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<Chapter><Heading> Finite metric spaces and their filtered complexes </Heading>

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<Table Align="|l|" >

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<Row>

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<Item>

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<Index> CayleyMetric</Index>

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<C>CayleyMetric(g,h,N) </C>

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<C>CayleyMetric(g,h) </C>

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<P/>

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Inputs two permutations <M>g,h</M> and optionally the degree <M>N</M> of a symmetric group containing them. It returns the minimum number of transpositions needed to 

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 express <M>g*h^-1</M> as a product of transpositions. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> HammingMetric</Index>

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<C>HammingMetric(g,h,N) </C>

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<C>HammingMetric(g,h) </C>

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<P/>

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Inputs two permutations <M>g,h</M> and optionally the degree <M>N</M> of a symmetric group containing them. It returns the  number of integers moved by the permutation 

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  <M>g*h^-1</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> KendallMetric</Index>

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<C>KendallMetric(g,h,N) </C>

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<C>KendallMetric(g,h) </C>

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<P/>

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

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> EuclideanSquaredMetric</Index>

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<C>EuclideanSquaredMetric(v,w) </C>

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<P/>

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Inputs two vectors <M>v,w</M> of equal length and returns 

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the sum of the squares of the components of <M>v-w</M>. In other words, it returns the square of the Euclidean distance between <M>v</M> and <M>w</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> EuclideanApproximatedMetric</Index>

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<C>EuclideanApproximatedMetric(v,w) </C>

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<P/>

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Inputs two vectors <M>v,w</M> of equal length and returns

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a rational approximation to the square root of the sum of the squares of the components of <M>v-w</M>. In other words, it returns an approximation to the Euclidean distance between <M>v</M> and <M>w</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> ManhattanMetric</Index>

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<C>ManhattanMetric(v,w) </C>

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<P/>

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Inputs two vectors <M>v,w</M> of equal length and returns

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the sum of the absolute values of the components of <M>v-w</M>. 

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 This is often referred to as the taxi-cab distance between <M>v</M> and <M>w</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> VectorsToSymmetricMatrix</Index>

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<C>VectorsToSymmetricMatrix(L) </C>

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<C>VectorsToSymmetricMatrix(L,D) </C>

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<P/>

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

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> SymmetricMatDisplay</Index>

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<C>SymmetricMatDisplay(S) </C>

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<C>SymmetricMatDisplay(L,V) </C>

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<P/>

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Inputs an <M>n \times n</M>

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 symmetric matrix  <M>S</M> of non-negative integers and an integer <M>t</M> in <M>[0 .. 100]</M>. Optionally it inputs a list <M>V=[V_1, ... , V_k]</M>

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of disjoint subsets of <M>[1 .. n]</M>.  It displays the graph with vertex set <M>[1 .. n]</M> and with an edge between <M>i</M> and <M>j</M> if <M>S[i][j] &lt; t</M>. If the optional list <M>V</M> is input then the vertices in <M>V_i</M> will be given a common colour distinct from other vertices.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> SymmetricMatrixToFilteredGraph</Index>

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<C>SymmetricMatrixToFilteredGraph(S,t,m) </C>

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<P/>

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Inputs an integer symmetric matrix <M>S</M>, a positive 

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integer <M>t</M> and a positive integer <M>m</M>. The function returns a 

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filtered graph of filtration length <M>t</M>.  The <M>k</M>-th term of the 

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filtration is a graph with one vertex for each row of <M>S</M>. There is an edge in this graph between the <M>i</M>-th and <M>j</M>-th vertices if the entry

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<M>S[i][j]</M> is less than or equal to <M>k*m/t</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> PermGroupToFilteredGraph</Index>

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<C>PermGroupToFilteredGraph(S,D) </C>

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<P/>

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Inputs a permutation group <M>G</M>  and  a metric <M>D</M> defined 

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on permutations.

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 The function returns a 

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filtered graph.  The <M>k</M>-th term of the 

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filtration is a graph with one vertex for each element of the group

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 <M>G</M>. There is an edge in this graph between vertices 

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 <M>g</M> and <M>h</M> if 

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<M>D(g,h)</M> is less than some integer threshold <M>t_k</M>. The thresholds

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<M>t_1 &lt; t_2 &lt; ... &lt; t_N</M> are chosen to form as long a sequence as possible subject to  each term of the filtration being a distinct graph.

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</Item>

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</Row>

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</Table>

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</Chapter>

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