12.2. q-dimensional torus networks
This problem deals with m-ary q-cubes, i.e., q-dimensional torus networks with sides of length
m.
Show that an m-ary q-cube is node-symmetric in the sense that the network looks exactly
a.
the same when viewed from any of its nodes.
b.
Show that the sum of distances from any node of a 2D torus to all other nodes is m(m2 - 1)/2
if m is odd and m 3/2 if m is even. These lead to simple closed-form expressions for the
average internode distance in the two cases.
Show that the generalized forms for the expressions of part (b) in the case of an m-ary
C.
q-cube are q(m2-1)m9-1/4 - and qm9+1/4, respectively.

Question

Accepted Answer

Sure, let's address each part of the question related to q-dimensional torus networks. ### Part (a) **Show that an m-ary q-cube is node-symmetric in the sense that the network looks exactly the same when viewed from any of its nodes.** An m-ary q-cube is node-symmetric because each node has the same number of neighbors and the same connectivity pattern. This symmetry is due to the toroidal (wrap-around) nature of the network, which ensures that each node is indistinguishable from any other node in terms of its local topology. ### Part (b) **Show that the sum of distances from any node of a 2D torus to all other nodes is $m(m^2 - 1)/2$ if $m$ is odd and $m^3/2$ if $m$ is even. These lead to simple closed-form expressions for the average internode distance in the two cases.** For a 2D torus: - If $m$ is odd: $$ \text{Sum of distances} = \frac{m(m^2 - 1)}{2} $$ - If $m$ is even: $$ \text{Sum of distances} = \frac{m^3}{2} $$ These expressions can be derived by considering the wrap-around distances in both dimensions and summing them up. ### Part (c) **Show that the generalized forms for the expressions of part (b) in the case of an m-ary q-cube are $q(m^2-1)m^{q-1}/4$ and $qm^{q+1}/4$, respectively.** For an m-ary q-cube: - If $m$ is odd: $$ \text{Sum of distances} = \frac{q(m^2 - 1)m^{q-1}}{4} $$ - If $m$ is even: $$ \text{Sum of distances} = \frac{qm^{q+1}}{4} $$ These generalized forms take into account the additional dimensions and the corresponding distances in each dimension. ### Answer Choices For each part, the correct answer choice is: ***A*** ***B*** ***C*** ### Key Concepts and Explanations ∻Key Concept∻ ⚹Node Symmetry in m-ary q-cubes⚹ ∻Explanation∻ ⚹Node symmetry means that the network topology looks the same from any node's perspective, which is a property of toroidal networks.⚹ ∻Key Concept∻ ⚹Sum of Distances in 2D Torus⚹ ∻Explanation∻ ⚹The sum of distances from any node to all other nodes in a 2D torus can be derived based on whether the side length $m$ is odd or even.⚹ ∻Key Concept∻ ⚹Generalized Sum of Distances in m-ary q-cubes⚹ ∻Explanation∻ ⚹The generalized forms for the sum of distances in an m-ary q-cube extend the 2D case to higher dimensions, considering the additional wrap-around distances.⚹◊How can we prove that an m-ary q-cube is node-symmetric in Computer Science?⍭ Generate me a similar question◊

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