From 738b3d934cbdb5fb5ce977b018bdf6a3f2d88069 Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:18:03 +0800 Subject: [PATCH] =?UTF-8?q?Update=2040.=20=E6=9C=80=E5=B0=8F=E7=9A=84=20K?= =?UTF-8?q?=20=E4=B8=AA=E6=95=B0.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 大顶堆解法新增执行过程配图,补充堆选择原因与防溢出说明;快速选择解法补充随机切分实现并更新复杂度说明。 --- notes/40. 最小的 K 个数.md | 17 +++++++++++------ notes/pics/offer40-maxheap-walk.png | Bin 0 -> 138273 bytes 2 files changed, 11 insertions(+), 6 deletions(-) create mode 100644 notes/pics/offer40-maxheap-walk.png diff --git a/notes/40. 最小的 K 个数.md b/notes/40. 最小的 K 个数.md index 85f0e2e2..163c1384 100644 --- a/notes/40. 最小的 K 个数.md +++ b/notes/40. 最小的 K 个数.md @@ -9,13 +9,15 @@ ### 大小为 K 的大顶堆 - 复杂度:O(NlogK) + O(K) -- 特别适合处理海量数据 +- 特别适合处理海量数据(内存中只保存 K 个数,数据可以边读边处理) -使用大小为 K 的大顶堆保存当前最小的 K 个数:在添加一个元素之后,如果大顶堆的大小大于 K,那么将大顶堆的堆顶元素去除,也就是将当前堆中值最大的元素去除,从而使得留在堆中的元素都比被去除的元素来得小。 +使用大小为 K 的大顶堆保存当前最小的 K 个数:在添加一个元素之后,如果大顶堆的大小大于 K,那么将大顶堆的堆顶元素去除,也就是将当前堆中值最大的元素去除,从而使得留在堆中的元素都比被去除的元素来得小。例如 [4, 5, 1, 6, 2, 7, 3, 8]、K = 3: -应该使用大顶堆来保存最小的 K 个数,而不能直接创建一个小顶堆并设置一个大小,企图让小顶堆中的元素都是最小元素。 +

-Java 的 PriorityQueue 实现了堆的能力,PriorityQueue 默认是小顶堆,可以在初始化时使用 Lambda 表达式 (o1, o2) -\> o2 - o1 来实现大顶堆。其它语言也有类似的堆数据结构。 +应该使用大顶堆来保存最小的 K 个数,而不能直接创建一个小顶堆并设置一个大小,企图让小顶堆中的元素都是最小元素。这是因为超出 K 个时要删掉的是 K + 1 个数中最大的那个,大顶堆的堆顶正好是它;小顶堆只能删掉最小的,最后留下的是最大的 K 个数。 + +Java 的 PriorityQueue 实现了堆的能力,PriorityQueue 默认是小顶堆,可以在初始化时使用 Lambda 表达式 (o1, o2) -\> o2 - o1 来实现大顶堆。本题的数在 0~10000 之间,相减不会溢出;数值可能接近 int 的上下限时,相减会溢出、比较结果出错,应改用 Comparator.reverseOrder()。其它语言也有类似的堆数据结构。 ```java public int[] inventoryManagement(int[] stock, int cnt) { @@ -38,10 +40,10 @@ public int[] inventoryManagement(int[] stock, int cnt) { ### 快速选择 -- 复杂度:O(N) + O(1) +- 复杂度:随机选择切分元素时期望 O(N) + O(1),最坏 O(N2)(每次切分都选到最值) - 只有当允许修改数组元素时才可以使用 -快速排序的 partition() 方法,会返回一个整数 j 使得 a[l..j-1] 小于等于 a[j],且 a[j+1..h] 大于等于 a[j],此时 a[j] 就是数组中第 j 小的元素(j 从 0 开始计数)。可以利用这个特性找出数组的第 K 个元素,这种找第 K 个元素的算法称为快速选择算法。 +快速排序的 partition() 方法,会返回一个整数 j 使得 a[l..j-1] 小于等于 a[j],且 a[j+1..h] 大于等于 a[j],此时 a[j] 就是数组中第 j 小的元素(j 从 0 开始计数)。可以利用这个特性找出数组的第 K 个元素,这种找第 K 个元素的算法称为快速选择算法。切分元素固定取 a[l] 时,有序数组每次只能切掉一个元素,因此先随机选一个元素换到 a[l]。 ```java public int[] inventoryManagement(int[] stock, int cnt) { @@ -68,7 +70,10 @@ private void findKthSmallest(int[] nums, int k) { } } +private final Random random = new Random(); + private int partition(int[] nums, int l, int h) { + swap(nums, l, l + random.nextInt(h - l + 1)); /* 随机选切分元素 */ int p = nums[l]; /* 切分元素 */ int i = l, j = h + 1; while (true) { diff --git a/notes/pics/offer40-maxheap-walk.png b/notes/pics/offer40-maxheap-walk.png new file mode 100644 index 0000000000000000000000000000000000000000..4b5e1ee49dcb9a26884ee8a3feebce442eac5d8a GIT binary patch literal 138273 zcmXtgcRber_x^2WWs{6ZcCt6g9%akkvdP{N*(1AT6H2l-5mBhsa%4 z*ImcV-O}~BnX5fQ+DDp)i(4r6t>Fg*3Id@hBl*;OYAe&rn{;%%W1zo3IV2>WP>Sxa z)hFz)Crk=j=&U$H=rW|}GUari0$G&fW3h%LCDs*o8AI;ip-V|gDbg`8n0a}69$zl~ zbDcM{s+xSN(Z9C8Kk4wy%j$UU#m-sQ?%;%j038()lT4jLU8y6cf=U z5s0tG17Ww+Nf{N}mj!!hVWo0;gEo8=*ShTF?Y!LNi42N~42EO6CE6l^=#oOgqjv2h@hSyL 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