Merge branch 'master' of github.com:lucashc/programmingCheatSheet

stholtrop · stholtrop · commit 975e4b3d9501 · 2019-09-17T10:26:33.000+02:00
diff --git a/algorithms/Extended Euclidean algorithm.py b/algorithms/Extended Euclidean algorithm.py
@@ -0,0 +1,9 @@
+# Extended Euclidean algorithm
+def xgcd(a, b):
+    """return (g, x, y) such that a*x + b*y = g = gcd(a, b)"""
+    x0, x1, y0, y1 = 0, 1, 1, 0
+    while a != 0:
+        q, b, a = b // a, a, b % a
+        y0, y1 = y1, y0 - q * y1
+        x0, x1 = x1, x0 - q * x1
+    return b, x0, y0
diff --git a/algorithms/chinese_remainder_theorem.py b/algorithms/chinese_remainder_theorem.py
@@ -0,0 +1,24 @@
+# Chinese Remainder Theorem
+## \note only works for coprime numbers
+from functools import reduce
+def chinese_remainder(n, a):
+    sum=0
+    prod=reduce(lambda a, b: a*b, n)
+    for n_i, a_i in zip(n,a):
+        p=prod/n_i
+        sum += a_i* mul_inv(p, n_i)*p
+    return sum % prod
+
+
+def mul_inv(a, b):
+    b0= b
+    x0, x1= 0,1
+    if b== 1: return 1
+    while a>1 :
+        q=a// b
+        a, b= b, a%b
+        x0, x1=x1 -q *x0, x0
+    if x1<0 : x1+= b0
+    return x1
+
+
diff --git a/algorithms/gcd.py b/algorithms/gcd.py
diff --git a/algorithms/maxflow.py b/algorithms/maxflow.py
@@ -0,0 +1,37 @@
+# Max Flow (Ford-Fulkerson)
+## \note Using a matrix $g$ with source $s$ and sink $t$
+from queue import Queue
+import math
+def max_flow(s, t, g):
+    m = 0
+    path = bfs(s, t, g)
+    while path[t] != -1:
+        flow = math.inf
+        current = t
+        while current != s:
+            flow = min(flow, g[path[current]][current])
+            current = path[current]
+        current = t
+        while current != s:
+            g[path[current]][current] -= flow
+            g[current][path[current]] += flow
+            current = path[current]
+        m += flow
+        path = bfs(s, t, g)
+    return m
+
+def bfs(s, t, g):
+    visited = [False for i in range(len(g))]
+    q = Queue()
+    q.put(s)
+    visited[s] = True
+    path = [-1 for i in range(len(g))]
+    while not q.empty():
+        n = q.get()
+        if n == t: break
+        for i in range(len(g)):
+            if not visited[i] and g[n][i] > 0:
+                q.put(i)
+                path[i] = n
+                visited[i] = True
+    return path
diff --git a/main.tex b/main.tex
@@ -57,6 +57,8 @@ \section{Modules}
 \subsection{Python}
 \begin{itemize}
 	\item Priority queue \arrow \ms{queue.PriorityQueue}
+	\item Python base switching \arrow \ms{int(x, base=b)}
+	\item Deque \arrow \ms{collections.deque}
 \end{itemize}
 \subsection{C++}
 \begin{itemize}
@@ -121,7 +123,7 @@ \subsection*{Links of rechts ombuigen}\vspace{-0.5em}
 	\draw[->, >=latex] (0mm, 0mm) node[left] {A} -- (15mm, 10mm) node[above left] {B};
 	\draw[->, >=latex] (15mm, 10mm) -- (40mm, 20mm) node[above] {C};
 	\end{tikzpicture}
-\end{minipage}%
+\end{minipage}
 \begin{minipage}{0.25\linewidth}
 	\begin{align*}
 	\vec{AB} &= \begin{bmatrix} p \\ q \end{bmatrix} \\
@@ -179,6 +181,31 @@ \subsection*{Point to line distance}
 \begin{equation*}
 	d(ax+by+c=0, (x_0, y_0)) = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}
 \end{equation*}
+\subsection*{Number theory}
+$d(n)$ is het aantal positieve delers van een positief geheel getal $n$, met 1 en $n$ inbegrepen.
+\begin{align*}
+	d(1) + d(2) + \ldots + d(n) &= \lfloor \frac{n}{1} \rfloor + \lfloor \frac{n}{2} \rfloor + \ldots + \lfloor \frac{n}{n} \rfloor
+\end{align*}
+
+Aantal factoren van een priemgetal $a$ in $n!$ is:
+\begin{align*}
+	\lfloor \frac{n}{a} \rfloor + \rfloor \frac{n}{a^2} \rfloor + \ldots
+\end{align*}
+Kleine stelling van Fermat, $p$ priem en $a>0$:
+\begin{align*}
+	a^p \equiv a \mod p
+\end{align*}
+Als $a$ en $p$ copriem zijn:
+\begin{align*}
+	a^{p-1} \equiv 1 \mod p
+\end{align*}
+\subsection*{Series}
+Geometric series with $r$:
+\begin{align*}
+	a + ar^2 + \ldots + ar^{n-1} &= \sum^{n-1}_{k=0} ar^{k} = a\left(\frac{1-r^n}{1-r}\right) \\
+	\lim_{n\rightarrow \infty} S &= \frac{a}{1-r} \quad \forall |r| < 1
+\end{align*}
+
 \section{Algorithms}
 
 %PLACEHOLDER%
