Search tree

Figure 6-3. Search tree of mappings obtained by applying the backtracking algorithm for the pair of structures Cq and Qt (see the graphs in Figure 6-2). Array (M, M2, Mj, M4) denotes the mapping 1 —> M, 2 —> M2, 3 —> M3, 4 —> M4.

In many real-world applications, the isomorphism wotild be found before all 16 mappings were checked. For the example from Figure 6-3, many algorithms would find the isomorphism at the fourth mapping following the leftmost path in the search tree the bold line in Figure 6-4) ... 

Figure 6-4. Backtracking approach realized as depth-first search aigorithm. Dotted arrows trace the route used for traversing all mappings in the search tree. Each node in the tree corresponds to a mapping between Cq and C-p (Figure 6-2).

In the worst case, the backtracking algorithm will form a search tree of depth n, where n is the number of atoms in the query graph. Also, in this case a separate sub-tree search process for each atom of the target graph will be initiated. That is why the linear multiplier m is apphed to Eq. (7). 

Search trees are widely used to represent the different states that a problem cem adopt, example is shown in Figure 9.4 from which it should be clear where the name deri especially if the page is turned upside down. A tree contains nodes that are connected edges. The presence of an edge indicates that the two nodes it connects ctre related in so way. Each node represents a state that the system may adopt. The root node represents initial state of the system. Terminal nodes have no child nodes. A goal node is a special k of terminal node that corresponds to Em acceptable solution to the problem. 

The formulation for this scenario entails 1411 constraints, 511 continuous and 120 binary variables. The reduction in continuous variables compared to scenario 1 is due to the absence of linearization variables, since no attempt was made to linearize the scenario 2 model as explained in Section 4.3. An average of 1100 nodes were explored in the branch and bound search tree during the three major iterations between the MILP master problem and the NLP subproblem. The problem was solved in 6.54 CPU seconds resulting in an optimal objective of 2052.31 kg, which corresponds to 13% reduction in freshwater requirement. The corresponding water recycle/reuse network is shown in Fig. 4.10. 

The corresponding mathematical formulation entails 5534 constraints, 1217 continuous and 280 binary variables. An average of 4000 nodes were explored in the branch and bound search tree. The solution required three major iterations and took 309.41 CPU seconds to obtain the optimal solution of 1285.50 kg. This corresponds to 45.53% reduction in freshwater demand. A water reuse/recycle network that corresponds to this solution is shown in Fig. 4.11. 

The overall model for this scenario involves 5614 constraints, 1132 continuous 280 binary variables. Three major iterations with an average of 1200 nodes in the branch and bound search tree were required in the solution. The objective value of 1560 kg, which corresponds to 33.89% reduction in freshwater requirement, was obtained in 60.24 CPU seconds. An equivalent of this scenario, without reusable water storage, i.e. scenario 2, resulted in 13% reduction in fresh water. Figure 4.12 shows the water recycle/reuse network corresponding to this solution. 

More generally, MILPs are solved with branch and bound algorithms, similar to the spatial branch and bound method of the previous section, that explore the search space. As seen in Fig. 3-61, binary variables are used to define the search tree, and a number of bounding properties can be noted from the structure of (3-110). 

These properties lead to pruning of the search tree. Branching then continues in the tree until the upper and lower bounds converge. 

In principal the generation of the quant network is done by decomposing the overall decision problem into smaller sub-problems by looping around nested recursive functions that are used to divide the search tree into the parts that are useful to... 

The central recursion traverses the product flow net. At arbitrary points of the explosion additional search trees to examine alternative explosions can be built up... 

Fig. 14 The schematic representation of the search tree. Red nodes indicate energetically unfavorable conformations during the ligand building.

Regarding the optimal solutions obtained versus CPU effort, meanwhile in the ATF the optimal solution is found in less then 2 s, the DTP took about 18 min to prove optimality. However, the optimal solution was obtained relatively early in the search tree analysis ( 5 s of computation), which means that the majority of the CPU effort is used to prove optimality. It should also be pointed out that the relaxed solution is equal between the three strategies, representing a good accuracy between formulations. 

Calculation of Software Accuracy at each node of the search tree... 

In the interest of pruning the search tree, we can use what is known about the sources and sinks to limit the alternatives to the more reasonable choices. 

The QPS system solves this problem using the strategy shown by the search tree in Figure 2. Answers are produced by the QPS system in MgXformat, and the solution report for the above example is shown in Figure 3. 

Flexible ligand docking by incremental construction. The search tree resulting after the steps of the algorithm as... 

T o enhance the performance of grid searches, two advantages of the tree representation can be utilized. Firstly, parts of the tree that correspond to conformations with atom clashes or close contacts can be detected and pruned. Fig. 18 illustrates this for a conformation of n-heptane. Neither changes to the torsion xn nor to rn, will avoid the steric clash exhibited at the marked atoms. Thus, the search tree can be pruned above the node of xn i-... 

Secondly, the perception of symmetries within the molecule under investigation may also partly restrict the number of nodes and edges of the search tree that have to be processed. 

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