diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index d674b3ba0..92ba6d1e6 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -10,7 +10,7 @@ comments: true - 「小顶堆 Min Heap」,任意结点的值 $\leq$ 其子结点的值;
- ![min_heap_and_max_heap](heap.assets/min_heap_and_max_heap.png){ width="600" } + ![min_heap_and_max_heap](heap.assets/min_heap_and_max_heap.png)
## 8.1.1. 堆术语与性质 @@ -217,9 +217,7 @@ comments: true 具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。 -
- ![representation_of_heap](heap.assets/representation_of_heap.png){ width="600" } -
+![representation_of_heap](heap.assets/representation_of_heap.png) 我们将索引映射公式封装成函数,以便后续使用。 @@ -419,23 +417,23 @@ comments: true === "Step 1"
- ![heap_push_step1](heap.assets/heap_push_step1.png){ width="600" } + ![heap_push_step1](heap.assets/heap_push_step1.png)
=== "Step 2" - ![heap_push_step2](heap.assets/heap_push_step2.png){ width="600" } + ![heap_push_step2](heap.assets/heap_push_step2.png) === "Step 3" - ![heap_push_step3](heap.assets/heap_push_step3.png){ width="600" } + ![heap_push_step3](heap.assets/heap_push_step3.png) === "Step 4" - ![heap_push_step4](heap.assets/heap_push_step4.png){ width="600" } + ![heap_push_step4](heap.assets/heap_push_step4.png) === "Step 5" - ![heap_push_step5](heap.assets/heap_push_step5.png){ width="600" } + ![heap_push_step5](heap.assets/heap_push_step5.png) === "Step 6" - ![heap_push_step6](heap.assets/heap_push_step6.png){ width="600" } + ![heap_push_step6](heap.assets/heap_push_step6.png) 设结点总数为 $n$ ,则树的高度为 $O(\log n)$ ,易得堆化操作的循环轮数最多为 $O(\log n)$ ,**因而元素入堆操作的时间复杂度为 $O(\log n)$** 。 @@ -570,34 +568,34 @@ comments: true 顾名思义,**从顶至底堆化的操作方向与从底至顶堆化相反**,我们比较根结点的值与其两个子结点的值,将最大的子结点与根结点执行交换,并循环以上操作,直到越过叶结点时结束,或当遇到无需交换的结点时提前结束。 === "Step 1" - ![heap_poll_step1](heap.assets/heap_poll_step1.png){ width="600" } + ![heap_poll_step1](heap.assets/heap_poll_step1.png) === "Step 2" - ![heap_poll_step2](heap.assets/heap_poll_step2.png){ width="600" } + ![heap_poll_step2](heap.assets/heap_poll_step2.png) === "Step 3" - ![heap_poll_step3](heap.assets/heap_poll_step3.png){ width="600" } + ![heap_poll_step3](heap.assets/heap_poll_step3.png) === "Step 4" - ![heap_poll_step4](heap.assets/heap_poll_step4.png){ width="600" } + ![heap_poll_step4](heap.assets/heap_poll_step4.png) === "Step 5" - ![heap_poll_step5](heap.assets/heap_poll_step5.png){ width="600" } + ![heap_poll_step5](heap.assets/heap_poll_step5.png) === "Step 6" - ![heap_poll_step6](heap.assets/heap_poll_step6.png){ width="600" } + ![heap_poll_step6](heap.assets/heap_poll_step6.png) === "Step 7" - ![heap_poll_step7](heap.assets/heap_poll_step7.png){ width="600" } + ![heap_poll_step7](heap.assets/heap_poll_step7.png) === "Step 8" - ![heap_poll_step8](heap.assets/heap_poll_step8.png){ width="600" } + ![heap_poll_step8](heap.assets/heap_poll_step8.png) === "Step 9" - ![heap_poll_step9](heap.assets/heap_poll_step9.png){ width="600" } + ![heap_poll_step9](heap.assets/heap_poll_step9.png) === "Step 10" - ![heap_poll_step10](heap.assets/heap_poll_step10.png){ width="600" } + ![heap_poll_step10](heap.assets/heap_poll_step10.png) 与元素入堆操作类似,**堆顶元素出堆操作的时间复杂度为 $O(\log n)$** 。 @@ -862,7 +860,7 @@ $$ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 $$ -![heapify_count](heap.assets/heapify_count.png){ width="600" } +![heapify_count](heap.assets/heapify_count.png) 化简上式需要借助中学的数列知识,先对 $T(h)$ 乘以 $2$ ,易得