From 96eec573066d77e2b5555fc7cc421462893e4ae5 Mon Sep 17 00:00:00 2001
From: Rupanshu Yadav <58172798+zyrch@users.noreply.github.com>
Date: Tue, 13 Oct 2020 22:05:41 +0530
Subject: [PATCH] Update fenwick_tree_2d.cpp

---
 range_queries/fenwick_tree_2d.cpp | 22 +++++++++++++++++-----
 1 file changed, 17 insertions(+), 5 deletions(-)

diff --git a/range_queries/fenwick_tree_2d.cpp b/range_queries/fenwick_tree_2d.cpp
index 233bcd375e7..6e5b83e177e 100644
--- a/range_queries/fenwick_tree_2d.cpp
+++ b/range_queries/fenwick_tree_2d.cpp
@@ -5,9 +5,11 @@
 
 template <class T>
 struct fenwick {
+    // siz - size of the fenwick tree
+    // arr - fenwick tree array
     int siz;
     std::vector<T> arr;
-    
+
     // Constructor 
     // n - size of the array on which fenwick tree is used
     // creates a new array with all entries as 0
@@ -40,13 +42,17 @@ struct fenwick {
 
     // find sum in range l to r
     T query(int l, int r) {
+	// sum l--r == sum 0--r - sum 0--(l-1)
 	return sum(r) - sum(l - 1);
     }
 };
 
+// implements 2D fenwick using a array of 1-D fenwicks
 template<class T>
 struct fenwick2D{
-
+    
+    // siz - size of the fenwick tree
+    // arr - fenwick tree array
     int siz;
     std::vector<fenwick<T>> arr;
 
@@ -65,7 +71,9 @@ struct fenwick2D{
     // position y
     void update(int x, int y, T delta) {
 	while(x < siz + 1) {
+	    // update in 1-D fenwick
 	    arr[x].update(y, delta);
+	    // add the last bit of x to x itself
 	    x += x & -x;
 	}
     }
@@ -76,12 +84,16 @@ struct fenwick2D{
 	while(x > 0) {
 	    // for each fenwick tree in range calucate sum till y
 	    an += arr[x].sum(y);
+	    // remove the last bit of x from x
 	    x -= x & -x;
 	}
 	return an;
     }
 
     // sum of region {x1 to x} and {y1 to y}
+    // sum in region x1 to x and y1 to y can written as
+    //    - sum (x, y) - sum(x1 - 1, y) - sum(x, y - 1) + sum(x1 - 1, y1 - 1)
+    // last part as is add rectangle {x1, y1} is subtracted twice
     T query(int x1, int y1, int x, int y) {
 	return sum(x, y) - sum(x1 - 1, y) - sum(x, y1 - 1) + sum(x1 - 1, y1 - 1);
     } 
@@ -91,9 +103,9 @@ struct fenwick2D{
 int main() {
 
     int n = 5;
-    
+
     fenwick2D<int> fenwick_tree_2D(n);
-    
+
     fenwick_tree_2D.update(1, 1, 2);
     fenwick_tree_2D.update(1, 2, 3);
     fenwick_tree_2D.update(2, 1, 3);
@@ -104,7 +116,7 @@ int main() {
     fenwick_tree_2D.update(2, 3, 5);
     fenwick_tree_2D.update(3, 2, 5);
     fenwick_tree_2D.update(3, 3, 6);
-    
+
     assert(fenwick_tree_2D.query(1, 1, 2, 2) == 12);
     assert(fenwick_tree_2D.query(1, 2, 3, 3) == 27);
     fenwick_tree_2D.update(1, 1, -3);