Dealing with Multiple Blocked Intersections
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Introduction to Grid Paths
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Welcome everyone! Today, we're exploring grid paths. Does anyone know what a grid path is?
Is it a way of moving on a grid?
Exactly! We can only move up or right. For example, starting from (0,0) to (5,10), how many paths can we find?
I think we can use combinations since we must make a certain number of right and up moves.
Right! We’re looking at 15 total moves: 5 right and 10 up. This leads us to calculate combinations like '15 choose 5'.
How do we calculate that?
Remember the formula: n choose k = n! / (k! * (n-k)!). So, for our example, that’s 15! / (5! * 10!).
Can you show us why this works?
Sure! Every path is a sequence of moves, and each combination of right and up gives a unique path.
In summary, understanding combinations helps us determine how many paths exist on a grid.
Blocked Intersections
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Now, what happens if we have a blocked intersection, like (2,4)?
We would need to exclude paths that cross it.
That's correct! We can find the total paths that cross this blocked intersection and subtract them.
How do we calculate the paths that go through (2,4)?
Great question! We calculate those paths from (0,0) to (2,4) and then from (2,4) to (5,10).
So we find the two counts and multiply them?
Yes! We find the paths to (2,4) as well as those from (2,4) to (5,10), then combine them.
Then we subtract that from the original count, right?
Exactly! This method lets us effectively manage blocked intersections.
To recap, we find paths separately and adjust our total by subtracting the 'bad' paths.
Inclusion-Exclusion Principle
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Now let's say we have two intersections that are blocked; how would that change our calculations?
We'd count paths that go through each blocked intersection separately?
Right! But we also need to be careful about paths that may include both intersections.
How do we deal with that?
We use the inclusion-exclusion principle. First as we add the paths corresponding to each intersection, we then need to subtract those counted twice.
So we have to keep track of overlaps?
Exactly! If we compute paths through both intersections, we need to ensure they aren't double-counted.
This seems complicated!
It can be! But this systematic approach helps us find the right path count with precision.
Summarizing, the inclusion-exclusion principle helps us manage overlapping counts effectively.
Dynamic Programming Overview
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Let's shift to dynamic programming. How does it enhance our pathfinding?
Doesn't it help avoid redundant calculations?
Exactly! In dynamic programming, we build solutions incrementally and store previously computed results.
How does that work with grid paths?
We compute paths iteratively and store those values, which prevents recalculating the same paths multiple times.
So we can create a table of paths based on our previous results?
Correct! Each cell in the table uses the values from the left and below to update its own value.
How about dealing with blocked paths?
If a cell is blocked, we set its value to 0, and it ensures paths cannot go through it.
In summary, dynamic programming efficiently computes paths while addressing constraints like blockages.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section begins by introducing grid paths, where movement is constrained to only up or right from the bottom-left corner to the top-right corner of a grid. It further explores the combinatorial solutions to calculate the number of paths and how to adapt these calculations when intersections on the grid are blocked. The section also describes the inclusion-exclusion principle and dynamic programming strategies to efficiently count valid paths.
Detailed
Dealing with Multiple Blocked Intersections
The section focuses on a grid path problem in which movements are restricted to only the upward and rightward directions. The grid spans from point (0,0) to (5,10). The total paths from the bottom-left to top-right corner can be calculated using combinations, specifically the binomial coefficient, represented as 'n choose k', applying the formula to count the distinct permutations of moves.
When intersections become blocked, such as (2,4), paths that cross this point need to be excluded from the total count. A structured combinatorial approach is employed where the paths that are affected by the blockage can be counted separately and subtracted from the total.
Moreover, the text introduces the inclusion-exclusion principle to manage scenarios with multiple blocked intersections, ensuring that paths that may cover both intersections are not double-counted. The importance of dynamic programming is highlighted as a method to optimize path calculation, particularly in the presence of blocked intersections, thereby allowing an efficient resolution of the paths while addressing overlapping conditions.
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Introduction to Blocked Intersections
Chapter 1 of 5
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Chapter Content
But the problem becomes more interesting, if we constrain it by saying that some of these intersections are blocked for instance, supposing there is some road work going on and we cannot go through this intersection (2, 4). This is the intersection 2 comma 4 second column and the fourth row counting from below. It’s actually 2 comma 3, but 1, 2, 3, 4 yeah 2 comma 4. Now, if we cannot go through this then any path which goes through this particular block intersection should no longer be counted. Out to those 3003 some paths are no longer valid paths.
Detailed Explanation
This chunk introduces the concept of blocked intersections in the context of grid paths. When we previously calculated a total of 3003 paths from the bottom left corner (0,0) to the top right corner (5,10) with no constraints, we now must consider that certain paths may not be valid if they pass through blocked intersections like (2, 4). Thus, any path that passes through this blocked point cannot contribute to our count of valid paths.
Examples & Analogies
Imagine a city where certain roads are closed for construction. If you usually take a certain route that includes these roads, you must find alternative paths to reach your destination. This is similar to how we must now consider alternative routes in the grid problem when certain intersections are blocked.
Counting Bad Paths
Chapter 2 of 5
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For instance, in the earlier thing the blue path that we had drawn actually goes through this, the red path does not, where the yellow path overlapped with the blue path unfortunately in this bad section. It also passes through this. There are some paths which are allowed from the 3003 and some which are not. So, how do we determine how many paths survived this kind of block. So, again we can use a combinatorial argument in order to be blocked a path must go to (2, 4) and then from (2, 4) to (5, 5). If we could only count how many paths go from (0, 0) to (2, 4) and then how many paths go from (2, 4) to (5, 10), these are all the bad paths.
Detailed Explanation
The explanation here revolves around identifying paths that pass through the blocked intersection. To find out how many paths are 'bad' (i.e., paths that don't count because they pass through (2, 4)), we can use a combinatorial approach: Count how many ways there are to get from the starting point (0, 0) to (2, 4), and then how many ways from (2, 4) to the endpoint (5, 10). The total number of paths going through the blocked intersection can then be found by multiplying these two counts.
Examples & Analogies
Think of this as having to count the number of ways to reach a destination when a certain building in your path is closed. First, you find out how many ways you can get to the building, and then from there, how you can reach your ultimate destination. The total ways that go through the closed building are the paths you need to deduct from the total.
Inclusion-Exclusion Principle
Chapter 3 of 5
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Now, what happens if we put 2 such intersections? So, we will you can do the same thing we can count all the paths which get blocked because of the first intersection, we can count all the paths which pass through in this case (4, 4) is the second intersection which has been blocked. So, we can count all these paths which pass through (4, 4). This we know how to do: we just computed it for (2, 4), but the problem is that there are some paths like the yellow paths which pass through both (2, 4) and (4, 4). So, we need a third count we need to count paths which pass through both of these and make sure we do not double count them.
Detailed Explanation
This chunk introduces the complexity of having multiple blocked intersections on the grid. When there are two blocked intersections, the approach is to count the paths that pass through the first blocked intersection, the paths through the second, and then subtract the paths that have been double-counted because they go through both intersections. This is a direct application of the inclusion-exclusion principle in combinatorics, where we account for overlaps in counts.
Examples & Analogies
Consider a road map where two routes are both being worked on. If you would normally count routes going through each construction point to get to your destination, you must also ensure that you do not count any route that goes through both constructions twice.
Using the Recursive Formula
Chapter 4 of 5
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Let us look at the inductive structure of the problem, suppose we say we want to get in one step to intersection (i, j). How can we reach this in one step since our roads only go left to right and bottom to top, the only way we can reach (i, j) is by taking a right edge from its left neighbor. So, we can go from (i-1, j) to (i, j) or we can go from below from (i, j-1). Notice that if a path comes from the left it must be different from a path that comes from below. So, we can just add these up.
Detailed Explanation
This chunk lays out the inductive approach to counting paths. It asserts that to calculate the number of paths to a point (i, j), we can only arrive there from the points directly left (i-1, j) and below (i, j-1). This gives us a recursive formula: paths(i,j) = paths(i-1, j) + paths(i, j-1). This means that the total number of paths to a certain intersection can be derived from the sum of the paths leading into that intersection from both the left and the bottom.
Examples & Analogies
Imagine navigating through a maze with only two ways to get into a room: from the left or from below. The number of ways you can reach that room is simply the total of the different ways you can get to the left room plus those to the below room.
Handling Blockages with Paths Calculation
Chapter 5 of 5
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This gives us a direct way to actually compute this even with holes because, the only difference now is that if, there is a hole we just declare that no paths can reach that place. So, we just add an extra clause which says paths(i, j) is 0 if there is a hole at (i,j); otherwise we use exactly the same inductive formulation and now what happens is, if I have a hole below me, if I have a hole below me, no paths can come from that direction because by definition paths(i, j) at that point is 0.
Detailed Explanation
In this part, we learn how to integrate the concept of 'holes' or 'blocked intersections' into our path-counting framework. We simply stipulate that if a location (i, j) is blocked, the number of paths reaching that location is zero. This means that when evaluating paths, if an intersection is blocked, it has no paths leading into it, effectively eliminating that route from our calculations.
Examples & Analogies
Consider a game where players can only move on certain tiles. If a tile is blocked, players cannot step on it. Consequently, you're left to find routes that navigate around those obstacles while counting valid moves.
Key Concepts
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Grid Paths: A structured way to navigate through a grid using specified directions.
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Blocked Intersections: Points within the grid that restrict movement and open up complications in path counting.
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Combinatorial Counting: Utilizing combinations to count possible paths efficiently.
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Dynamic Programming: A technique for optimizing the path calculation process to avoid redundant work.
Examples & Applications
To go from (0,0) to (5,10), one can visualize the possible routes as combinations of 10 upward and 5 right movements.
If (2,4) is blocked, we find how many paths lead through it by calculating paths from (0,0) to (2,4) and (2,4) to (5,10) then subtracting those from the total.
Memory Aids
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Rhymes
In a grid we move with flair, right or up, here and there. Count the paths, don't you fuss, block the points, but count with trust.
Stories
Imagine two builders trying to reach a garden on a grid path. They each bring their own tools but one path has a hole! They must find different ways to bypass it, adjusting how they calculate their route together.
Memory Tools
Grid paths can be counted as 'RU' for Right and Up moves.
Acronyms
BIP - Blocked Intersection Principle, helps you remember to adjust for paths in grids.
Flash Cards
Glossary
- Grid Path
A path from the bottom-left corner to the top-right corner of a grid, constrained to moving only up or right.
- Combinatorial Solution
An approach that uses combinations to determine the number of possible paths without enumeration.
- Blocked Intersection
A point on the grid that cannot be traversed, affecting the total path count.
- InclusionExclusion Principle
A combinatorial method for calculating the size of a union of overlapping sets.
- Dynamic Programming
A programming paradigm that solves problems by breaking them down into simpler subproblems and storing results to avoid repetition.
Reference links
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