Odd Queries
Contest: Codeforces Round 859 (Div. 4)
Difficulty rating: 900
Tags: data structures, implementation

You have an array $$$a_1, a_2, \dots, a_n$$$. Answer $$$q$$$ queries of the following form:

- If we change all elements in the range $$$a_l, a_{l+1}, \dots, a_r$$$ of the array to $$$k$$$, will the sum of the entire array be odd?

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.

The first line of each test case consists of $$$2$$$ integers $$$n$$$ and $$$q$$$ ($$$1 \le n \le 2 \cdot 10^5$$$; $$$1 \le q \le 2 \cdot 10^5$$$) — the length of the array and the number of queries.

The second line of each test case consists of $$$n$$$ integers $$$a_i$$$ ($$$1 \le a_i \le 10^9$$$) — the array $$$a$$$.

The next $$$q$$$ lines of each test case consists of $$$3$$$ integers $$$l,r,k$$$ ($$$1 \le l \le r \le n$$$; $$$1 \le k \le 10^9$$$) — the queries.

It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$, and the sum of $$$q$$$ doesn't exceed $$$2 \cdot 10^5$$$.

Output format:
For each query, output "YES" if the sum of the entire array becomes odd, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
For the first test case:

- If the elements in the range $$$(2, 3)$$$ would get set to $$$3$$$ the array would become $$$\{2, 3, 3, 3, 2\}$$$, the sum would be $$$2+3+3+3+2 = 13$$$ which is odd, so the answer is "YES".
- If the elements in the range $$$(2, 3)$$$ would get set to $$$4$$$ the array would become $$$\{2, 4, 4, 3, 2\}$$$, the sum would be $$$2+4+4+3+2 = 15$$$ which is odd, so the answer is "YES".
- If the elements in the range $$$(1, 5)$$$ would get set to $$$5$$$ the array would become $$$\{5, 5, 5, 5, 5\}$$$, the sum would be $$$5+5+5+5+5 = 25$$$ which is odd, so the answer is "YES".
- If the elements in the range $$$(1, 4)$$$ would get set to $$$9$$$ the array would become $$$\{9, 9, 9, 9, 2\}$$$, the sum would be $$$9+9+9+9+2 = 38$$$ which is even, so the answer is "NO".
- If the elements in the range $$$(2, 4)$$$ would get set to $$$3$$$ the array would become $$$\{2, 3, 3, 3, 2\}$$$, the sum would be $$$2+3+3+3+2 = 13$$$ which is odd, so the answer is "YES".

Example input:
2
5 5
2 2 1 3 2
2 3 3
2 3 4
1 5 5
1 4 9
2 4 3
10 5
1 1 1 1 1 1 1 1 1 1
3 8 13
2 5 10
3 8 10
1 10 2
1 9 100

Example output:
YES
YES
YES
NO
YES
NO
NO
NO
NO
YES

Editorial:
Note that for each question, the resulting array is $$$$$$[a_1, a_2, \dots, a_{l-1}, k, \dots, k, a_{r+1}, a_{r+2}, \dots, a_n].$$$$$$
So, the sum of the elements of the new array after each question is $$$$$$a_1 + \dots + a_{l-1} + (r-l+1) \cdot k + a_{r+1} + \dots + a_n.$$$$$$
We can compute $$$a_1 + \dots + a_{l-1}$$$ and $$$a_{r+1} + \dots + a_n$$$ in $$$\mathcal{O}(1)$$$ time by precomputing the sum of all prefixes and suffixes, or alternatively by using the prefix sums technique. So we can find the sum each time in $$$\mathcal{O}(1)$$$ per question, and just check if it's odd or not. The time complexity is $$$\mathcal{O}(n+q)$$$.