TAOCP 7.2.2.1 Exercise 81

Section 7.2.2.1: Dancing Links

Exercise 81. [21] [21] True or false: An exact cover problem that has no color assignments has exactly the same running time for Algorithms X and C.

Verified: yes
Solve time: 1m36s

Solution

The statement is false.

Algorithm X and Algorithm C solve different problems. Algorithm X maintains the lists of active items and options by the operations in (12) and (13). Algorithm C extends this procedure to handle color assignments, so its state contains additional information and its recursive processing includes additional work for compatibility checks.

Consider an exact cover instance with no color assignments. Every option either contains an item in the ordinary sense or does not contain that item. In this case, the compatibility tests required by Algorithm C never eliminate a choice because there are no colors to compare. Therefore the choices made by Algorithm C generate the same search tree as Algorithm X, provided that both algorithms make the same choice of item at each recursive call and examine options in the same order.

The equality of search trees does not imply equality of running time. At each node of the search tree, Algorithm X performs only the operations required for ordinary covering and uncovering. Algorithm C must additionally execute the general machinery that determines whether options with color information are compatible. When the instance has no color assignments, those tests have trivial outcomes, but the instructions for performing them are still executed.

For example, a recursive call of Algorithm C that chooses an option must inspect the color data associated with that option, determine that no color restriction applies, and continue. Algorithm X has no corresponding operation because the data structure contains no color information. Hence there are executions on which Algorithm C performs all operations performed by Algorithm X together with additional operations.

The only case in which the running times would coincide is a special implementation in which Algorithm C detects that the problem has no color assignments and switches to Algorithm X. Such a modification is not part of the definition of Algorithm C.

Thus the absence of color assignments makes the algorithms search the same solution space, but it does not make their running times identical.

$$ \boxed{\text{False}} $$

This completes the proof.

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