Referring to the method in Figure 7-1, explain the operation process of PARTITION on the array A=(13, 19, 9, 5, 12, 8, 7, 4, 21, 2, 6, 11).
In a word:
PARTITION is a common quicksort algorithm whose purpose is to divide an array into two parts such that all elements of one part are smaller than all elements of the other part. During this process, the smallest element is placed in the last position of the array.
The following is the process of performing the PARTITION operation on the array A=(13, 19, 9, 5, 12, 8, 7, 4, 21, 2, 6, 11):
Choose an element as the "pivot", here we choose the first element of the array, which is 13.
Divide the array into two parts, one part has elements smaller than the pivot, and the other part has elements larger than the pivot. In this example, we choose 13 as the pivot. From this choice, we can reconstruct array A as (9, 5, 8, 7, 4, 2, 6, 11) and (19, 12, 21).
Now, we need to merge these two parts together and make sure that the first element is the smallest element in the entire array. In this example, the smallest element is 2.
Exchange the pivot (13) and the smallest element (2). Now the array becomes (2, 9, 5, 8, 7, 4, 6, 11) and (19, 12, 21).
In this way, we have completed the PARTITION operation on the array A.
Xunfei Spark:
According to the method in Figure 7-1, we can perform the PARTITION operation according to the following steps:
1. First, sort the array A. The sorted array is: (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 19, 21).
2. Initialize three pointers i, j and k, where i points to the current element position to be processed, j points to the end of the array, and k is used to record the length of the array after partitioning.
3. When i is less than j, perform the following operations:
a. Compare A[i] with A[j-1]. If A[i] is smaller than A[j-1], exchange A[i] with A[k].
b. If A[i] is greater than or equal to A[j-1], move j one bit backward.
4. When i is greater than or equal to j, end the loop. At this point, the array A has been divided into three parts: A[0:k-1], A[k:j-1] and A[j:n], where k is the largest index value.
5. Perform further processing or output on these three parts as required. For example, if you need to solve the maximum subarray sum problem, you can treat these three parts as three independent subarrays; if you only need to find the maximum or minimum value of each part, etc.
In this example, the array A=(13,19,9,5,12,8,7,4,21,2,6,11), after performing the PARTITION operation according to the above method, the three parts obtained are: (2,4), (9), (13) (5), (8), (7), (6), (19), (2), (12), (11).
360 Smart Brain:
First of all, we need to make it clear that the function of the PARTITION function is to divide the interval of the array. In this question, we want to divide the array A according to the midpoint of the numbers 13 and 21, that is, the first interval of the array A contains A[0]-A[6], and the second interval contains A[7]-A [12].
Proceed as follows:
1. Calculate the length N=12 of the array A, and determine the left and right pointers, which are 0 and N-1 respectively, indicating that the initial range of the array A is [0, 0] to [0, 11].
2. Calculate the middle position mid of the array A, mid=(left+right)/2, and the corresponding element value is A[mid]=8.
3. Determine whether the element value is equal to 13 or 21. Since the midpoints of 13 and 21 are 8 and 5 respectively, we need to re-divide the array A into two intervals: interval 1 contains A[0]-A[6], and interval 2 contains A[7]-A[12 ]. Element 8 is at the dividing point of the two intervals, so we put A[mid] in interval 2.
4. Update the left pointer left and the right pointer right to point to the starting positions of the two new intervals of the array A respectively. At this time, the left pointer left=0 and the right pointer right=6, indicating that the new range of array A is [0, 0] to [0, 6], [7, 8] to [12, 21].
5. Repeat steps 2-4 until the left pointer left and the right pointer right meet. In this example, left pointer left=0, right pointer right=6, meet at A[6]=19. At this point, the array A has been divided into two intervals: [0, 0] to [0, 6] and [7, 8] to [12, 21].
6. Complete the PARTITION operation, and finally get two intervals: [0, 0] to [0, 6] and [7, 8] to [12, 21].
Note: The above operation process is only a possible method, and the specific implementation method may vary due to factors such as programming language and algorithm implementation.
chatglm：
The PARTITION algorithm is part of a commonly used quicksort algorithm that divides an array into two subarrays, where all elements of one subarray are smaller than all elements of the other subarray. The following is the operation process of the PARTITION algorithm on the array A=(13,19,9,5,12,8,7,4,21,2,6,11):
1. Select the last element 11 in the array A as the pivot element.
2. Arrange the elements in the array A in ascending order, and you can use recursive sorting or other sorting algorithms for sorting. After sorting, the array A becomes A=(2,6,7,8,9,12,13,19,21,5,11).
3. Starting from the first element of the array A, compare the elements with the reference element 11 one by one. If the element is smaller than the reference element, place it to the left of the reference element, otherwise place it to the right of the reference element. The specific operation is as follows:
-
Comparing the first element 2 with the base element 11, 2 is less than 11, so 2 is placed to the left of the base element 11.
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Comparing the second element 6 with the base element 11, 6 is less than 11, so the 6 is placed to the left of the base element 11.
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Comparing the third element 7 with the base element 11, 7 is less than 11, so the 7 is placed to the left of the base element 11.
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Comparing the fourth element 8 with the base element 11, 8 is less than 11, so the 8 is placed to the left of the base element 11.
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Comparing the fifth element 9 with the base element 11, 9 is less than 11, so the 9 is placed to the left of the base element 11.
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The sixth element 12 is compared with the base element 11, 12 is greater than 11, so 12 is placed to the right of the base element 11.
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The seventh element 13 is compared with the base element 11, 13 is greater than 11, so 13 is placed to the right of the base element 11.
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The eighth element 19 is compared with the base element 11, 19 is greater than 11, so 19 is placed to the right of the base element 11.
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The ninth element 21 is compared with the base element 11, 21 is greater than 11, so 21 is placed to the right of the base element 11.
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Comparing the tenth element 5 with the base element 11, 5 is less than 11, so 5 is placed to the left of the base element 11.
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Comparing the eleventh element 11 with the base element 11, 11 is equal to 11, so 11 is placed to the left of the base element 11.
4. After the above operations, the elements in array A have been divided into two sub-arrays, and all elements in one sub-array are smaller than all elements in the other sub-array. The specific results are as follows:
A=(2,6,7,8,9,12,13,19,21,5,11)
Among them, the left sub-array is A=(2,6,7,8,9,12,13), and the right sub-array is A=(19,21,5,11).
Tongyi Qianqian:
Assume that the array A is to be divided into two parts so that the sum of the elements of these two parts is the largest. We can use the PARTITION function to achieve this goal.
First, we need to choose an appropriate element as the "pivot", which can be any element in the array. The principle of selecting the benchmark is to hope that this element can make the sum of the elements of the two parts as close as possible when the array is divided into two parts.
In this example, we choose element 13 as the datum. Using the PARTITION function, the array A can be divided into two parts so that the sum of the elements of the two parts is the largest. Specifically, the following SQL statement can be written:
SELECT PARTITION(A BY 13) AS max_sum
FROM (
SELECT A, SUM(A) AS sum_A
FROM A
GROUP BY A
) AS temp
WHERE sum_A > (SELECT SUM(A) FROM A GROUP BY A) / 2;
Among them, the PARTITION function divides the array A into two parts, so that the sum of the elements of these two parts is the largest. In this example, we divide the array A by element 13 and get two parts [13, 19, 9, 5, 12, 8] and [7, 4, 21, 2, 6, 11]. We then use the SUM function to calculate the sum of the elements in each part, compare these sums to the sum of all elements in array A, and select the larger value as the final result.
After executing the above SQL statement, you can get the following results:
+-------+
| max_sum |
+-------+
| 24 |
+-------+
This result shows that when the array A is divided into two parts [13, 19, 9, 5, 12, 8] and [7, 4, 21, 2, 6, 11], the sum of the elements of these two parts is the largest , for 24.
Tiangong:
OK, according to the method in Figure 7-1, we can perform PARTITION operation on array A in the following steps:
1. Initialize the variable k as 0.
2. Initialize the variable i as 0.
3. Initialize variable j as 0.
4. Initialize the variable count to 0.
5. For each element a[i] in array A:
a. If a[i] > k, do the following:
i. 将 k 更新为 a[i]。
j. 将 count 更新为 count + 1。
b. If a[i] <= k, do the following:
i. 将 i 更新为 i + 1。
j. 将 count 更新为 count + 1。
6. Finally, the elements in the array A are divided according to the rules of the PARTITION operation.
Specifically, according to the above steps, we can get the following results:
1.k = 13。
2.i = 0。
3.j = 0。
4.count = 0。
5. For each element a[i] in array A:
a. If a[i] > 13, do the following:
i. 将 k 更新为 a[i]。
j. 将 count 更新为 count + 1。
b. If a[i] <= 13, do the following:
i. 将 i 更新为 i + 1。
j. 将 count 更新为 count + 1。
6. Finally, the array A is divided into three sub-arrays:
A[0] = (13, 19, 9, 5, 12, 8, 7, 4, 21, 6, 11)；
A[1] = (9, 5, 12, 8, 7, 4, 21, 6)；
A[2] = (5, 12, 8, 7, 4, 21, 6)。
Hope this answer helps you understand PARTITION operation.
chatgpt：
Before performing the PARTITION operation, we need to select a pivot element. Here, I choose to use the last element of the array, which is 11, as the pivot.
First, we set two pointers, i and j. Initially, i points to the starting position of the array -1, and j points to the starting position of the array. Then, we start from the starting position of the array, compare the size relationship between the elements and the pivot one by one, and perform the exchange operation.
1. Initial state:
A = (13, 19, 9, 5, 12, 8, 7, 4, 21, 2, 6, 11)
i = -1
j = 0
pivot = 11
2. The first step:
The current element A[j] = 13 is compared with the pivot 11, 13 > 11, so there is no need to exchange, and j moves backward one bit.
3. The second step:
The current element A[j] = 19 is compared with the pivot 11, 19 > 11, so there is no need to exchange, and j moves backward one bit.
4. The third step:
The current element A[j] = 9 is compared with the pivot 11, 9 < 11, so exchange 9 with the element 19 at the position of A[i+1], and then both i and j move backward one bit.
A = (13, 9, 19, 5, 12, 8, 7, 4, 21, 2, 6, 11)
i = 0
j = 1
5. The fourth step:
The current element A[j] = 5 is compared with the pivot 11, 5 < 11, so exchange 5 with the element 9 at the position of A[i+1], and then both i and j move backward one bit.
A = (13, 5, 19, 9, 12, 8, 7, 4, 21, 2, 6, 11)
i = 1
j = 2
6. The fifth step:
The current element A[j] = 19 is compared with the pivot 11, 19 > 11, so there is no need to exchange, and j moves backward one bit.
7. The sixth step:
The current element A[j] = 9 is compared with the pivot 11, 9 < 11, so exchange 9 with the element 19 at the position of A[i+1], and then both i and j move backward one bit.
A = (13, 5, 9, 19, 12, 8, 7, 4, 21, 2, 6, 11)
i = 2
j = 3
8. Repeat the above steps, continuously move the pointer and exchange elements until j has traversed the entire array.
Finally, exchange the pivot and the element at the position of A[i+1], and place the pivot in a suitable position.
9. Final result:
A = (5, 9, 4, 19, 12, 8, 7, 13, 21, 2, 6, 11)
In this process, the main purpose is to move elements smaller than the pivot to the left, elements larger than the pivot to the right, and finally complete the division of the array.
Summarize
