Matrix chain Multiplication Part - 3

Duration: 2 min

This video lesson is available to enrolled students.

Return to /learn/COAL-INDIA-MANAGEMENT-TRAINEE/algorithms/dynamic-programming/matrix-chain-order/asset-matrix-chain-multiplication-part-3 after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Paper-2 | Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  3. Paper-2 | Databases / DBMS & SQL: 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
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: 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 Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. Paper-2 | Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. 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
  11. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video introduces the Matrix Chain Multiplication problem, focusing on finding the optimal order of multiplication to minimize cost. The instructor explains that the goal is not to perform the multiplication but to determine the sequence that yields the lowest computational expense. He illustrates this with matrix dimension examples and scalar multiplication calculations. The lecture then details the recursive structure, defining $A_{ij}$ as the product of matrices from $i$ to $j$. He explains that any parenthesization splits the product between $A_k$ and $A_{k+1}$ for some $k$ in the range $i \le k < j$. This leads to the recursive cost function $m[i, j]$, which is the minimum of the sum of sub-product costs plus the cost of multiplying the resulting matrices.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins by clarifying that in the matrix-chain multiplication problem, the objective is not to perform the multiplication but to find the optimal order that minimizes cost. He uses visual aids showing matrix dimensions like 2x5 and 5x2, calculating the scalar multiplications as $2 imes 5 imes 2$. He then formalizes the problem structure, defining $A_{ij}$ as the matrix resulting from the product $A_i \dots A_j$. He explains that for any optimal parenthesization, the product must be split between $A_k$ and $A_{k+1}$ for some $k$ in the range $i \le k < j$. This leads to the recursive definition where the cost $m[i, j]$ is the minimum of the sum of costs of the two sub-products plus the cost of multiplying the resulting matrices. He writes the formula $m[i, j] = \min \{ m[i, k] + m[k+1, j] + p_{i-1} p_k p_j \}$ on the board.

  2. 2:00 – 2:16 02:00-02:16

    The lecture transitions to a practical example involving five matrices $A_1$ through $A_5$ with dimensions 30x35, 35x15, 15x5, 5x10, and 10x20. The instructor lists the dimension array $p$ as 30, 35, 15, 5, 10, 20. He starts solving for the minimum number of scalar multiplications using the previously derived recursive formula. He writes out the initial steps, calculating $m[1, 2]$ and $m[2, 3]$ to demonstrate how the algorithm proceeds from the base cases upwards.

The video effectively bridges the theoretical foundation of dynamic programming with a concrete application. It starts by defining the problem's goal—minimizing scalar multiplications through optimal parenthesization—and derives the recursive cost function. It then immediately applies this theory to a specific numerical example, setting up the dimension array and beginning the calculation process to find the minimum cost for a chain of five matrices. This progression from abstract definition to concrete calculation helps students understand how to implement the algorithm.

Loading lesson…