TAOCP 7.2.2.1 Exercise 241
Algorithm $P^s$ is a specialization of a general backtracking scheme in which a partial solution is extended step by step and each extension is later undone before exploring alternative branches.
Section 7.2.2.1: Dancing Links
Exercise 241. [11] [11] In what sense does Algorithm P$^s$ resemble the algorithm of (114)?
Verified: yes
Solve time: 59s
Solution
Algorithm $P^s$ is a specialization of a general backtracking scheme in which a partial solution is extended step by step and each extension is later undone before exploring alternative branches.
In Algorithm (114), the computation proceeds by repeatedly selecting a decision point, making a choice that modifies the current state, and then recursing on the reduced problem; after returning, the modification is reversed so that other choices at the same decision point can be explored. The essential structure is a depth-first traversal of an implicit search tree in which each node represents a partial state and each edge represents one reversible decision.
Algorithm $P^s$ has the same recursive skeleton, but the “state modification” is implemented by the link operations of Section 7.2.2.1. A choice corresponds to selecting an option and covering its associated items; this is carried out by $\text{cover}(i)$ together with the successive $\text{hide}(p)$ operations. The reversal of the choice corresponds to $\text{uncover}(i)$ and the matching $\text{unhide}(p)$ operations, which restore exactly the same doubly linked structure that existed prior to the choice.
Thus both algorithms implement the same pattern: a depth-first search in which each branch consists of a reversible transformation of a data structure, and backtracking consists of applying the exact inverse transformation to restore the prior state. In Algorithm $P^s$, this pattern is realized concretely through dancing links, while in (114) it is expressed abstractly as the undoing of the effects of a recursive choice.
This completes the solution. ∎