Design & Analysis of Algorithms - Vol 3 | 2. Inductive Formulation of the Grid Path by Abraham | Learn Smarter
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2. Inductive Formulation of the Grid Path

The chapter focuses on dynamic programming and its application in solving grid path problems. It explains how paths from the origin to various points can be calculated using recursive relations, while also addressing the concepts of memoization and the significance of handling obstacles or holes in the paths. The distinction between recursive calculations and dynamic programming is highlighted alongside practical examples.

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Sections

  • 2.1

    Inductive Formulation Of The Grid Path

    This section introduces the inductive formulation for finding the number of paths in a grid, outlining how paths are derived based on previous points and boundary conditions.

    Learning Practice

  • 2.1.1

    Paths To (I,j)

    This section explores the inductive formulation of paths on a grid, detailing how to traverse from the origin to a point (i,j) and the implications of obstacles in this path.

    Learning Practice

  • 2.1.2

    Boundary Conditions

    This section discusses the concept of boundary conditions in grid path problems, introducing inductive formulations and dynamic programming strategies.

    Learning Practice

  • 2.1.3

    Initial Conditions

    This section explores how to calculate paths in a grid using recursive and inductive methods, emphasizing the role of initial conditions and boundary conditions.

    Learning Practice

  • 2.1.4

    Handling Holes

    This section discusses how to calculate paths in a grid, taking into account obstacles (or 'holes') that restrict movement.

    Learning Practice

  • 2.1.5

    Challenges With Recursion

    This section discusses the challenges faced when using recursion in grid path calculations and introduces solutions like memoization and dynamic programming to resolve these issues.

    Learning Practice

  • 2.2

    Using Dynamic Programming On The Grid

    This section explores dynamic programming, focusing on how to calculate paths in a grid using an inductive approach and accounting for obstacles.

    Learning Practice

  • 2.2.1

    Dag Structure Of Dependencies

    The section introduces the inductive formulation for navigating a grid and extends this concept to account for obstructions, using dynamic programming techniques.

    Learning Practice

  • 2.2.2

    Row By Row Computation

    This section discusses the inductive formulation of grid paths and the application of dynamic programming to efficiently compute the number of paths from the bottom-left to the top-right of a grid.

    Learning Practice

  • 2.2.3

    Handling Holes In Computation

    This section discusses how to calculate paths in a grid with obstacles and explores the concepts of dynamic programming and memoization.

    Learning Practice

  • 2.2.4

    Column By Column Computation

    The section discusses the computation of grid paths using dynamic programming, focusing on inductive formulations and techniques to efficiently handle obstacles.

    Learning Practice

  • 2.2.5

    Topological Sorting

    This section explores the concept of topological sorting as it relates to grid path calculations and dynamic programming.

    Learning Practice

  • 2.3

    Illustration Of Memoization Vs Dynamic Programming

    This section explores the differences between memoization and dynamic programming through the context of grid paths.

    Learning Practice

  • 2.3.1

    Effect Of Holes On Computation

    This section examines how placing holes within a grid affects the computation of paths and introduces concepts of inductive reasoning and dynamic programming.

    Learning Practice

  • 2.3.2

    Efficiency Of Memoization

    Memoization optimally solves recursive problems by storing previously computed values, thereby reducing redundant calculations.

    Learning Practice

  • 2.3.3

    Conclusion On Dynamic Programming

    This section summarizes the concepts of dynamic programming, particularly focusing on calculating paths in a grid while considering obstacles.

    Learning Practice

References

ch46 part b.pdf

Class Notes

Memorization

What we have learnt

  • Paths to any point in a gri...
  • Memoization avoids redundan...
  • Handling obstacles in a pat...

Final Test

Revision Tests