#P1718A2. Burenka and Traditions (hard version)
Burenka and Traditions (hard version)
Description
This is the hard version of this problem. The difference between easy and hard versions is only the constraints on $a_i$ and on $n$. You can make hacks only if both versions of the problem are solved.
Burenka is the crown princess of Buryatia, and soon she will become the $n$-th queen of the country. There is an ancient tradition in Buryatia — before the coronation, the ruler must show their strength to the inhabitants. To determine the strength of the $n$-th ruler, the inhabitants of the country give them an array of $a$ of exactly $n$ numbers, after which the ruler must turn all the elements of the array into zeros in the shortest time. The ruler can do the following two-step operation any number of times:
- select two indices $l$ and $r$, so that $1 \le l \le r \le n$ and a non-negative integer $x$, then
- for all $l \leq i \leq r$ assign $a_i := a_i \oplus x$, where $\oplus$ denotes the bitwise XOR operation. It takes $\left\lceil \frac{r-l+1}{2} \right\rceil$ seconds to do this operation, where $\lceil y \rceil$ denotes $y$ rounded up to the nearest integer.
Help Burenka calculate how much time she will need.
The first line contains a single integer $t$ $(1 \le t \le 500)$ — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ $(1 \le n \le 10^5)$ - the size of the array
The second line of each test case contains $n$ integers $a_1, a_2, \cdots , a_n$ $(0 \le a_i < 2^{30})$ — elements of the array.
It is guaranteed that the sum of $n$ in all tests does not exceed $10^5$.
For each test case, output a single number — the minimum time that Burenka will need.
Input
The first line contains a single integer $t$ $(1 \le t \le 500)$ — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ $(1 \le n \le 10^5)$ - the size of the array
The second line of each test case contains $n$ integers $a_1, a_2, \cdots , a_n$ $(0 \le a_i < 2^{30})$ — elements of the array.
It is guaranteed that the sum of $n$ in all tests does not exceed $10^5$.
Output
For each test case, output a single number — the minimum time that Burenka will need.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">7</div><div class="test-example-line test-example-line-odd test-example-line-1">4</div><div class="test-example-line test-example-line-odd test-example-line-1">5 5 5 5</div><div class="test-example-line test-example-line-even test-example-line-2">3</div><div class="test-example-line test-example-line-even test-example-line-2">1 3 2</div><div class="test-example-line test-example-line-odd test-example-line-3">2</div><div class="test-example-line test-example-line-odd test-example-line-3">0 0</div><div class="test-example-line test-example-line-even test-example-line-4">3</div><div class="test-example-line test-example-line-even test-example-line-4">2 5 7</div><div class="test-example-line test-example-line-odd test-example-line-5">6</div><div class="test-example-line test-example-line-odd test-example-line-5">1 2 3 3 2 1</div><div class="test-example-line test-example-line-even test-example-line-6">10</div><div class="test-example-line test-example-line-even test-example-line-6">27 27 34 32 2 31 23 56 52 4</div><div class="test-example-line test-example-line-odd test-example-line-7">5</div><div class="test-example-line test-example-line-odd test-example-line-7">1822 1799 57 23 55</div><div class="test-example-line test-example-line-odd test-example-line-7"></div>
2
2
0
2
4
7
4
Note
In the first test case, Burenka can choose segment $l = 1$, $r = 4$, and $x=5$. so it will fill the array with zeros in $2$ seconds.
In the second test case, Burenka first selects segment $l = 1$, $r = 2$, and $x = 1$, after which $a = [0, 2, 2]$, and then the segment $l = 2$, $r = 3$, and $x=2$, which fills the array with zeros. In total, Burenka will spend $2$ seconds.