Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding the Inductive Formulation of Grid Paths
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Welcome, everyone! Today we are discussing how to find paths in a grid. Let’s consider reaching a point denoted as (i, j). Can anyone tell me how we might get there?
Would we go right or up from a previous point?
Exactly! We can arrive at (i, j) either from the left at (i, j-1) or from below at (i-1, j). This leads us to our inductive formulation. Can anyone summarize that?
The number of paths to (i, j) equals the number of paths to its neighbors, which are (i-1, j) and (i, j-1).
Very good! It's essential to remember this formula for defining paths as it builds the foundation for further calculations. The formula is `paths(i, j) = paths(i-1, j) + paths(i, j-1)`.
Exploring Boundary Conditions
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now, let’s discuss boundary conditions. How do we compute paths when we are at the edge of the grid?
I think paths can only come from one direction at the edges, right?
Correct! In the leftmost column at (i, 0), we can only come from (i-1, 0), while in the bottom row at (0, j), movement is only from (0, j-1). What happens at the starting point (0, 0)?
There’s only one path because we just stay there!
Exactly! So the value at the origin is one path, and this is critical when calculating potential paths in the grid.
Handling Holes in the Grid
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Moving on, what happens if we encounter holes in our grid?
Those cells would have zero paths contributing to them, right?
Exactly! If there’s a hole at (i, j), it gets assigned a value of 0, meaning we ignore paths coming from neighbors. Let’s see how this propagates.
So if there’s a hole below a point, that point won’t count paths coming from below?
Right! The propagating nature of zero means any further calculations will account for these limitations. Let's summarize this key point: holes directly influence neighboring calculations.
Memoization and Dynamic Programming
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Lastly, let's differentiate between memoization and dynamic programming. Can anyone explain how they differ?
Memoization stores results of expensive function calls, while dynamic programming builds up solutions iteratively.
Exactly! Memoization is top-down, and it only computes the value as needed by storing previously evaluated values. Dynamic programming, on the other hand, computes all subproblems using a systematic order.
Is one more efficient than the other?
Well, generally, dynamic programming can be more efficient overall, especially in large problems, as it ensures all subproblems are addressed. Just keep in mind the trade-off between memory use and computation speed in your design.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section delves into the inductive formulation of grid path calculations, explaining how paths can be determined through recursion. It discusses initial conditions, boundary conditions, and the distinction between memoization and dynamic programming in managing computational efficiency.
Detailed
Initial Conditions
This section discusses the inductive formulation of calculating paths in a grid while focusing on the implications of initial and boundary conditions. It begins by framing the problem of reaching a point (i, j) in a grid, highlighting that the only possible movements are to the right or upward. From this, it establishes an inductive relationship where the number of paths to (i, j) is computed as the sum of paths to its left and below neighbors: specifically, paths(i, j) = paths(i-1, j) + paths(i, j-1).
However, special attention is given to boundary conditions, particularly at the edges and corners of the grid, where paths can only originate from one direction. For instance, at the leftmost column and bottom row, the paths are determined primarily from their respective adjacent cells, starting from the origin (0, 0) which has a singular path unto itself.
Furthermore, the treatment of obstacles (or holes) in paths is explained, noting that a cell with a hole will register zero paths, effectively propagating this limitation to neighboring cells. This ensures that blocked paths are appropriately ignored in calculations. The section also addresses the potential inefficiency of recursive calculations when dealing with overlapping subproblems, introducing memoization and dynamic programming as strategies to optimize this computation. The discussion details how to construct a Directed Acyclic Graph (DAG) to visualize dependencies between grid cells, and demonstrates how a systematic filling approach—whether row-wise or column-wise—can yield the correct number of paths efficiently. In conclusion, the section emphasizes that while dynamic programming is more comprehensive in evaluating all potential subproblems, it is also somewhat less efficiently used in cases with significant holes compared to memoization, thus presenting a trade-off in computational strategies.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Grid Paths
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, let us look for a better solution. So, as you might guess, since we are looking at these kind of inductive formulations and recursive programs, what we are really looking for is the inductive formulation of the grid path. So, let us ask ourselves the following question. How do I get to a point (i,j) on the grid? So, I claim, that given our structure, which is, that we can go right or we can go up, there are only two ways I can come here. I can either come right from the neighbor on my left. So, from (i,j - 1) I can make one step right and come to (i,j) or from (i - 1, j), I can go up once there. So, if I have any paths, which starts at (0,0), right, any path, which somehow comes to (i,j -1), then by taking one, exactly one step, that path becomes one path to (i,j) right. So, every path from (0,0) to i minus, (i,j -1) can be extended in a unique way to (i,j). Likewise, any path, which comes from (0,0) to (i-1,j) can be extended in the unique way, right. So, if I count the task coming to the two neighbours, then each of those paths can be extended to reach the current node.
Detailed Explanation
In this chunk, we explore how to analyze paths on a grid. There are two allowed movements: right and up. To understand how to get to a point (i, j), we recognize that we can arrive there either from the left neighbor at (i, j - 1) or from the below neighbor at (i - 1, j). Each path extending into (i, j) comes from these two points. Thus, the total paths to (i, j) can be calculated by considering all paths arriving from these neighbors. This understanding sets up a foundation for recursive and inductive reasoning which will be used to calculate routes through a grid.
Examples & Analogies
Imagine navigating through a city grid. You can move east or north, and you're trying to figure out how many different routes you can take from your starting position (0,0) to reach a specific destination, let's say (3,2). You can visualize each position on the grid as an intersection and each way of reaching it as a different route. To get to your destination, you'd count all the routes available from the intersections just before yours.
Establishing Path Definitions
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, let us write paths (i,j) to denote the number of paths from (0,0) to the current point (i,j). So, what we have just seen from our inductive analysis on the problem is, that if you are at (i,j), then the paths come from left or from below. So, the paths to (i,j) are the sum of paths to (i -1,j) and the paths to (i,j -1). So, there are, of course, some boundary conditions. So, if you look at our grid, in general, then if we look at the leftmost column, let us start the bottom row. So, we start the bottom row, right, then we know, that this is of the form (0,0), then (1,0) and so on to (5,0), right. So, anything of the form (i,0) derives its value only from the left because there is no corresponding row from the left.
Detailed Explanation
In this section, we define paths(i, j) as the total number of ways to reach point (i, j) from the start point (0, 0). Using the inductive analysis, we see that paths to (i, j) can be computed by summing the paths that arrive from the point directly below it (i - 1, j) and the point directly to the left (i, j - 1). This formula will be used to calculate paths throughout the grid, taking into account that the edges of the grid have specific conditions, such as the leftmost column only being reachable from the left.
Examples & Analogies
Think of this as building pathways in a community park. Each point represents a spot in the park, and you can only walk from one spot to another either straight from your left or below. The number of paths to reach a picnic area (i,j) depends on how many paths lead to the spot directly beside it and the spot directly below it. If a path leads nowhere, it's like having a dead end – it contributes zero toward reaching the picnic area.
Initial Conditions in Grid Paths
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
And now, there is nothing from the left. So, I can only get it from j-1 from the row below. And finally, you have to ask ourselves what happens at the initial conditions. So, if I am at (0,0) and I want to go to (0,0), how many ways are there? Well, this is a trivial path, there is only one way, by just staying there. It is important that it is not 0 because remember, that if it is 0, then everywhere you are just adding things and nothing will come, you will get no paths.
Detailed Explanation
In this part, we discuss the initial condition at the starting point (0, 0). The only way to stay put is to simply remain where you are, which means there's exactly one way to be at this point. This is crucial because if we incorrectly assigned this to be zero, it would disrupt the calculations across the entire grid, leading to no valid paths. This initial condition thus serves as the foundation upon which further calculations will be built.
Examples & Analogies
Imagine a game where you're standing still at your starting line, and the only option is to remain there. You cannot move in any direction, but this 'no movement' state is still considered a valid position – there is just one way to be there: to stay. If we consider that there could be zero ways to be at the start, it would feel like you never truly began the game.
Handling Holes in the Grid
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, how do we deal with holes in this setup? That is easy, we just say, that whenever there is a hole at a given point we just declare parts of paths(i,j) to be 0. In other words, if there is a hole at some point, then this thing is going to contribute 0 regardless of what is above or below.
Detailed Explanation
In this section, we address how to manage 'holes' or unavailable paths in our grid. If a specific point (i, j) is considered a hole, it will contribute 0 possible paths. This means that despite the paths that might lead to this point from its neighbors (both above and side), it cannot be traversed, and thus should not be counted toward the total. This rule will propagate through the grid and impact how we count paths leading to other points potentially still reachable.
Examples & Analogies
Consider a construction area in a park with blocked walkways where workers are doing maintenance. If you encounter one of these areas, you can't proceed through it. No matter how many paths exist leading to this point, since it’s blocked, all paths leading to it effectively count as 0 because you can’t walk through it.
Dynamic Programming vs. Recursion
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The difficulty with this calculation is the same as involved with the Fibonacci, which is, that if I have a particular calculation, say for example, supposing we start a recursive calculation at (5,10), then this will ask for (4,10) and (5,9). Now, these in turn will both ask for (4,9), so (4,9). if I just evaluate this paths function recursively, the way it is returning, the inductive definition, it will end up calling paths of (4,9) at least twice from this and this wasteful recomputation will occur throughout and we will get an exponentiation number of calls to paths.
Detailed Explanation
This chunk introduces the inefficiencies associated with purely recursive approaches, specifically illustrating how paths can be redundantly recalculated. For example, if we recursively find the number of paths to (5, 10), we may end up recalculating paths to (4, 9) multiple times, leading to an exponentially growing number of calculations. This reiterates the need for optimization techniques in managing computational complexity, such as dynamic programming methods.
Examples & Analogies
Imagine you're trying to calculate the total number of routes to your friend's birthday party (5, 10), but as you explore all paths, you realize you keep getting stuck at the same traffic intersections (4, 9). Each time you hit that intersection, you remember how many ways led to it. Instead of counting it anew each time, you should jot down that number – this will save time and effort on your journey.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Paths Calculation: The method of determining how many different ways one can reach a specific point in a grid.
-
Boundary Conditions: Conditions that define how paths can originate along the edges of the grid.
-
Holes: Points in the grid that block paths and must be accounted for in calculations.
-
Memoization vs. Dynamic Programming: Two approaches to optimize path calculations where memoization saves previously computed results and dynamic programming iteratively computes all values.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
To calculate paths to point (3,2) in a 5x5 grid, you can sum paths from (2,2) and (3,1). If the grid has a hole at (2,2), the number of paths to (3,2) would then come solely from (3,1).
-
In a grid with a hole at point (2,4) and (4,4), the paths to (3,4) must consider that the computation path to (2,4) is zero.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Holes in the grid, paths that won't fit, zero the count, don't throw a fit!
📖 Fascinating Stories
-
Imagine paths in a maze where certain spaces are blocked. As you navigate, you count ways to the exit, but if paths encounter obstructions, you simply ignore them, remembering to count only valid routes.
🧠 Other Memory Gems
-
Remember: P = L + B for Paths. (P = Paths, L = Left Neighbor, B = Below Neighbor)
🎯 Super Acronyms
BHP - remember to check Boundary conditions, handle Holes, and formulate Paths.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Inductive Formulation
Definition:
A method of defining functions or sequences based on the concept that if the function holds for certain conditions, it can be proven to hold for the next step.
-
Term: Boundary Conditions
Definition:
Conditions at the edges of a grid that affect path calculations, determining how many paths can originate from these locations.
-
Term: Memoization
Definition:
An optimization technique where results of expensive function calls are stored, allowing for faster retrieval on repeated calls.
-
Term: Dynamic Programming
Definition:
A method for solving complex problems by breaking them down into simpler subproblems and solving those first, storing their results.
-
Term: Holes
Definition:
Obstacles in a grid path that prevent new paths from being counted, resulting in a value of zero for those points.