Tree Assigner & Placer (TAP) Algorithm

To optimally span a tree in a weighted graph, low complexity algorithms are well known, e.g., Kruskal’s one [Kru56]. These greedy algorithms, however, investigate the links one by one and they do not consider paths at all. Then, the trees can be formed in such a way that not all links of a path will be added to the trees. This would cause malfunctions since the traffic cannot flow through the torn paths. Moreover, the algorithm to be pro-posed should define not only one but more trees at the same time, and should perform the assignment between these trees and the demands.

If Steiner trees are considered instead of spanning trees (as it was the case in Chapter 3) the problem can be reduced to the Minimal Set Cover problem. Let U be a set of basic elements andS = {s0...s_n|si ⊂ U}be a cover forU: Sn

i=1s_i = U. The goal is to find minimalS^∗ ⊂S. To adopt this formulation letU be the set of paths given by the Demand Routing algorithm, andSbe the set of all allowed Steiner trees (their roots reside at ANs).

This problem, however, is known to be NP-hard [Sla96], therefore heuristic methods are required.

The proposed novel heuristic method is called Tree Assigner and Placer (TAP). TAP ex-ploits that the paths terminate at the edge nodes (ENs) where the root of the trees are also placed. It enables a further decomposition of the problem: TAP considers paths belonging to a single edge node at the same time. This decomposition leads also to the same optimal solution because the paths terminating different edge nodes will surely use different trees.

The aim of the proposed algorithm is, therefore, to generate a minimal set of spanning trees whose roots are at the same edge node. Like SAL, this method generates a trajectory of a discrete time stochastic process, where the sets of trees, of assigned and unassigned paths areT rees, Assigned andU nAssigned. In each step either an unassigned path is appended to a tree (tree extension), or a tree having the least path is removed releasing the assigned paths as well (tree removal). The selection occurs with fixed probabilities:

Ptree_extension = 0.8andPtree_removal = 0.2. When all paths are assigned to trees (called maximal state) the solution is stored. Finally, after a pre-defined number of steps (here:

1000 iterations), the solution using the least trees will be selected. The tree construction algorithm consist of the following major phases:

• Initialization: Here, the sets of the assigned and unassigned paths and of the trees are initialized as the name of the phase implies. The counter for remaining steps (step) is set to 1000.

• Main Loop: In each iteration random selection between the two operators is per-formed with the defined probabilities. Then the selected step is executed. If tree removal is chosen the tree containing the least paths is removed from the tree set, and the paths that are previously assigned to the erased tree, are released again. The last step in each iteration is to check whether the new solution obtained after the execution of the operations is a full solution or not. If the new solution is full and the number of used trees is less than the number of trees used by the stored best solution, the new solution will be stored as the best one. Finally, thestep variable is decreased, and if it reaches zero the algorithm terminates.

Tree Extension

During the tree extension process an unassigned or free path is randomly selected with uniform probability distribution. Then the algorithm looks for a tree from setT reesthat fits to the path, i.e. it does not form a cycle with the selected path. This loop check is performed in one step. All existing tree instances are checked one by one and the first fitting tree is selected and the edges forming the path are added to this tree, as well. If there is no such a tree that fits to the current path a new tree will be created containing the edges of the path and obviously it will be inserted into the tree set. The pseudo-code of tree extension is shown on Figure 4.2.

Checking the Loops

In the proposed algorithms a method is required ensuring that after extending a tree with a path the considered will not contain loops. I propose a cycle-check mechanism with the complexity ofo(n)via exploiting that all the terminations of the considered demands

CHAPTER 4. SCALABLE TREE OPTIMIZATION 50

procedure Tree_Extension() a←F ALSE

if|T rees|>0then

Select pathprandomly fromU nassigned

foreacht∈ {T :T ∈T rees∧root(T) =target(p)} do if¬a∧ ¬FormsCycle(p,t)

t←t∪ {e|e∈p}

a←T RU E

Assigned←Assigned∪ {p}

end end end if¬a

Create a new treet_new t_new ← {e|e∈p}

T rees←T rees∪t_new return

Figure 4.2: Pseudo code of the tree extension subroutine.

are at a single edge node; moreover, these nodes are the roots of the considered trees.

Therefore, the algorithm has the following major steps:

• Step 1 gets the edges of the path one by one from the access node to the edge node, and for each edge

• Step 2 checks the following logical statement: the actual edge would form a cycle with the tree if both adjacent nodes are in the tree, however, the edge itself is not in the tree. If a loop is identified it returns true.

• Step 3 appends the edge and both adjacent nodes into the tree and if there are unchecked edges of the path then goto step 1.

• Step 4 returns false.