Data structures 约束条件下数组中的最大k元素

给定一个整数数组，如何打印O（k）中最大的k个元素？
重要提示：数组初始化为零。每当程序调用函数

Add（i）

时，第i个单元格将递增1。在调用print函数之前，如何利用这一事实来维护数组内部的秩序

所需空间复杂度：O（n），其中n是给定数组的大小。

---

我正在寻找一个不涉及堆的解决方案。

这里有一个可能的解决方案，它可以通过使用大量的链接列表来工作

基本思想如下。对于每个数组位置，创建一个表示该数组元素的双链接列表单元格。然后，按数值对数组中的所有元素进行分组。例如，当您第一次创建数组时，所有的值都为零，您将有一个包含所有数组位置的双链接列表，如下所示：

+---+     +---+     +---+     +---+     +---+
| 0 | <-> | 1 | <-> | 2 | <-> | 3 | <-> | 4 |
+---+     +---+     +---+     +---+     +---+

                   +---+     +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 2 | <-> | 4 |
                   +---+     +---+     +---+     +---+

                   +---+
Cells with value 1 | 3 |
                   +---+

                   +---+     +---+
Cells with value 0 | 1 | <-> | 4 |
                   +---+     +---+

                   +---+
Cells with value 1 | 0 |
                   +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

                   +--------+---------+
                   |        |         | 
                   v        |         |
+--------------------+    +---+     +---+
| Cells with value 0 | -> | 1 | <-> | 4 |
+--------------------+    +---+     +---+
          ^
          |   +-------------+
          |   |             |
          v   v             |
+--------------------+    +---+
| Cells with value 1 | -> | 0 |
+--------------------+    +---+
          ^
          |        +--------+---------+
          |        |        |         | 
          v        v        |         |
+--------------------+    +---+     +---+
| Cells with value 2 | -> | 2 | <-> | 3 |
+--------------------+    +---+     +---+
          ^
          |
          +---- max

+--+++--+++--+++--++---+
| 0 |  | 1 |  | 2 |  | 3 |  | 4 |
+---+     +---+     +---+     +---+     +---+

每当您将某个值增加1时，请将其从其所在的列表中拼接出来，并将其添加到表示比上一个值大1的值的新列表中。例如，递增数组索引3会使情况如下所示：

+---+     +---+     +---+     +---+     +---+
| 0 | <-> | 1 | <-> | 2 | <-> | 3 | <-> | 4 |
+---+     +---+     +---+     +---+     +---+

                   +---+     +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 2 | <-> | 4 |
                   +---+     +---+     +---+     +---+

                   +---+
Cells with value 1 | 3 |
                   +---+

                   +---+     +---+
Cells with value 0 | 1 | <-> | 4 |
                   +---+     +---+

                   +---+
Cells with value 1 | 0 |
                   +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

                   +--------+---------+
                   |        |         | 
                   v        |         |
+--------------------+    +---+     +---+
| Cells with value 0 | -> | 1 | <-> | 4 |
+--------------------+    +---+     +---+
          ^
          |   +-------------+
          |   |             |
          v   v             |
+--------------------+    +---+
| Cells with value 1 | -> | 0 |
+--------------------+    +---+
          ^
          |        +--------+---------+
          |        |        |         | 
          v        v        |         |
+--------------------+    +---+     +---+
| Cells with value 2 | -> | 2 | <-> | 3 |
+--------------------+    +---+     +---+
          ^
          |
          +---- max

+--+++--++--++---+
值为0 | 0 | 1 | 2 | 4的单元格|
+---+     +---+     +---+     +---+
+---+
值为1 | 3的单元格|
+---+

然后递增单元格2将得出以下结果：

                   +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 4 |
                   +---+     +---+     +---+

                   +---+     +---+
Cells with value 1 | 2 | <-> | 3 |
                   +---+     +---+

                   +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 4 |
                   +---+     +---+     +---+

                   +---+
Cells with value 1 | 2 |
                   +---+

                   +---+
Cells with value 2 | 3 |
                   +---+

+--++--++---+
值为0 | 0 | 1 | 4的单元格|
+---+     +---+     +---+
+---+     +---+
值为1 | 2 | 3的单元格|
+---+     +---+

再次递增单元格3将得出以下结果：

                   +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 4 |
                   +---+     +---+     +---+

                   +---+     +---+
Cells with value 1 | 2 | <-> | 3 |
                   +---+     +---+

                   +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 4 |
                   +---+     +---+     +---+

                   +---+
Cells with value 1 | 2 |
                   +---+

                   +---+
Cells with value 2 | 3 |
                   +---+

+--++--++---+
值为0 | 0 | 1 | 4的单元格|
+---+     +---+     +---+
+---+
值为1 | 2的单元格|
+---+
+---+
值为2 | 3的单元格|
+---+

现在，假设您再次递增单元格（2）。这会将单元格（2）移动到值为2的单元格的链接列表中。由于不再需要值为（1）的单元格列表，我们可以删除它：

                   +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 4 |
                   +---+     +---+     +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

+--++--++---+
值为0 | 0 | 1 | 4的单元格|
+---+     +---+     +---+
+---+     +---+
值为2 | 2 | 3的单元格|
+---+     +---+

为了增加0，我们将重新创建值1的列表，如下所示：

                   +---+     +---+
Cells with value 0 | 1 | <-> | 4 |
                   +---+     +---+

                   +---+
Cells with value 1 | 0 |
                   +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

+--++---+
值为0 | 1 | 4的单元格|
+---+     +---+
+---+
值为1 | 0的单元格|
+---+
+---+     +---+
值为2 | 2 | 3的单元格|
+---+     +---+

要在时间O（k）中打印出前k个元素，请从最高值的链表开始，并在那里打印出元素。如果这还不够，那么转到第二高值列表并打印其内容，然后是下一个列表，然后是下一个列表，以此类推

挑战在于如何维护这些单独的链接列表。我们需要能够做到以下几点：

确定特定单元格所在的链接列表，以便我们可以将其提升到下一个级别

确定与特定链表关联的数值，以便在将一个单元格提升为更高的值时，可以确定是将其移动到序列中的下一个列表，还是需要创建新列表

删除其值不再使用的空列表

向后迭代列表，以便打印前k个元素

这个问题的一个解决方案是使用另一个双链表表示列表列表。具体来说，我们将执行以下操作：

• 维护列表的双链接列表。“列表列表”中的每个单元格将存储与该列表中的单元格关联的编号

• 保持指向“列表列表”中最后一个单元格的指针，这样我们就可以高效地打印前k个元素

• 用指向“列表列表”单元格的指针注释每个现有单元格，这样，当我们进行增量时，我们可以确定它属于哪个列表

例如，我画的是这样的：

+---+     +---+     +---+     +---+     +---+
| 0 | <-> | 1 | <-> | 2 | <-> | 3 | <-> | 4 |
+---+     +---+     +---+     +---+     +---+

                   +---+     +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 2 | <-> | 4 |
                   +---+     +---+     +---+     +---+

                   +---+
Cells with value 1 | 3 |
                   +---+

                   +---+     +---+
Cells with value 0 | 1 | <-> | 4 |
                   +---+     +---+

                   +---+
Cells with value 1 | 0 |
                   +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

                   +--------+---------+
                   |        |         | 
                   v        |         |
+--------------------+    +---+     +---+
| Cells with value 0 | -> | 1 | <-> | 4 |
+--------------------+    +---+     +---+
          ^
          |   +-------------+
          |   |             |
          v   v             |
+--------------------+    +---+
| Cells with value 1 | -> | 0 |
+--------------------+    +---+
          ^
          |        +--------+---------+
          |        |        |         | 
          v        v        |         |
+--------------------+    +---+     +---+
| Cells with value 2 | -> | 2 | <-> | 3 |
+--------------------+    +---+     +---+
          ^
          |
          +---- max

+--++---+
值为0 | 1 | 4的单元格|
+---+     +---+
+---+
值为1 | 0的单元格|
+---+
+---+     +---+
值为2 | 2 | 3的单元格|
+---+     +---+

实际上是这样的：

+---+     +---+     +---+     +---+     +---+
| 0 | <-> | 1 | <-> | 2 | <-> | 3 | <-> | 4 |
+---+     +---+     +---+     +---+     +---+

                   +---+     +---+     +---+     +---+
Cells with value 0 | 0 | <-> | 1 | <-> | 2 | <-> | 4 |
                   +---+     +---+     +---+     +---+

                   +---+
Cells with value 1 | 3 |
                   +---+

                   +---+     +---+
Cells with value 0 | 1 | <-> | 4 |
                   +---+     +---+

                   +---+
Cells with value 1 | 0 |
                   +---+

                   +---+     +---+
Cells with value 2 | 2 | <-> | 3 |
                   +---+     +---+

                   +--------+---------+
                   |        |         | 
                   v        |         |
+--------------------+    +---+     +---+
| Cells with value 0 | -> | 1 | <-> | 4 |
+--------------------+    +---+     +---+
          ^
          |   +-------------+
          |   |             |
          v   v             |
+--------------------+    +---+
| Cells with value 1 | -> | 0 |
+--------------------+    +---+
          ^
          |        +--------+---------+
          |        |        |         | 
          v        v        |         |
+--------------------+    +---+     +---+
| Cells with value 2 | -> | 2 | <-> | 3 |
+--------------------+    +---+     +---+
          ^
          |
          +---- max

+--------+---------+
|        |         | 
v||
+--------------------+    +---+     +---+
|值为0 |->1 | 4的单元格|
+--------------------+    +---+     +---+
^
|   +-------------+
|   |             |
v v|
+--------------------+    +---+
|值为1 |->0的单元格|
+--------------------+    +---+
^
|        +--------+---------+
|        |        |         | 
v||
+--------------------+    +---+     +---+
|值为2 |->2 | 3的单元格|
+--------------------+    +---+     +---+
^
|
+----马克斯

这允许我们将元素从列表中拼接出来，找出它们属于哪个列表，然后确定是将它们拼接到我们上面的列表中，还是在两个列表之间创建一个新列表

数组中的每个元素都单独使用空间O（1）。每个“列表列表”单元格占用空间O（1）a