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codebook/4_Flow_Matching/Model.tex

@@ -1,34 +1,34 @@
 % \normalsize
 \begin{itemize}
-    \itemsep-0.3em
+    %\itemsep-0.3em
     \item Maximum/Minimum flow with lower bound / Circulation problem
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-        \itemsep-0.3em
+        %\itemsep-0.3em
         \item Construct super source $S$ and sink $T$.
         \item For each edge $(x, y, l, u)$, connect $x \rightarrow y$ with capacity $u - l$.
         \item For each vertex $v$, denote by $in(v)$ the difference between the sum of incoming lower bounds and the sum of outgoing lower bounds.
         \item If $in(v) > 0$, connect $S \rightarrow v$ with capacity $in(v)$, otherwise, connect $v \rightarrow T$ with capacity $-in(v)$.
         \begin{itemize}
-            \itemsep-0.2em
+            %\itemsep-0.2em
             \item To maximize, connect $t \rightarrow s$ with capacity $\infty$ (skip this in circulation problem), and let $f$ be the maximum flow from $S$ to $T$. If $f \neq \sum_{v \in V, in(v) > 0}{in(v)}$, there's no solution. Otherwise, the maximum flow from $s$ to $t$ is the answer.
             \item To minimize, let $f$ be the maximum flow from $S$ to $T$. Connect $t \rightarrow s$ with capacity $\infty$ and let the flow from $S$ to $T$ be $f^\prime$. If $f + f^\prime \neq \sum_{v \in V, in(v) > 0}{in(v)}$, there's no solution. Otherwise, $f^\prime$ is the answer.
         \end{itemize}
         \item The solution of each edge $e$ is $l_e + f_e$, where $f_e$ corresponds to the flow of edge $e$ on the graph.
     \end{enumerate}
     \item Construct minimum vertex cover from maximum matching $M$ on bipartite graph $(X, Y)$
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-        \itemsep-0.3em
+        %\itemsep-0.3em
         \item Redirect every edge: $y \rightarrow x$ if $(x, y) \in M$, $x \rightarrow y$ otherwise.
         \item DFS from unmatched vertices in $X$.
         \item $x \in X$ is chosen iff $x$ is unvisited.
         \item $y \in Y$ is chosen iff $y$ is visited.
     \end{enumerate}
     \item Minimum cost cyclic flow
-    \vspace{-0.5em}
+    %\vspace{-0.5em}
     \begin{enumerate}
-        \itemsep-0.3em
+        %\itemsep-0.3em
         \item Consruct super source $S$ and sink $T$
         \item For each edge $(x, y, c)$, connect $x \rightarrow y$ with $(cost, cap) = (c, 1)$ if $c > 0$, otherwise connect $y \rightarrow x$ with $(cost, cap) = (-c, 1)$
         \item For each edge with $c < 0$, sum these cost as $K$, then increase $d(y)$ by 1, decrease $d(x)$ by 1
@@ -37,9 +37,9 @@
         \item Flow from $S$ to $T$, the answer is the cost of the flow $C + K$
     \end{enumerate}
     \item Maximum density induced subgraph
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-        \itemsep-0.3em
+        %\itemsep-0.3em
         \item Binary search on answer, suppose we're checking answer $T$
         \item Construct a max flow model, let $K$ be the sum of all weights
         \item Connect source $s \rightarrow v$, $v \in G$ with capacity $K$
@@ -48,25 +48,25 @@
         \item $T$ is a valid answer if the maximum flow $f < K \lvert V \rvert$
     \end{enumerate}
     \item Minimum weight edge cover
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-        \itemsep-0.3em
+        %\itemsep-0.3em
       \item For each $v \in V$ create a copy $v^\prime$, and connect $u^\prime \to v^\prime$ with weight $w(u, v)$.
       \item Connect $v \to v^\prime$ with weight $2\mu(v)$, where $\mu(v)$ is the cost of the cheapest edge incident to $v$.
       \item Find the minimum weight perfect matching on $G^\prime$.
     \end{enumerate}
     \item Project selection problem
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-      \itemsep-0.3em
+      %\itemsep-0.3em
       \item If $p_v > 0$, create edge $(s, v)$ with capacity $p_v$; otherwise, create edge $(v, t)$ with capacity $-p_v$.
       \item Create edge $(u, v)$ with capacity $w$ with $w$ being the cost of choosing $u$ without choosing $v$.
       \item The mincut is equivalent to the maximum profit of a subset of projects.
     \end{enumerate}
     \item Dual of minimum cost maximum flow
-    \vspace{-1em}
+    %\vspace{-1em}
     \begin{enumerate}
-      \itemsep-0.3em
+      %\itemsep-0.3em
       \item Capacity $c_{uv}$, Flow $f_{uv}$, Cost $w_{uv}$, Required Flow difference for vertex $b_u$.
       \item If all $w_{uv}$ are integers, then optimal solution can happen when all $p_u$ are integers.
     \end{enumerate}
@@ -76,11 +76,11 @@
     \begin{aligned}\min\sum_{u} b_up_u + \sum_{uv}c_{uv}\max(0, p_v - p_u - w_{uv}) \\ p_u \geq 0 \end{aligned}
     $$
     %\item 0/1 quadratic programming
-    %\vspace{-1em}
+    %%\vspace{-1em}
     %\[ \sum_x{c_xx} + \sum_y{c_y\bar{y}} + \sum_{xy}c_{xy}x\bar{y} + \sum_{xyx^\prime y^\prime}c_{xyx^\prime y^\prime}(x\bar{y} + x^\prime\bar{y^\prime}) \]
     %can be minimized by the mincut of the following graph:
     %\begin{enumerate}
-    %  \itemsep-0.3em
+    %  %\itemsep-0.3em
     %  \item Create edge $(x, t)$ with capacity $c_x$ and create edge $(s, y)$ with capacity $c_y$.
     %  \item Create edge $(x, y)$ with capacity $c_{xy}$.
     %  \item Create edge $(x, y)$ and edge $(x^\prime, y^\prime)$ with capacity $c_{xyx^\prime y^\prime}$.

codebook/6_Math/Estimation.tex

@@ -2,7 +2,7 @@
 \item Estimation
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item The number of divisors of $n$ is at most around $100$ for $n<5e4$, $500$ for $n<1e7$, $2000$ for $n<1e10$, $200000$ for $n<1e19$.
     \item The number of ways of writing $n$ as a sum of positive integers, disregarding the order of the summands. $1, 1, 2, 3, 5, 7, 11, 15, 22, 30$ for $n=0\sim 9$, $627$ for $n=20$, $\sim 2e5$ for $n=50$, $\sim 2e8$ for $n=100$.
     \item Total number of partitions of $n$ distinct elements: $B(n)=1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597,$\\

codebook/6_Math/Euclidean.tex

@@ -1,5 +1,5 @@
 \begin{itemize}
-  \itemsep-0.5em
+  %\itemsep-0.5em
   \item $m = \lfloor\frac{an + b}{c}\rfloor$
   \item Time complexity: $O(\log{n})$
 \end{itemize}

codebook/6_Math/Generating_function.tex

@@ -2,7 +2,7 @@
 \item Ordinary Generating Function
 $A(x) = \sum_{i\ge 0} a_ix^i$
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item $A(rx)             \Rightarrow r^na_n$
     \item $A(x) + B(x)       \Rightarrow a_n + b_n$
     \item $A(x)B(x)          \Rightarrow \sum_{i=0}^{n} a_ib_{n-i}$
@@ -13,7 +13,7 @@
 \item Exponential Generating Function
 $A(x) = \sum_{i\ge 0} \frac{a_i}{i!}x_i$
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item $A(x) + B(x)       \Rightarrow a_n + b_n$
     \item $A^{(k)}(x)        \Rightarrow a_{n+k}$
     \item $A(x)B(x)          \Rightarrow \sum_{i=0}^{n} \binom{n}{i}a_ib_{n-i}$
@@ -22,7 +22,7 @@
 \end{itemize}
 \item Special Generating Function
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item $(1+x)^n           = \sum_{i\ge 0} \binom{n}{i}x^i$
     \item $\frac{1}{(1-x)^n} = \sum_{i\ge 0} \binom{i}{n-1}x^i$
 \end{itemize}

codebook/6_Math/SimplexConstruction.tex

@@ -4,13 +4,13 @@
 $\bar{\mathbf{x}}$ and $\bar{\mathbf{y}}$ are optimal if and only if for all $i \in [1, n]$, either $\bar{x}_i = 0$ or $\sum_{j=1}^{m}A_{ji}\bar{y}_j = c_i$ holds and for all $i \in [1, m]$ either $\bar{y}_i = 0$ or $\sum_{j=1}^{n}A_{ij}\bar{x}_j = b_j$ holds.
 
 \begin{enumerate}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item In case of minimization, let $c^\prime_i = -c_i$
     \item $\sum_{1 \leq i \leq n}{A_{ji}x_i} \geq b_j \rightarrow \sum_{1 \leq i \leq n}{-A_{ji}x_i} \leq -b_j$
     \item $\sum_{1 \leq i \leq n}{A_{ji}x_i} = b_j$ 
-        \vspace{-0.5em}
+        %\vspace{-0.5em}
         \begin{itemize}
-            \itemsep-0.5em
+            %\itemsep-0.5em
             \item $\sum_{1 \leq i \leq n}{A_{ji}x_i} \leq b_j$
             \item $\sum_{1 \leq i \leq n}{A_{ji}x_i} \geq b_j$
         \end{itemize}

codebook/6_Math/Theorem.tex

@@ -15,7 +15,7 @@
 
 Denote $L$ be a $n \times n$ matrix as the Laplacian matrix of graph $G$, where $L_{ii} = d(i)$, $L_{ij} = -c$ where $c$ is the number of edge $(i, j)$ in $G$.
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item The number of undirected spanning in $G$ is $\lvert \det(\tilde{L}_{11}) \rvert$.
     \item The number of directed spanning tree rooted at $r$ in $G$ is $\lvert \det(\tilde{L}_{rr}) \rvert$.
 \end{itemize}
@@ -27,7 +27,7 @@
 \item Cayley's Formula
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item Given a degree sequence $d_1, d_2, \ldots, d_n$ for each \textit{labeled} vertices, there are $\frac{(n - 2)!}{(d_1 - 1)!(d_2 - 1)!\cdots(d_n - 1)!}$ spanning trees.
   \item Let $T_{n, k}$ be the number of \textit{labeled} forests on $n$ vertices with $k$ components, such that vertex $1, 2, \ldots, k$ belong to different components. Then $T_{n, k} = kn^{n - k - 1}$.
 \end{itemize}
@@ -47,15 +47,15 @@
 \item Möbius inversion formula
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item $f(n)=\sum_{d\mid n}g(d)\Leftrightarrow g(n)=\sum_{d\mid n}\mu(d)f(\frac{n}{d})$
   \item $f(n)=\sum_{n\mid d}g(d)\Leftrightarrow g(n)=\sum_{n\mid d}\mu(\frac{d}{n})f(d)$
 \end{itemize}
 
 \item Spherical cap
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item A portion of a sphere cut off by a plane.
   \item $r$: sphere radius, $a$: radius of the base of the cap, $h$: height of the cap, $\theta$: $\arcsin(a/r)$.
   \item Volume $=\pi h^2(3r-h)/3=\pi h(3a^2+h^2)/6=\pi r^3(2+\cos\theta)(1-\cos\theta)^2/3$.
@@ -65,7 +65,7 @@
 \item Lagrange multiplier
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item Optimize $f(x_1, \ldots, x_n)$ when $k$ constraints $g_i(x_1, \ldots, x_n)=0$.
   \item Lagrangian function $\mathcal{L}(x_1, \ldots, x_n, \lambda_1, \ldots, \lambda_k) = f(x_1, \ldots, x_n) = \sum^{k}_{i=1}\lambda_i g_i(x_1, \ldots, x_n)$.
   \item The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function.

codebook/9_Else/Matroid.tex

@@ -1,14 +1,14 @@
 Start from $S = \emptyset$. In each iteration, let 
-\vspace{-0.5em}
+%\vspace{-0.5em}
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item $Y_1 = \{x \not\in S \mid S \cup \{x\} \in I_1 \}$
   \item $Y_2 = \{x \not\in S \mid S \cup \{x\} \in I_2 \}$
 \end{itemize}
 If there exists $x \in Y_1 \cap Y_2$, insert $x$ into $S$. Otherwise for each $x \in S, y \not\in S$, create edges
-\vspace{-0.5em}
+%\vspace{-0.5em}
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
   \item $x \to y$ if $S - \{x\} \cup \{y\} \in I_1$.
   \item $y \to x$ if $S - \{x\} \cup \{y\} \in I_2$.
 \end{itemize}

codebook/9_Else/Mos_Algorithm.tex

@@ -2,7 +2,7 @@
 \item Mo's Algorithm With Addition Only
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item Sort querys same as the normal Mo's algorithm.
     \item For each query $[l, r]$:
     \item If $l / blk = r / blk$, brute-force.
@@ -14,7 +14,7 @@
 \item Mo's Algorithm With Offline Second Time
 
 \begin{itemize}
-    \itemsep-0.5em
+    %\itemsep-0.5em
     \item Require: Changing answer $\equiv$ adding $f([l, r], r + 1)$.
     \item Require: $f([l, r], r + 1) = f([1, r], r + 1) - f([1, l), r + 1)$.
     \item Part1: Answer all $f([1, r], r + 1)$ first.

codebook/codebook.tex

@@ -13,9 +13,11 @@
 \usepackage{fancyhdr}								%設定頁首頁尾
 \usepackage{graphicx}								%Graphic
 \usepackage{enumerate}
+\usepackage{multicol}
 \usepackage{titlesec}
 \usepackage{amsmath}
 \usepackage[CheckSingle, CJKmath]{xeCJK}
+\usepackage{savetrees}
 % \usepackage{CJKulem}
 
 %\usepackage[T1]{fontenc}
@@ -33,20 +35,17 @@
 \evensidemargin=0pt
 \hoffset=-42pt
 
-\titlespacing\subsection{0pt}{4pt plus 2pt minus 2pt}{0pt plus 2pt minus 2pt}
+\titlespacing\section{0pt}{0pt plus 2pt minus 2pt}{0pt plus 2pt minus 2pt}
+\titlespacing\subsection{0pt}{0pt plus 2pt minus 2pt}{0pt plus 2pt minus 2pt}
 
 
 %\renewcommand\listfigurename{圖目錄}
 %\renewcommand\listtablename{表目錄} 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\setmainfont{Consolas}				%主要字型
-%\setmonofont{Monaco}				%主要字型
-\setmonofont{Consolas}
-\setCJKmainfont{Noto Sans CJK TC}
-% \setCJKmainfont{Consolas}			%中文字型
-%\setmainfont{sourcecodepro}
+\setmainfont{Ubuntu}				%主要字型
+\setmonofont{Ubuntu Mono}
 \XeTeXlinebreaklocale "zh"						%中文自動換行
 \XeTeXlinebreakskip = 0pt plus 1pt				%設定段落之間的距離
 \setcounter{secnumdepth}{3}						%目錄顯示第三層
@@ -125,7 +124,9 @@
 \renewcommand{\contentsname}{Contents} 
 
 \scriptsize
-\tableofcontents
+\begin{multicols}{2}
+  \tableofcontents
+\end{multicols}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %\newpage