On the 2-MRS Problem in a Tree with Unreliable Edges

Algorithm 1

Algorithm SUM-MAX.

Input: an undirected tree 𝑇=(𝑉,𝐸,𝑝) with each edge 𝑒∈𝐸 having

a probability weight 𝑝(𝑒)∈(0,1);

Output: a Sum-Max 2-MRS ⟨𝑢^∗,𝑣^∗⟩ of 𝑇.

Step 0 𝑖←1;

Step 1 Use DFS to traverse 𝑇 with 𝑢_𝑖 as the origin, store the

    resultant rooted tree 𝑇_𝑢𝑖, record 𝑓_𝑢𝑖(𝑢_𝑗) and compute

    Pr(𝑢_𝑖,𝑢_𝑗) for all 𝑢_𝑗∈𝑉, and store 𝐷_𝑢𝑖;

    if  𝑖≤|𝑉|  then  𝑖←𝑖+1; goto Step 1;

    else  𝑖←1; goto Step 2; endif

Step 2  for  ℎ=𝐻_𝑢𝑖,𝐻_𝑢𝑖−1,…,1,0  do

   for all nodes 𝑢_𝑗 on the ℎ-level of 𝑇_𝑢𝑖  do

    𝒳^𝜆_𝑢𝑖(𝑢_𝑗)←1;

    if  𝑢_𝑗∉𝐷_𝑢𝑖  then

     Compute 𝒳^𝜆_𝑢𝑖(𝑢_𝑗) by (30);

    else break; endif

   endfor

  endfor

Step 3  for  ℎ=𝐻_𝑢𝑖,𝐻_𝑢𝑖−1,…,1,0  do

   for all nodes 𝑢_𝑗 on the ℎ-level of 𝑇_𝑢𝑖  do

    𝒴^𝜆_𝑢𝑖(𝑢_𝑗)←0;

    if  𝑢_𝑗≠𝑢_𝑖  then

     𝑤←𝑢_𝑗; 𝑘←ℎ;

     while  𝑘≥1  do

      𝑤^′←𝑤; 𝑤←𝑓_𝑢𝑖(𝑤); 𝑠^𝑢𝑗_𝑢𝑖(𝑤)←𝑤^′; 𝑘←𝑘−1;

      Compute ℱ_𝜆(𝑢_𝑖,𝑢_𝑗;𝑤) by (19) or (20);

      𝒴^𝜆_𝑢𝑖(𝑢_𝑗)←𝒴^𝜆_𝑢𝑖(𝑢_𝑗)+ℱ_𝜆(𝑢_𝑖,𝑢_𝑗;𝑤)×

        (𝒳^𝜆_𝑢𝑖(𝑤)−𝑝(𝑤,𝑠^𝑢𝑗_𝑢𝑖(𝑤))𝒳^𝜆_𝑢𝑖(𝑠^𝑢𝑗_𝑢𝑖(𝑤)));

     endwhile

    else  goto Step 4;  endif

   endfor

  endfor

Step 4  for  𝑗=1,2,…,|𝑉|  do

   if  𝑗≠𝑖  then

    Compute 𝔼^𝜆_𝑢𝑖[𝑢_𝑗] by (28);

   else break;  endif

    endfor

    if  𝑖<|𝑉|  then

   𝑖←𝑖+1; goto Step 2;

    else

   Find the maximum of all the values of 𝔼^𝜆_𝑢𝑖[𝑢_𝑗] and

   then determine 𝑢^∗_𝑖 and 𝑢^∗_𝑗 by (34);

    endif