Update Eulers_Formula.mdx #4801

inmbtr · 2024-09-25T14:37:02Z

Gave Introduction and modified the examples.

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Gave Introduction and modified the examples.

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SansPapyrus683 · 2024-09-25T14:38:05Z

content/6_Advanced/Eulers_Formula.mdx

+<Resources>
+    <Resource
+        source="MIT"
+        title="    MIT Course Notes"


weird formatting

SansPapyrus683 · 2024-09-25T14:38:12Z

content/6_Advanced/Eulers_Formula.mdx

@@ -11,19 +11,215 @@ frequency: 1

 ## Introduction

-<IncompleteSection />
+A planar graph is a graph that can be drawn on a plane without any edges crossing. In other words, it can be embedded in the plane such that no two edges intersect except at their endpoints. Planar graphs can be represented in a two-dimensional space without overlaps between edges.


third sentence is redundant

SansPapyrus683 · 2024-09-25T14:38:34Z

content/6_Advanced/Eulers_Formula.mdx

+
+Euler's Formula states that any correct embedding of a connected planar graph satistfies:-
+
+         <center><h1>V − E + F = 2</h1></center>


use double dollar signs for latex

SansPapyrus683 · 2024-09-25T14:38:41Z

content/6_Advanced/Eulers_Formula.mdx

+
+         <center><h1>V − E + F = 2</h1></center>
+
+Where V is the no of vertices, E is the number of edges and F is the number of faces.


use latex for these variables

SansPapyrus683 · 2024-09-25T14:38:53Z

content/6_Advanced/Eulers_Formula.mdx

+**Memory Complexity:** $\mathcal{O}(N \log N)$
+
+```cpp
+#include "rainbow.h"


SansPapyrus683 · 2024-09-25T14:39:00Z

content/6_Advanced/Eulers_Formula.mdx

+```cpp
+#include "rainbow.h"
+#include <bits/stdc++.h>
+#define FOR(i, x, y) for (int i = x; i < y; i++)


SansPapyrus683 · 2024-09-25T14:39:07Z

content/6_Advanced/Eulers_Formula.mdx

+#define FOR(i, x, y) for (int i = x; i < y; i++)
+using namespace std;
+
+const int MAXN = 2e5, MAXSEG = (6e5 + 9) * 19 + 1;


SansPapyrus683 · 2024-09-25T14:39:25Z

content/6_Advanced/Eulers_Formula.mdx

+	int E =
+	    edges_horiz.query(ar + 1, br, ac, bc) + edges_vert.query(ar, br, ac + 1, bc);
+	int V = vertices.query(ar + 1, br, ac + 1, bc);
+	int R = rivers.query(ar, br, ac, bc);
+	int C = (ar >= mn_r || br <= mx_r || ac >= mn_c || bc <= mx_c ? 1 : 2);
+	return E - V + C - R;


use lowercase variable names

SansPapyrus683 · 2024-09-25T14:39:38Z

content/6_Advanced/Eulers_Formula.mdx


 ## Example 2

 <Problems problems="e2" />

-<IncompleteSection />
+# Explanation 


why is this h1

content/6_Advanced/Eulers_Formula.mdx

Made changes according to your advice.

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SansPapyrus683 · 2024-09-25T15:55:20Z

content/6_Advanced/Eulers_Formula.mdx

+    <center>     $$ V − E + F = 2 $$  </center>
+
+$$ V = Vertices $$
+
+$$ E = Edges $$
+
+$$ F = Faces $$


do you...check how this looks in the markdown renderer

Yeah I will change it.

Any other changes?

ryanchou-dev

thank you for your contribution!

ryanchou-dev · 2024-09-27T05:22:14Z

content/6_Advanced/Eulers_Formula.mdx

+In this problem, we're asked to count the number of contiguous areas of cells on
+several flat rectangles. Such areas are separated by river segments and
+rectangle boundaries.
+
+Where else do we count the number of areas on a flat surface?


Suggested change

In this problem, we're asked to count the number of contiguous areas of cells on

several flat rectangles. Such areas are separated by river segments and

rectangle boundaries.

Where else do we count the number of areas on a flat surface?

In this problem, we're given a 2D grid with a river segment (path) running through it. Our task is to count the number of contiguous areas of non-river cells on $Q$ subrectangles in this grid.

That's the definition of a planar graph! How can we calculate the number of segments on this plane?

wording is confusing on this part. it might be better to not restate the problem at all

ryanchou-dev · 2024-09-27T05:23:52Z

content/6_Advanced/Eulers_Formula.mdx

+Finding $E$, $V$, and $R$ is a lot more complicated though.
+
+### Finding $E$, $V$, and $R$
+
+To find $E$, $V$, and $R$, we can use a data structure that can handle 2D range
+queries efficiently.
+
+However, the coordinates of the grid can get very large, so a simple 2D BIT or
+segment tree won't work here.
+
+To work around this, we can either use a 2D BIT with coordinate compression or a
+persistent segment tree. See the sections on
+[offline 2D sum queries](/plat/2DRQ#2d-offline-sum-queries) or
+[persistent segment trees](/adv/persistent) for more details.


would be useful to tell the user why we need 2d range queries.

also you don't need to split this into another section.

ryanchou-dev · 2024-09-27T05:24:02Z

content/6_Advanced/Eulers_Formula.mdx


-<IncompleteSection />
+**Memory Complexity:** $\mathcal{O}(N \log N)$


Suggested change

**Memory Complexity:** $\mathcal{O}(N \log N)$

ryanchou-dev · 2024-09-27T05:24:18Z

content/6_Advanced/Eulers_Formula.mdx

+
+```cpp
+
+const int MAXN = 2e5, MAXSEGMENT = (6e5 + 9) * 19 + 1;


separate with underscore: MAX_N

ryanchou-dev · 2024-09-27T05:24:51Z

content/6_Advanced/Eulers_Formula.mdx

+
+int cnt = 1, segtree[MAXSEGMENT], left_c[MAXSEGMENT], right_c[MAXSEGMENT];
+
+struct Segtree {


can you use the one from the module & codesnip it?

ryanchou-dev · 2024-09-27T05:26:10Z

content/6_Advanced/Eulers_Formula.mdx

+
+**Time Complexity:** $\mathcal{O}(N^2 \log N)$
+
+**Memory Complexity:** $\mathcal{O}(N^2)$

 ```cpp


ok this entire code block needs to be rewritten -- we want the code to be as readable /easy-to-understand as possible. so this means removing templates/macros

Update Eulers_Formula.mdx

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Update Eulers_Formula.mdx

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Update Eulers_Formula.mdx #4801

Update Eulers_Formula.mdx #4801

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		Euler's Formula states that any correct embedding of a connected planar graph satistfies:-

		<center><h1>V − E + F = 2</h1></center>


		<center><h1>V − E + F = 2</h1></center>

		Where V is the no of vertices, E is the number of edges and F is the number of faces.


		<IncompleteSection />
		Memory Complexity: $\mathcal{O}(N \log N)$


		```cpp

		const int MAXN = 2e5, MAXSEGMENT = (6e5 + 9) * 19 + 1;


		int cnt = 1, segtree[MAXSEGMENT], left_c[MAXSEGMENT], right_c[MAXSEGMENT];

		struct Segtree {

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