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ashawkey · Dec 29, 2023 · d695e23 · d695e23
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diff --git a/docs/algorithm/advanced/后缀数组.md b/docs/algorithm/advanced/后缀数组.md
@@ -194,9 +194,7 @@ LCPL(i, j) = min{height[i+1, ..., j]} // RMQ O(1)
 
 $$
 \displaylines{
-
   LCPL(i, j) = min\{LCPL(k, k+1)| k=i, ..., j-1\}
-  
 }
 $$
 
@@ -206,10 +204,8 @@ $$
 
 $$
 \displaylines{
-
   \forall i \le k \lt j \\
   LCPL(k, j) \ge LCPL(i, j)
-  
 }
 $$
 
@@ -225,9 +221,7 @@ $$
 
 $$
 \displaylines{
-
   H[i] \ge H[i-1] -1
-  
 }
 $$
 

diff --git a/docs/algorithm/advanced/最短路算法.md b/docs/algorithm/advanced/最短路算法.md
@@ -18,7 +18,6 @@
 $$
 \displaylines{
   T(n) = O(E)*dk_Q + O(V)*em_Q
-  
 }
 $$
 

diff --git a/docs/algorithm/advanced/线段树.md b/docs/algorithm/advanced/线段树.md
@@ -29,9 +29,7 @@ struct node{
 
 $$
 \displaylines{
-
   2*2^{ceil(log_2(n))}-1 \le 4n-1
-  
 }
 $$
 

diff --git a/docs/algorithm/basic/DA01_Intro.md b/docs/algorithm/basic/DA01_Intro.md
@@ -75,11 +75,9 @@
 
 $$
 \displaylines{
-
     \exist C,N_0 \gt 0 \ \ s.t. \ \forall n \ge N_0:\\
     f(n) \le C \cdot g(n) \\
     \Rightarrow f(n) \in O(g(n))
-    
 }
 $$
 
@@ -93,11 +91,9 @@ $$
 
 $$
 \displaylines{
-
     \exist C,N_0 \gt 0 \ \ s.t. \ \forall n \ge N_0:\\
     f(n) \ge C \cdot g(n) \\
     \Rightarrow f(n) \in \Omega(g(n))
-    
 }
 $$
 
@@ -109,10 +105,8 @@ $$
 
 $$
 \displaylines{
-
     f(n) \in O(g(n)) \ and \ f(n) \in \Omega(g(n)) \\
     \Rightarrow f(n) \in \Theta(g(n))
-    
 }
 $$
 

diff --git a/docs/algorithm/basic/DA02_LinearTable.md b/docs/algorithm/basic/DA02_LinearTable.md
@@ -95,11 +95,9 @@ Link the head and tail.
 
 $$
 \displaylines{
-
   2t - t = km \\
   2t'-t' = (k+1)m \\
   \therefore t'-t = m
-  
 }
 $$
 
@@ -111,12 +109,10 @@ $$
 
 $$
 \displaylines{
-
   t = n + x \\
   t = km \\
   \therefore km = n +x \\
   (k-1)m + (m-x) = n \\
-  
 }
 $$
 

diff --git a/docs/algorithm/basic/DA03_StackAndQueue.md b/docs/algorithm/basic/DA03_StackAndQueue.md
@@ -17,7 +17,6 @@
   * **性质**：
 
 
-
 $$
 \displaylines{
 C_0 = 1 \ \  and \ \ C_{n+1} = \sum_{i=0}^nC_iC_{n-i}  \\

diff --git a/docs/algorithm/basic/DA05_BinaryTree.md b/docs/algorithm/basic/DA05_BinaryTree.md
@@ -24,7 +24,6 @@ $$
   ... \\
   f(n) = f(n-1)f(0) + f(n-2)f(1) + ... + f(1)f(n-2) + f(n-1)f(0) \\
   the\ solution\ is \ f(n) = \frac{C_{2n}^n}{n+1}
-  
 }
 $$
 
@@ -93,7 +92,6 @@ $$
         I_{n+1} = I_n + k \\
         E_{n+1} = E_n -k+2(k+1) = E_n+k+2 \\
         E_{n+1} = I_{n+1}+2(n+1)
-        
 }
 $$
 
@@ -110,7 +108,6 @@ $$
 $$
 \displaylines{
     P=NK - (N-1) = N(K-1)+1
-    
 }
 $$
 
@@ -123,7 +120,6 @@ $$
     N = N_0 + N_1 + N_2  = E +1\\
     E = N_1 + 2N_2 \\
     \Rightarrow N_0 = N_2 + 1
-    
 }
 $$
 
@@ -515,7 +511,6 @@ $$
   \frac {D(N)} {N+1} = \frac {D(N-1)}{N} + \frac{2(N-1)}{N(N+1)} \\
   f(n) = f(n-1)+O(1/n) \\
   f(n) \in \sum_{i=1}^nO(1/n) = O(log\ n)
-  
 }
 $$
 
@@ -530,7 +525,6 @@ $$
   \sum_{i=1}^{N}\frac 1 i \in O(logN) \\
   this \ follows \ continuous \ condition: \\
   \int \frac 1 xdx = lnx
-  
 }
 $$
 
@@ -627,7 +621,6 @@ $$
 $$
 \displaylines{
   \sum_{i=0}^{logn}2^i(logn - i) = \sum_{j=0}^{logn}n\frac j {2^j} \lt 2n
-  
 }
 $$
 

diff --git a/docs/algorithm/basic/DA10_Search.md b/docs/algorithm/basic/DA10_Search.md
@@ -31,9 +31,7 @@ $$
 
 $$
 \displaylines{
-
   \sum_{i=1}^n\frac 1 n(n-i+1) = \frac {n+1} 2
-  
 }
 $$
 
@@ -133,9 +131,7 @@ $$
 
 $$
 \displaylines{
-
   x_{(a)} \rightarrow x_{(b)} = y_{(a)}
-  
 }
 $$
 
@@ -219,10 +215,8 @@ $$
 
 $$
 \displaylines{
-
   ASL_{succ} = \frac 1 {11} (1+5+1+2+2+1+1+1+1+2+3) = 20/11 \\
   ASL_{fail} = \frac 1 {13} (8+7+6+5+4+3+2+1+1+1+2+1+11) = 4
-  
 }
 $$
 
@@ -236,10 +230,8 @@ $$
 
 $$
 \displaylines{
-
     p(K, 2i-1) = i^2 \\
     p(K, 2i) = -i^2 
-    
 }
 $$
 
@@ -251,9 +243,7 @@ $$
 
 $$
 \displaylines{
-
     p(K, i) = perm[i-1]
-    
 }
 $$
 

diff --git a/docs/algorithm/basic/DA11_Index.md b/docs/algorithm/basic/DA11_Index.md
@@ -212,7 +212,6 @@
 $$
 \displaylines{
   m = l + m' = l + n - 1 = \sum_{i=1}^n(x_i + 1) = N + n
-  
 }
 $$
 
@@ -346,7 +345,6 @@ $$
   h \le 2k+1 \\
   n \ge 2^k - 1 \\
   \Rightarrow h\le 2log_2(n+1)+1
-  
 }
 $$
 

diff --git a/docs/algorithm/basic/DA12_Advanced.md b/docs/algorithm/basic/DA12_Advanced.md
@@ -11,10 +11,8 @@
 
 $$
 \displaylines{
-
   loc(A[j_0, ..., j_{n-1}]) = loc(A[0,...,0])\\
   + size*[\sum_{i=0}^{n-2}(j_i\prod_{k=i+1}^{n-1}d_k)+j_{n-1}]
-  
 }
 $$
 
@@ -105,12 +103,10 @@ TAIL: 广义表中**第一个元素以外的元素构成的(!)广义表(!)**。
 
 $$
 \displaylines{
-
   \frac {p_i} W = \frac {q_j} W = \frac 1 {2n+1} \\
   ASL(n)  = \frac 1 {2n+1} (\sum_{i=1}^n(H(p_i)+1)+\sum_{j=0}^nH(q_j)) \\
   = \frac 1 {2n+1} (I+n+E)\\
   = \frac {2I+3n} {2n+1}
-
 }
 $$
 

diff --git a/docs/algorithm/review/Python.md b/docs/algorithm/review/Python.md
@@ -70,7 +70,6 @@ l = [x in l if x != val]
   Not well supported. Use list comprehension or filter.
 
 
-
 ### set
 
 ```python

diff --git a/docs/algorithm/review/old/字符串.md b/docs/algorithm/review/old/字符串.md
@@ -173,7 +173,6 @@
   ```
 
 
-
 ### 最长公共子串
 
 求N个字符串的最长公共子串（每个字符串最长为L）

diff --git a/docs/algorithm/review/problem_series/search_2d_matrix.md b/docs/algorithm/review/problem_series/search_2d_matrix.md
@@ -3,7 +3,6 @@
 在一个 n * m 的二维数组中，每一行都按照从左到右递增的顺序排序，每一列都按照从上到下递增的顺序排序。请完成一个高效的函数，输入这样的一个二维数组和一个整数，判断数组中是否含有该整数。
 
 
-
 并不是二维二分！没有二维二分法！
 
 ```

diff --git a/docs/algorithm/review/problem_series/【滑窗哈希】重复子串.md b/docs/algorithm/review/problem_series/【滑窗哈希】重复子串.md
@@ -88,9 +88,7 @@ properties:
 
 $$
 \displaylines{
-
   hash(s[i:j+1]) = hash(s[:j]) - hash(s[:i-1])\cdot P^{j-i+1}
-
 }
 $$
 

diff --git a/docs/algorithm/review/区间问题.md b/docs/algorithm/review/区间问题.md
@@ -73,7 +73,6 @@ Q[i, j] = sums[j] - sums[i-1]
   ```
 
 
-
 #### 差分
 
 支持$O(1)$的区间修改，以及均摊$O(1)$的单点查询（需要查询所有单点的值）。
@@ -114,7 +113,6 @@ arr[i] = diff[i] + arr[i-1]
   ```
   
   
-
 * [1D discretized array](https://leetcode.cn/problems/my-calendar-iii/)
 
   区间范围远大于区间数量时，需要使用map模拟稀疏数组。
@@ -140,8 +138,6 @@ arr[i] = diff[i] + arr[i-1]
   ```
 
 
-
-
 * 2D array
 
   通常用`s[i][j]`表示**右下子矩阵`x[i:][j:]`**的增量，这样可以正向循环求原始矩阵值。
@@ -168,8 +164,6 @@ arr[i] = diff[i] + arr[i-1]
   ```
 
   
-
-
 #### 树状数组
 
 支持$O(\log n)$时间的单点更新与区间求和。
@@ -219,8 +213,6 @@ arr[i] = diff[i] + arr[i-1]
   ```
 
 
-
-
 #### 线段树
 
 支持$O(\log n)$时间的区间更新与区间求和，但常数较大。
@@ -439,15 +431,13 @@ int query(int i, int j) {
   ```
 
 
-
 * [增量](https://leetcode.cn/problems/my-calendar-iii/)
 
   ```cpp
 
   ```
 
 
-
 * 动态
 
   ```cpp

diff --git a/docs/algorithm/review/双指针.md b/docs/algorithm/review/双指针.md
@@ -15,8 +15,6 @@
   若要找到环头，则相遇之后再从起点跑一个指针，此指针与slow指针相遇时即到达环头。
 
 
-
-
 ### 有序数组中的搜索
 
 ##### 两数之和

diff --git a/docs/algorithm/review/基础.md b/docs/algorithm/review/基础.md
@@ -66,7 +66,6 @@
   ```
 
   
-
 * fast power
 
   ```c++
@@ -96,7 +95,6 @@
   ```
 
 
-
 * Fibonacci
 
   ```c++
@@ -265,7 +263,6 @@
   ```
 
   
-
 * GCD/LCM
 
   ```c++