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Design Patterns
In software engineering, a software design pattern is a general, reusable solution to a commonly occurring problem within a given context in software design. It is not a finished design that can be transformed directly into source or machine code. Rather, it is a description or template for how to
Disjoint Set
A disjoint-set data structure maintains a collection \(\mathcal{S} = \{S_{1}, S_{2}, \dots, S_{k}\}\) of disjoint dynamic sets.
A weighted-union heuristic
Theorem 1
Using linked-list representation of disjoint sets and the weighted union heuristic, a sequence of \(m\) MAKE-SET, UNION, and FIND-SET operations, \(n\) of which are MAKE-SET
Graph
Breadth-first search
Lemma 1
Let \(G = (V, E)\) be a directed or undirected graph, and let \(s \in V\) be an arbitrary vertex. Then for any edge \((u, v) \in E\),\( \delta (s, v) \leq \delta (s, u) + 1\).
Lemma 2
Let \(G = (V, E)\) be a directed or undirected
Tree
Binary Tree
A binary tree \(T\) is a structure defined on a finite set of nodes that either
* contains no nodes, or
* is composed of three disjoint set of nodes: a root node, a binary tree called its left subtree, and a binary tree called its right subtree.
The binary
Divide-and-Conquer
Divide the problem into a number of sub-problems that are smaller instances of the same problem.
Conquer the sub-problems by solving them recursively. If the sub-problem sizes are small enough, however, just solve the sub-problems in a straightforward manner.
Combine the solutions to the sub-problems into a solution for the
Curiously Recurring Template Pattern
The curiously recurring template pattern (CRTP) is an idiom in C++ in which a class X derives from a class template instantiation using X itself as template argument. It is a form of F-bounded Quantification.
// The Curiously Recurring Template Pattern (CRTP)
template <class T>
class Base
{
// methods within
Calculus
Limits
\[ \lim_{x \to c} f(x) = L \]
Discontinuity
* Removable Discontinuity
* Jump Discontinuity
* Essential Discontinuity
Derivatives