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Bug fixes and improvements (#1348)

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* Add "reference" for EN version. Bug fixes.

* Unify the figure reference as "the figure below" and "the figure above".
Bug fixes.

* Format the EN markdown files.

* Replace "" with <u></u> for EN version and bug fixes

* Fix biary_tree_dfs.png

* Fix biary_tree_dfs.png

* Fix zh-hant/biary_tree_dfs.png

* Fix heap_sort_step1.png

* Sync zh and zh-hant versions.

* Bug fixes

* Fix EN figures

* Bug fixes

* Fix the figure labels for EN version
This commit is contained in:
Yudong Jin
2024-05-06 14:44:48 +08:00
committed by GitHub
parent 8e60d12151
commit c4a7966882
99 changed files with 615 additions and 259 deletions
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14
en/docs/chapter_computational_complexity/space_complexity.md
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@@ -1,6 +1,6 @@
# Space complexity
"Space complexity" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".
<u>Space complexity</u> is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".
## Space related to algorithms
@@ -725,12 +725,12 @@ The time complexity of both `loop()` and `recur()` functions is $O(n)$, but thei
## Common types
Let the size of the input data be $n$, the following chart displays common types of space complexities (arranged from low to high).
Let the size of the input data be $n$, the figure below displays common types of space complexities (arranged from low to high).
$$
\begin{aligned}
O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
\text{Constant Order} < \text{Logarithmic Order} < \text{Linear Order} < \text{Quadratic Order} < \text{Exponential Order}
& O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
& \text{Constant} < \text{Logarithmic} < \text{Linear} < \text{Quadratic} < \text{Exponential}
\end{aligned}
$$
@@ -754,7 +754,7 @@ Linear order is common in arrays, linked lists, stacks, queues, etc., where the
[file]{space_complexity}-[class]{}-[func]{linear}
```
As shown below, this function's recursive depth is $n$, meaning there are $n$ instances of unreturned `linear_recur()` function, using $O(n)$ size of stack frame space:
As shown in the figure below, this function's recursive depth is $n$, meaning there are $n$ instances of unreturned `linear_recur()` function, using $O(n)$ size of stack frame space:
```src
[file]{space_complexity}-[class]{}-[func]{linear_recur}
@@ -770,7 +770,7 @@ Quadratic order is common in matrices and graphs, where the number of elements i
[file]{space_complexity}-[class]{}-[func]{quadratic}
```
As shown below, the recursive depth of this function is $n$, and in each recursive call, an array is initialized with lengths $n$, $n-1$, $\dots$, $2$, $1$, averaging $n/2$, thus overall occupying $O(n^2)$ space:
As shown in the figure below, the recursive depth of this function is $n$, and in each recursive call, an array is initialized with lengths $n$, $n-1$, $\dots$, $2$, $1$, averaging $n/2$, thus overall occupying $O(n^2)$ space:
```src
[file]{space_complexity}-[class]{}-[func]{quadratic_recur}
@@ -780,7 +780,7 @@ As shown below, the recursive depth of this function is $n$, and in each recursi
### Exponential order $O(2^n)$
Exponential order is common in binary trees. Observe the below image, a "full binary tree" with $n$ levels has $2^n - 1$ nodes, occupying $O(2^n)$ space:
Exponential order is common in binary trees. Observe the figure below, a "full binary tree" with $n$ levels has $2^n - 1$ nodes, occupying $O(2^n)$ space:
```src
[file]{space_complexity}-[class]{}-[func]{build_tree}