Backtracking condition

As shown in Fig. 7, atom 1 in structure 1 is first tried to be matched onto atom 2 in structure 2. Following this search branch, a substructure with five atoms and four bonds is found before any backtracking is performed (bold lines in structures 1 and 2 of Fig. 7). The size of the common structure found so far is used as a backtrack condition. With this backtracking condition, no branches were cut off before the next more extended substructure containing six atoms and five bonds is found. Then, this new best number of bonds is used as a new backtrack condition for further search. It can be easily seen from structures 1 and 2 in Fig. 7 that the correct MCSS will not be found until atom 1 in structure 1 is matched against atom 8 or 9 in structure 2. This example reveals that most CPU time is wasted on investigating many useless searches on bad branches within the search tree. 

Thus the backtrack condition immediately obtains its maximal value of 8. Using this value, as long as one atom in structure 1 fails to match any atom in structure 2, the corresponding branch will be immediately cut off. 

Assigning A Larger Initial Value to the Backtrack Condition... 

As previously discussed, for general MCSS problems, it is impossible to predict how many atoms and bonds MCSSs will have. Therefore an initial value of 0 is assigned to the backtrack condition. In certain cases, some useless trees may be searched first, and in such cases the backtracking condition will not obtain a larger value until a search... 

Fig. 4. Illustrative model of paths between two trap sites embedded in a three-dimensional cubic lattice. The dashed 24-link line has 7 unnecessary kinks which reduce its contribution to the path sum, but there are many of them (Table 2) note that the kinks in the figure are two-dimensional but the count in Eq. 17 is three-dimensional. The paths corresponding to terms in Eq. 14 may in general cross over themselves and backtrack, but may not visit the initial or final sites twice. The latter condition does not arise directly from Eq. 13 but rather from the irreversibility concept underlying the theory of the rate constant...

The next entry in the hrf list is hrf(acetanilide,(tb,35)), which is matched to hrf(A,T,X), resulting in A = acetanilide, T = tb and X = 35. The second condition then becomes hrf(B,(tb,Y)). Beginning again at the start of the list of hrf entries in the data base, the first match is with hrf(aconitine,(tb,45)), which sets B to aconitine and Y to 35. The third condition is satisfied with 45 not equal 35, and PROLOG has found another solution. From the chemist s point of view this solution is the same as the first one. However, from the point of view of PROLOG, the two solutions are distinct, as we did not supply PROLOG with an appropriate rule. Further backtracking does not produce other solutions. 

Once a new X and corresponding trial point x(X) have been determined in a line search iteration, conditions of sufficient progress with respect to the objective function are tested. If these conditions are not satisfied, a new value for X is sought in another line search step of interpolation, following a backtracking strategy (i.e., reduction of X2). 

Computing the Parity-check Matrix (H). After determining + and a. the algorithm has to find an H matrix that satisfies the conditions (2) and (3) with the previously selected set of error vectors. As all the matrices have to be considered, it may require a huge computational effort. A recursive backtracking algorithm has been developed to lighten the process. It checks partial matrices and adds a new column only if the previous matrix satisfies the requirements. 

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