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*h^-1 as a product of adjacent transpositions. An adjacent transposition has the for (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

##### 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

##### 33.1-8 SymmetricMatDisplay

`‣ SymmetricMatDisplay`( S ) | ( function )

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

Inputs an n × 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

##### 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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