GAP (HAP commands) - Chapter 33: Finite metric spaces and their filtered complexes

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33 Finite metric spaces and their filtered complexes
33.1
33.1-1 CayleyMetric
33.1-2 HammingMetric
33.1-3 KendallMetric
33.1-4 EuclideanSquaredMetric
33.1-5 EuclideanApproximatedMetric
33.1-6 ManhattanMetric
33.1-7 VectorsToSymmetricMatrix
33.1-8 SymmetricMatDisplay
33.1-9 SymmetricMatrixToFilteredGraph
33.1-10 PermGroupToFilteredGraph

### 33 Finite metric spaces and their filtered complexes

#### 33.1

##### 33.1-1 CayleyMetric

`‣ CayleyMetric`( g, h, N ) | ( function )

`‣ CayleyMetric`( g, h ) | ( function )

Inputs two permutations \(g,h\) and optionally the degree \(N\) of a symmetric group containing them. It returns the minimum number of transpositions needed to express \(g*h^-1\) as a product of transpositions.

Examples: 1

##### 33.1-2 HammingMetric

`‣ HammingMetric`( g, h, N ) | ( function )

`‣ HammingMetric`( g, h ) | ( function )

Inputs two permutations \(g,h\) and optionally the degree \(N\) of a symmetric group containing them. It returns the number of integers moved by the permutation \(g*h^-1\).

Examples:

##### 33.1-3 KendallMetric

`‣ KendallMetric`( g, h, N ) | ( function )

`‣ KendallMetric`( g, h ) | ( function )

Inputs two permutations \(g,h\) and optionally the degree \(N\) of a symmetric group containing them. It returns the minimum number of adjacent transpositions needed to express \(g^-1*h\) as a product of adjacent transpositions. An adjacent transposition has the form \((i,i+1)\).

Examples:

##### 33.1-4 EuclideanSquaredMetric

`‣ EuclideanSquaredMetric`( v, w ) | ( function )

Inputs two vectors \(v,w\) of equal length and returns the sum of the squares of the components of \(v-w\). In other words, it returns the square of the Euclidean distance between \(v\) and \(w\).

Examples:

##### 33.1-5 EuclideanApproximatedMetric

`‣ EuclideanApproximatedMetric`( v, w ) | ( function )

Inputs two vectors \(v,w\) of equal length and returns a rational approximation to the square root of the sum of the squares of the components of \(v-w\). In other words, it returns an approximation to the Euclidean distance between \(v\) and \(w\).

Examples: 1 , 2

##### 33.1-6 ManhattanMetric

`‣ ManhattanMetric`( v, w ) | ( function )

Inputs two vectors \(v,w\) of equal length and returns the sum of the absolute values of the components of \(v-w\). This is often referred to as the taxi-cab distance between \(v\) and \(w\).

Examples: 1

##### 33.1-7 VectorsToSymmetricMatrix

`‣ VectorsToSymmetricMatrix`( L ) | ( function )

`‣ VectorsToSymmetricMatrix`( L, D ) | ( function )

Inputs a list \(L\) of vectors and optionally a metric \(D\). The default is \(D=ManhattanMetric\). It returns the symmetric matrix whose i-j-entry is \(S[i][j]=D(L[i],L[j])\).

Examples: 1 , 2

##### 33.1-8 SymmetricMatDisplay

`‣ SymmetricMatDisplay`( S ) | ( function )

`‣ SymmetricMatDisplay`( L, V ) | ( function )

Inputs an \(n \times n\) symmetric matrix \(S\) of non-negative integers and an integer \(t\) in \([0 .. 100]\). Optionally it inputs a list \(V=[V_1, ... , V_k]\) of disjoint subsets of \([1 .. n]\). It displays the graph with vertex set \([1 .. n]\) and with an edge between \(i\) and \(j\) if \(S[i][j] < t\). If the optional list \(V\) is input then the vertices in \(V_i\) will be given a common colour distinct from other vertices.

Examples: 1

##### 33.1-9 SymmetricMatrixToFilteredGraph

`‣ SymmetricMatrixToFilteredGraph`( S, t, m ) | ( function )

Inputs an integer symmetric matrix \(S\), a positive integer \(t\) and a positive integer \(m\). The function returns a filtered graph of filtration length \(t\). The \(k\)-th term of the filtration is a graph with one vertex for each row of \(S\). There is an edge in this graph between the \(i\)-th and \(j\)-th vertices if the entry \(S[i][j]\) is less than or equal to \(k*m/t\).

Examples: 1 , 2 , 3

##### 33.1-10 PermGroupToFilteredGraph

`‣ PermGroupToFilteredGraph`( S, D ) | ( function )

Inputs a permutation group \(G\) and a metric \(D\) defined on permutations. The function returns a filtered graph. The \(k\)-th term of the filtration is a graph with one vertex for each element of the group \(G\). There is an edge in this graph between vertices \(g\) and \(h\) if \(D(g,h)\) is less than some integer threshold \(t_k\). The thresholds \(t_1 < t_2 < ... < t_N\) are chosen to form as long a sequence as possible subject to each term of the filtration being a distinct graph.

Examples:

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