summing work across recursion levels where k equals log base 2 of n. This method highlights the importance of identifying patterns in recursive steps to determine overall efficiency.

Loading lesson…