Decreasing Functions

Duration: 9 min

This video lesson is available to enrolled students.

Return to /learn/RSSB-SENIOR-COMPUTER-INSTRUCTOR/algorithms/algorithm-analysis/growth-rate-comparisons/asset-decreasing-functions 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 introduces the concept of decreasing functions in algorithm analysis, defining them as functions where the value decreases as the input size n increases. The instructor explains that for decreasing functions, running time gets smaller as the problem size grows, which is counterintuitive compared to standard complexity analysis. The core educational objective is to arrange specific reciprocal and logarithmic functions in increasing order of growth as n approaches infinity. The video utilizes concrete examples, including the set {1/n, 1/n^2, 1/log n, 1/sqrt(n)}, to demonstrate how to compare these rates. The instructor employs tables to evaluate function values at specific integers (n=1, 2, 3, 4) and sketches graphs to visualize the asymptotic behavior where time approaches zero as n grows large. Key distinctions are made between terms like 1/n and 1/n^2, establishing that 1/n decreases slower than 1/n^2. The lecture emphasizes the hierarchy of decreasing functions, noting that (1/n) is asymptotically bigger than (1/n^2), and explores the relationship between polynomial reciprocals and logarithmic terms.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins by defining decreasing (decrement) functions, stating on-screen that a function f(n) is called decreasing if its value decreases as n increases. The instructor notes that for these functions, 'As input size grows, the running time keeps getting smaller.' An example problem is immediately presented to arrange four functions in increasing order of growth as n approaches infinity: 1/n, 1/n^2, 1/log n, and 1/sqrt(n). The instructor writes calculations on the right side of the screen, including arithmetic sums like '2 + 30 + 10 = 32 -> 1 sec' and '5 + 83 + 67 + 34 + 12 + 18 = 0 sec', likely illustrating timing concepts. Annotations such as 'n=3' and 'n=6' appear briefly to support the explanation of variable substitution.

  2. 2:00 – 5:00 02:00-05:00

    The instructor elaborates on the definition of decreasing functions by constructing a table for f(n) = 1/n. The visible text shows the relationship between n and 1/n, listing values: 'n | 1/1 = 1', '2 | 1/2', and '3 | 1/3'. This table demonstrates that as n increases from 1 to 4, the value of 1/n decreases. The instructor sketches a graph showing time decreasing as n approaches infinity, visually reinforcing the concept that larger inputs yield smaller running times. The focus remains on the inverse relationship between n and 1/n, preparing students to compare this behavior against other functions like 1/n^2.

  3. 5:00 – 9:02 05:00-09:02

    The lecture progresses to comparing the growth rates of reciprocal terms. The instructor fills a comparison table for 1/n and 1/n^2, calculating values for n=1 (where 1/1 = 1) and showing that for n >= 2, '1/n > 1/n^2'. This establishes that 1/n decreases slower than 1/n^2. The instructor writes the inequality '(1/n) Asymptotically bigger than (1/n^2)' on screen. The analysis extends to other terms, listing 'n', '1/n', '1/n^a', 'log n', and 'sqrt(n)'. The instructor writes '(1/n) < (1/log n)' and notes that 'log n > 1/n^0.5', boxing the final ordered list of functions to solve the initial problem.

The lecture systematically builds understanding of decreasing functions by first defining the concept, then applying it to specific examples. The teaching flow moves from abstract definition to concrete tabular evaluation, and finally to asymptotic comparison. Key evidence includes the on-screen text defining f(n) as decreasing if its value drops as n increases, and the specific inequality '(1/n) < (1/log n)' derived from comparing growth rates. The instructor uses the table method to prove that 1/n is greater than 1/n^2 for n > 1, which is crucial for ordering the functions. The visual graph of time versus n provides a geometric interpretation of the algebraic relationships. Students should note that for decreasing functions, 'asymptotically bigger' means the function value remains larger (closer to zero from above) compared to functions that decrease faster. The final ordering relies on recognizing that 1/n^2 decreases fastest, followed by 1/sqrt(n), then 1/log n, and finally 1/n decreases slowest among the given set.

Loading lesson…