single source shortest path practice question

Duration: 4 min

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Module outline

  1. HPSC Interview Preparation: Interview Preparation
  2. Python Language: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis, DSA using Python
  3. Data Structure & Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Sorting Algorithms, Minimum Spanning Trees, Time Complexity Analysis, Graph Theory, Shortest Path Algos
  4. Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Security
  5. Computer System & Organisation: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  6. Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, SQL, File Organization & Indexing, Transaction Management, Concurrency Control
  7. Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling
  8. Digital Logic & Circuits: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  9. Theory Of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Pushdown Automata & CFG, Turing Machines, Moore & Mealy Machines
  10. 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. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Data Warehousing & Mining
  12. Introduction to Emerging Trends: Emerging Tech Trends
  13. Digital Society & Cyber Law: Society, Law & Ethics
  14. HPSC PGT Previous Year Paper:
  15. HPSC Assistant Professor Previous Year Paper:
  16. Live Sessions: HPSC Live Sessions, EMRS TGT/PGT Live Class
  17. Full Mock Test:
  18. HPSC PGT - MOCK Test Series:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video addresses a theoretical question in graph theory concerning the stability of shortest paths under uniform weight modification. The instructor begins by presenting the problem statement displayed on the screen: "If we increase weight of every edge by 1, the shortest path always remains same." He asks if this statement is true or false. To investigate this, he constructs a counterexample on the whiteboard. He writes "ex." and draws a graph with a source node 'S' on the left and a destination node 't' on the right. He creates two distinct paths between them. The first path, S -> A -> B -> C -> D -> t, consists of 5 edges with weights 2, 4, 5, 6, and 7 respectively, drawn in the upper section. The second path, S -> E -> t, consists of 2 edges with weights 10 and 15, drawn in the lower section.

Initially, the total weight of the first path is calculated as 2 + 4 + 5 + 6 + 7 = 24. The total weight of the second path is 10 + 15 = 25. Since 24 is less than 25, the first path is the shortest path. The instructor then applies the condition of the problem: increasing every edge weight by 1. The new weights for the first path become 3, 5, 6, 7, and 8, summing to 29. The new weights for the second path become 11 and 16, summing to 27. Comparing the new totals, 27 is now less than 29. This demonstrates that the shortest path has shifted from the first path to the second path. Consequently, the instructor concludes that the statement is false, marking "No" as the correct answer. He circles the "No" option to emphasize the conclusion.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the problem statement regarding shortest paths and edge weight modification. He begins drawing a graph with source 'S' and destination 't', creating a multi-edge path S->A->B->C->D->t and a two-edge path S->E->t. He writes down the initial weights for each edge: 2, 4, 5, 6, 7 for the first path and 10, 15 for the second.

  2. 2:00 – 4:21 02:00-04:21

    The instructor calculates the initial total weights (24 vs 25) to establish the baseline shortest path. He then applies the transformation of adding 1 to every edge weight. He writes the new weights (3, 5, 6, 7, 8 and 11, 16) and calculates the new totals (29 vs 27). He demonstrates that the path with fewer edges becomes shorter, proving the statement false and selecting option (B) No.

The lesson demonstrates a critical property of shortest path algorithms: they are sensitive to uniform weight shifts if paths have different numbers of edges. By adding a constant to every edge, paths with more edges accumulate a larger total increase than paths with fewer edges. This example highlights that shortest paths are not invariant under such transformations, a key concept for understanding algorithm behavior in weighted graphs.

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