B. Arpa’s obvious problem and Mehrdad’s terrible solution(Hard)

Codeforces

IDCF10118B

Time2000ms

Memory512MB

Difficulty—

English · Original

Formal · Original

_There are some beautiful girls in Arpa’s land as mentioned before._ Once Arpa came up with an obvious problem: Given an array and a number x, count the number of pairs of indices i, j (1 ≤ i < j ≤ n) such that , where is bitwise _xor_ operation. Immediately, Mehrdad discovered a terrible solution that nobody trusted. Now Arpa needs your help to implement the solution to that problem. First line contains two integers n and x (1 ≤ n ≤ 2·106, 0 ≤ x ≤ 2107) — the number of elements in the array and the integer x. Second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 2107) — the elements of the array. It's guaranteed that . Input compression : x and ais are given in 32 base numerical system (i.e. "V" = 31, "H9" = 17·32 + 9 = 480). It's guaranteed that no number starts with zero in the input. Print a single integer: the answer to the problem. A bitwise _xor_ takes two bit integers of equal length and performs the logical _xor_ operation on each pair of corresponding bits. The result in each position is 1 if only the first bit is 1 or only the second bit is 1, but will be 0 if both are 0 or both are 1. You can read more about bitwise _xor_ operation here: https://en.wikipedia.org/wiki/Bitwise_operation#XOR. ## Input First line contains two integers n and x (1 ≤ n ≤ 2·106, 0 ≤ x ≤ 2107) — the number of elements in the array and the integer x.Second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 2107) — the elements of the array. It's guaranteed that .Input compression : x and ais are given in 32 base numerical system (i.e. "V" = 31, "H9" = 17·32 + 9 = 480).It's guaranteed that no number starts with zero in the input. ## Output Print a single integer: the answer to the problem. [samples] ## Note A bitwise _xor_ takes two bit integers of equal length and performs the logical _xor_ operation on each pair of corresponding bits. The result in each position is 1 if only the first bit is 1 or only the second bit is 1, but will be 0 if both are 0 or both are 1. You can read more about bitwise _xor_ operation here: https://en.wikipedia.org/wiki/Bitwise_operation#XOR.

**Definitions** Let $ n \in \mathbb{Z}^+ $, $ x \in \mathbb{Z}_{\geq 0} $. Let $ A = (a_1, a_2, \dots, a_n) $ be a sequence of non-negative integers, where each $ a_i \in \mathbb{Z}_{\geq 0} $. **Constraints** 1. $ 1 \leq n \leq 2 \cdot 10^6 $ 2. $ 0 \leq x \leq 2^{107} $ 3. $ 1 \leq a_i \leq 2^{107} $ for all $ i \in \{1, \dots, n\} $ 4. All values $ x $ and $ a_i $ are provided in base-32 representation; convert to base-10 before processing. **Objective** Count the number of ordered pairs $ (i, j) $ such that: $$ 1 \leq i < j \leq n \quad \text{and} \quad a_i \oplus a_j = x $$ where $ \oplus $ denotes the bitwise XOR operation.

API Response (JSON)

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