Building a feasible solution to . In order to construct a feasible solution to the current subproblem , we exploit a feasible solution to one of the subsequent subproblems , for a . Clearly, must satisfy the capacity constraint on the knapsacks . Moreover, notice that the radiation from the bit does not reach the knapsacks after , and then, for those knapsacks, satisfies the capacity constraint only if does. Therefore, we need to check the capacity constraint only for the knapsack . Furthermore, the bits after are not involved when we check the capacity constraint on that knapsack. Thus, in order to check the capacity constraint on the knapsack , it is enough to know the bits of the string . Notice that, for a signature , is the smallest width that holds the properties we showed above.