translation: Capitalize all the headers, list headers and figure captions (#1206)

* Capitalize all the headers, list headers and figure captions

* Fix the term "LRU"

* Fix the names of source code link in avl_tree.md

* Capitalize only first letter for nav trees in mkdocs.yml

* Update code comments

* Update linked_list.md

* Update linked_list.md

en/docs/chapter_hashing/hash_collision.md

@@ -1,4 +1,4 @@
-# Hash Collision
+# Hash collision
 
 As mentioned in the previous section, **usually the input space of a hash function is much larger than its output space**, making hash collisions theoretically inevitable. For example, if the input space consists of all integers and the output space is the size of the array capacity, multiple integers will inevitably map to the same bucket index.
 
@@ -9,22 +9,22 @@ Hash collisions can lead to incorrect query results, severely affecting the usab
 
 There are mainly two methods for improving the structure of hash tables: "Separate Chaining" and "Open Addressing".
 
-## Separate Chaining
+## Separate chaining
 
 In the original hash table, each bucket can store only one key-value pair. "Separate chaining" transforms individual elements into a linked list, with key-value pairs as list nodes, storing all colliding key-value pairs in the same list. The figure below shows an example of a hash table with separate chaining.
 
-![Separate Chaining Hash Table](hash_collision.assets/hash_table_chaining.png)
+![Separate chaining hash table](hash_collision.assets/hash_table_chaining.png)
 
 The operations of a hash table implemented with separate chaining have changed as follows:
 
-- **Querying Elements**: Input `key`, pass through the hash function to obtain the bucket index, access the head node of the list, then traverse the list and compare `key` to find the target key-value pair.
-- **Adding Elements**: First access the list head node via the hash function, then add the node (key-value pair) to the list.
-- **Deleting Elements**: Access the list head based on the hash function's result, then traverse the list to find and remove the target node.
+- **Querying elements**: Input `key`, pass through the hash function to obtain the bucket index, access the head node of the list, then traverse the list and compare `key` to find the target key-value pair.
+- **Adding elements**: First access the list head node via the hash function, then add the node (key-value pair) to the list.
+- **Deleting elements**: Access the list head based on the hash function's result, then traverse the list to find and remove the target node.
 
 Separate chaining has the following limitations:
 
-- **Increased Space Usage**: The linked list contains node pointers, which consume more memory space than arrays.
-- **Reduced Query Efficiency**: Due to the need for linear traversal of the list to find the corresponding element.
+- **Increased space usage**: The linked list contains node pointers, which consume more memory space than arrays.
+- **Reduced query efficiency**: Due to the need for linear traversal of the list to find the corresponding element.
 
 The code below provides a simple implementation of a separate chaining hash table, with two things to note:
 
@@ -37,28 +37,28 @@ The code below provides a simple implementation of a separate chaining hash tabl
 
 It's worth noting that when the list is very long, the query efficiency $O(n)$ is poor. **At this point, the list can be converted to an "AVL tree" or "Red-Black tree"** to optimize the time complexity of the query operation to $O(\log n)$.
 
-## Open Addressing
+## Open addressing
 
 "Open addressing" does not introduce additional data structures but uses "multiple probes" to handle hash collisions. The probing methods mainly include linear probing, quadratic probing, and double hashing.
 
 Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.
 
-### Linear Probing
+### Linear probing
 
 Linear probing uses a fixed-step linear search for probing, differing from ordinary hash tables.
 
-- **Inserting Elements**: Calculate the bucket index using the hash function. If the bucket already contains an element, linearly traverse forward from the conflict position (usually with a step size of $1$) until an empty bucket is found, then insert the element.
-- **Searching for Elements**: If a hash collision is found, use the same step size to linearly traverse forward until the corresponding element is found and return `value`; if an empty bucket is encountered, it means the target element is not in the hash table, so return `None`.
+- **Inserting elements**: Calculate the bucket index using the hash function. If the bucket already contains an element, linearly traverse forward from the conflict position (usually with a step size of $1$) until an empty bucket is found, then insert the element.
+- **Searching for elements**: If a hash collision is found, use the same step size to linearly traverse forward until the corresponding element is found and return `value`; if an empty bucket is encountered, it means the target element is not in the hash table, so return `None`.
 
 The figure below shows the distribution of key-value pairs in an open addressing (linear probing) hash table. According to this hash function, keys with the same last two digits will be mapped to the same bucket. Through linear probing, they are stored consecutively in that bucket and the buckets below it.
 
-![Distribution of Key-Value Pairs in Open Addressing (Linear Probing) Hash Table](hash_collision.assets/hash_table_linear_probing.png)
+![Distribution of key-value pairs in open addressing (linear probing) hash table](hash_collision.assets/hash_table_linear_probing.png)
 
 However, **linear probing tends to create "clustering"**. Specifically, the longer a continuous position in the array is occupied, the more likely these positions are to encounter hash collisions, further promoting the growth of these clusters and eventually leading to deterioration in the efficiency of operations.
 
 It's important to note that **we cannot directly delete elements in an open addressing hash table**. Deleting an element creates an empty bucket `None` in the array. When searching for elements, if linear probing encounters this empty bucket, it will return, making the elements below this bucket inaccessible. The program may incorrectly assume these elements do not exist, as shown in the figure below.
 
-![Query Issues Caused by Deletion in Open Addressing](hash_collision.assets/hash_table_open_addressing_deletion.png)
+![Query issues caused by deletion in open addressing](hash_collision.assets/hash_table_open_addressing_deletion.png)
 
 To solve this problem, we can use a "lazy deletion" mechanism: instead of directly removing elements from the hash table, **use a constant `TOMBSTONE` to mark the bucket**. In this mechanism, both `None` and `TOMBSTONE` represent empty buckets and can hold key-value pairs. However, when linear probing encounters `TOMBSTONE`, it should continue traversing since there may still be key-value pairs below it.
 
@@ -72,7 +72,7 @@ The code below implements an open addressing (linear probing) hash table with la
 [file]{hash_map_open_addressing}-[class]{hash_map_open_addressing}-[func]{}
 ```
 
-### Quadratic Probing
+### Quadratic probing
 
 Quadratic probing is similar to linear probing and is one of the common strategies of open addressing. When a collision occurs, quadratic probing does not simply skip a fixed number of steps but skips "the square of the number of probes," i.e., $1, 4, 9, \dots$ steps.
 
@@ -86,12 +86,12 @@ However, quadratic probing is not perfect:
 - Clustering still exists, i.e., some positions are more likely to be occupied than others.
 - Due to the growth of squares, quadratic probing may not probe the entire hash table, meaning it might not access empty buckets even if they exist in the hash table.
 
-### Double Hashing
+### Double hashing
 
 As the name suggests, the double hashing method uses multiple hash functions $f_1(x)$, $f_2(x)$, $f_3(x)$, $\dots$ for probing.
 
-- **Inserting Elements**: If hash function $f_1(x)$ encounters a conflict, try $f_2(x)$, and so on, until an empty position is found and the element is inserted.
-- **Searching for Elements**: Search in the same order of hash functions until the target element is found and returned; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return `None`.
+- **Inserting elements**: If hash function $f_1(x)$ encounters a conflict, try $f_2(x)$, and so on, until an empty position is found and the element is inserted.
+- **Searching for elements**: Search in the same order of hash functions until the target element is found and returned; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return `None`.
 
 Compared to linear probing, double hashing is less prone to clustering but involves additional computation for multiple hash functions.
 
@@ -99,7 +99,7 @@ Compared to linear probing, double hashing is less prone to clustering but invol
 
     Please note that open addressing (linear probing, quadratic probing, and double hashing) hash tables all have the issue of "not being able to directly delete elements."
 
-## Choice of Programming Languages
+## Choice of programming languages
 
 Various programming languages have adopted different hash table implementation strategies, here are a few examples: