Merge pull request #4558 from brebenelmihnea/difference-array

[Editorial] Difference Arrays

Author: SansPapyrus683

content/3_Silver/More_Prefix_Sums.mdx

@@ -3,7 +3,7 @@ id: more-prefix-sums
 redirects:
   - /silver/prefix-sums-2
 title: 'More on Prefix Sums'
-author: Darren Yao, Neo Wang, Qi Wang
+author: Darren Yao, Neo Wang, Qi Wang, Mihnea Brebenel
 contributors: Jesse Choe, Kevin Sheng, Brad Ma, Juheon Rhee
 description:
   'Max subarray sum, prefix sums in two dimensions, and a more complicated
@@ -652,6 +652,85 @@ for _ in range(query_num):
 
 <Problems problems="cum2" hideSuggestProblemButton />
 
+## Difference Arrays
+
+<Resources>
+	<Resource source="Codeforces" title="An Introduction To Difference Arrays" url="https://codeforces.com/blog/entry/78762"/>
+</Resources>
+
+<FocusProblem problem="sample3" />
+
+### Explanation - Greg and Array
+
+Let's create an array $s$, where $s[i]$ is the number of times operation $i$ is applied.
+The important step is how we update it.
+
+For an interval $[l, r]$, we can't loop through the interval and
+increment each value, as that would be $\mathcal{O}(MK)$ and too slow.
+Instead, we increment $s[l]$ by one and decrement $s[r+1]$ by one.
+
+Now, we get the *actual* array by computing its prefix sum array,
+resulting in $\mathcal{O}(M)$ time complexity.
+The second part, applying the operations, can be done exactly the same way.
+
+### Implementation - Greg and Array
+
+**Time Complexity:** $\mathcal{O}(N+M)$
+
+<LanguageSection>
+<CPPSection>
+
+```cpp
+#include <array>
+#include <iostream>
+#include <vector>
+
+using namespace std;
+
+int main() {
+	int n, m, k;
+	cin >> n >> m >> k;
+	vector<int> a(n + 1);
+	for (int i = 1; i <= n; i++) { cin >> a[i]; }
+	vector<array<int, 3>> updates(m);
+	for (array<int, 3> &update : updates) {
+		cin >> update[0] >> update[1] >> update[2];
+	}
+
+	vector<long long> s(m + 2);
+	vector<long long> add(n + 2, 0);
+	for (int i = 0; i < k; i++) {
+		int x, y;
+		cin >> x >> y;
+		s[x]++;
+		s[y + 1]--;
+	}
+
+	for (int i = 1; i <= m; i++) {
+		// Apply prefix sums
+		s[i] += s[i - 1];
+
+		// At the same time compute the second difference array
+		add[updates[i - 1][0]] += s[i] * updates[i - 1][2];
+		add[updates[i - 1][1] + 1] -= s[i] * updates[i - 1][2];
+	}
+
+	for (int i = 1; i <= n; i++) {
+		// Apply prefix sums
+		add[i] += add[i - 1];
+		cout << a[i] + add[i] << ' ';
+	}
+	cout << endl;
+}
+```
+
+</CPPSection>
+</LanguageSection>
+
+### Problems
+
+<Problems problems="difference-arrays-problemset" />
+
 ## Quiz
 
 <Quiz>

content/3_Silver/More_Prefix_Sums.problems.json

@@ -30,6 +30,49 @@
       }
     }
   ],
+  "sample3": [
+    {
+      "uniqueId": "cf-295A",
+      "name": "Greg and Array",
+      "url": "https://codeforces.com/contest/295/problem/A",
+      "source": "CF",
+      "difficulty": "Normal",
+      "isStarred": false,
+      "tags": ["Difference Array"],
+      "solutionMetadata": {
+        "kind": "in-module",
+        "moduleId": "more-prefix-sums"
+      }
+    }
+  ],
+
+  "difference-arrays-problemset": [
+    {
+      "uniqueId": "abc179_d",
+      "name": " Leaping Tak",
+      "url": "https://atcoder.jp/contests/abc179/tasks/abc179_d",
+      "source": "AtCoder",
+      "difficulty": "Normal",
+      "isStarred": false,
+      "tags": ["Difference Array", "DP"],
+      "solutionMetadata": {
+        "kind": "none"
+      }
+    },
+    {
+      "uniqueId": "cf-276C",
+      "name": "Little Girl and Maximum Sum",
+      "url": "https://codeforces.com/contest/276/problem/C",
+      "source": "CF",
+      "difficulty": "Normal",
+      "isStarred": false,
+      "tags": ["Difference Array"],
+      "solutionMetadata": {
+        "kind": "none"
+      }
+    }
+  ],
+
   "cum2": [
     {
       "uniqueId": "usaco-919",