Efficient Congestion Mitigation Using Congestion-Aware Steiner Trees and Network Coding Topologies

Algorithm 1

Algorithm NC.

Algorithm NC (𝐺,𝑠_𝑖,𝑠_𝑗,𝑇_𝑖𝑗,𝑇^𝑖,𝑇^𝑗):

 (1) Find a Steiner tree 𝜙_𝑖 connects 𝑠_𝑖 to terminals in 𝑇^𝑖

 (2) Find a Steiner tree 𝜙_𝑗 connects 𝑠_𝑗 to terminals in 𝑇^𝑗

 (3) For each link 𝑒∈𝑇^𝑖∪𝑇^𝑗 update 𝜂(𝑒)

 (4) Find a Steiner tree 𝜙 that connects s_𝑖 to terminals in 𝑇_𝑖𝑗

 (5) Assign zero cost to all edges in 𝜙

 (6) For each terminal 𝑡∈𝑇_𝑖𝑗 let 𝑑(𝑡) be the shortest distance between 𝑠_𝑗 and 𝑡

 (7) For all terminals 𝑡∈𝑇_𝑖𝑗 in the increasing order of 𝑑(𝑡) do:

 (8)  Let 𝐺^(𝑉^,𝐸^) be a graph formed from 𝐺(𝑉,𝐸) by reversing all edges in 𝑃_𝑖,𝑡, where 𝑃_𝑖,𝑡 is a path from 𝑠_𝑖 to 𝑡 in 𝜙

 (9)  Find shortest path 𝑃_𝑗,𝑡 from source 𝑠_𝑗 to terminal 𝑡 in 𝐺^(𝑉^,𝐸^)

 (10)   Assign zero cost to all edges of 𝑃_𝑗,𝑡

 (11) For each edge 𝑒(𝑣,𝑢)∈𝑃_𝑗,𝑡  do

 (12)   If there exists an edge 𝑒^(𝑢,𝑣)∈𝜙, remove 𝑒^(𝑢,𝑣) from 𝜙

 (13)  Otherwise, add 𝑒(𝑣,𝑢) to 𝜙

 (14) Return 𝜙,𝜙_𝑖, and 𝜙_𝑗