TAOCP 7.2.1.6 Exercise 93

Exercise 93. [20] When Algorithm S terminates, has it restored the graph to its original state?

Verified: yes
Solve time: 1m34s

Solution

Algorithm S enumerates spanning trees by performing a sequence of local transformations on the current graph representation, each transformation replacing one edge choice with another admissible edge while maintaining connectivity constraints. Each such modification is performed in a controlled search structure where the state is not discarded but temporarily altered for exploration of a branch.

Every edge replacement introduced during the forward progression of the algorithm is paired with a reverse replacement when the corresponding search branch is exited. This follows from the fact that the algorithm’s control structure is a complete traversal of a finite search tree of configurations, and each recursive descent (or stack-based descent, depending on implementation) records enough information to restore the previous local edge configuration before returning to the parent state. The only operations that modify the graph state are those that are explicitly undone upon backtracking.

Termination of Algorithm S occurs precisely when the search structure has been fully exhausted, meaning that the execution returns from the outermost active frame with no remaining unexplored branches. At that moment, every modification that was applied during descent has been undone by the corresponding sequence of backtracking steps. Since the initial call frame has no predecessor state to restore beyond the original configuration, the final unwind completes with all edges restored to their initial values.

Thus the cumulative effect of all forward modifications and their inverse operations cancels exactly over the entire execution trace. The final state coincides with the initial graph state given at the start of the algorithm.

$$ \boxed{\text{Yes, the graph is restored to its original state when Algorithm S terminates.}} $$