Triangle inequality in the context of Vector magnitude

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.

A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.

There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights. Currency is an acceptable unit for edge weight – there is no requirement for edge lengths to obey normal rules of geometry such as the triangle inequality. A spanning tree for that graph would be a subset of those paths that has no cycles but still connects every house; there might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost, representing the least expensive path for laying the cable.

In mathematics, a normed vector space or normed space is a vector space, typically over the real or complex numbers, on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If ${\displaystyle V}$ is a vector space over ${\displaystyle K}$, where ${\displaystyle K}$ is a field equal to ${\displaystyle \mathbb {R} }$ or to ${\displaystyle \mathbb {C} }$, then a norm on ${\displaystyle V}$ is a map ${\displaystyle V\to \mathbb {R} }$, typically denoted by ${\displaystyle \lVert \cdot \rVert }$, satisfying the following four axioms:

1. Non-negativity: for every ${\displaystyle x\in V}$,${\displaystyle \;\lVert x\rVert \geq 0}$.
2. Positive definiteness: for every ${\displaystyle x\in V}$, ${\displaystyle \;\lVert x\rVert =0}$ if and only if ${\displaystyle x}$ is the zero vector.
3. Absolute homogeneity: for every ${\displaystyle \lambda \in K}$ and ${\displaystyle x\in V}$,$${\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }$$
4. Triangle inequality: for every ${\displaystyle x\in V}$ and ${\displaystyle y\in V}$,$${\displaystyle \|x+y\|\leq \|x\|+\|y\|.}$$

If ${\displaystyle V}$ is a real or complex vector space as above, and ${\displaystyle \lVert \cdot \rVert }$ is a norm on ${\displaystyle V}$, then the ordered pair ${\displaystyle (V,\lVert \cdot \rVert )}$ is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by ${\displaystyle V}$.

In mathematics and computer science, a string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching or comparison and in fuzzy string searching. A requirement for a string metric (e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to be close. A string metric provides a number indicating an algorithm-specific indication of distance.

The most widely known string metric is a rudimentary one called the Levenshtein distance (also known as edit distance). It operates between two input strings, returning a number equivalent to the number of substitutions and deletions needed in order to transform one input string into another. Simplistic string metrics such as Levenshtein distance have expanded to include phonetic, token, grammatical and character-based methods of statistical comparisons.