42. Grid paths
This chapter discusses the combinatorial approach to counting grid paths and addressing obstacles in a grid. It explains how to determine the number of paths from the bottom left to the top right corner by using combinatorial mathematics and dynamic programming techniques. The chapter also covers how to adapt these methods when intersections are blocked, employing techniques like memoization and inclusion-exclusion principles for more complex scenarios.
Sections
Navigate through the learning materials and practice exercises.
What we have learnt
- Grid paths can be counted using combinatorial methods by choosing horizontal and vertical moves.
- Blocked intersections complicate path counting and require subtracting invalid paths from the total count.
- Dynamic programming offers an efficient way to compute paths, avoiding redundant calculations through memoization.
Key Concepts
- -- Grid Path
- A route taken from the bottom-left corner to the top-right corner of a grid, allowing only upwards and rightward moves.
- -- Combinatorial Counting
- A mathematical approach to count possible arrangements or pathways by determining combinations of moves needed across a defined space.
- -- Memoization
- An optimization technique that stores previously computed results to avoid redundant calculations in recursive functions.
- -- Dynamic Programming
- A method for solving complex problems by breaking them down into simpler subproblems, solving each just once and storing their solutions.
- -- InclusionExclusion Principle
- A combinatorial method used to calculate the total of overlapping sets by including the size of the sets and excluding the overlaps.
Additional Learning Materials
Supplementary resources to enhance your learning experience.