TAOCP 7.2.2.1 Exercise 81

**Answer: False.

taocpmathematicsalgorithmsvolume-4medium

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: no
Solve time: 1m49s

Answer: False.

We restate the claim precisely: whether an exact cover instance with no color assignments causes Algorithm X and Algorithm C to perform exactly the same sequence of primitive operations, hence have identical running time under the usual TAOCP cost model.

1. Restart of the argument

Algorithm C is an extension of Algorithm X in which items and options may carry additional color information, and in which the search procedure includes extra tests to enforce color compatibility. Algorithm X is obtained as the special case where no colors are present at all.

The key question is not whether the search tree is the same, but whether every primitive operation executed by Algorithm C coincides exactly with those executed by Algorithm X when the input contains no colors.

2. Structure of the search is identical

If no items are colored and no option contains any color annotations, then:

  • No option is ever rejected or modified due to color constraints.
  • No branching decision is affected by color compatibility tests.

Therefore, Algorithm C and Algorithm X generate the same recursion tree, visit the same nodes, and perform the same sequence of covers and uncovers on the underlying exact cover structure.

This part of the previous solution is correct.

3. Why running times are still not identical

However, the equality of the search tree does not imply equality of running time in TAOCP’s unit-cost model. Algorithm C still contains additional code that is executed even when colors are absent.

In particular, Algorithm C includes, for each option examined or each item interaction:

  • conditionals checking whether a color constraint applies,
  • accesses to color-related fields (even if empty or null),
  • control logic that determines whether to invoke color-processing steps.

When no colors exist, these tests and accesses do not change control flow, but they are still executed. A conditional that evaluates to false still costs time, and loop structures over empty or null color sets still incur loop overhead.

Algorithm X does not perform these operations at all.

Thus, even though both algorithms traverse the same search tree, Algorithm C performs strictly more primitive operations per visited node in its implementation.

4. Distinction between identical search and identical cost

We can separate two statements:

  • The recursion structure (sequence of recursive calls and cover/uncover operations) is identical.
  • The sequence of primitive instructions executed by the two algorithms is not identical.

The problem asks about exact running time equality, which depends on the second statement. Since Algorithm C executes additional fixed overhead instructions absent from Algorithm X, the total running times are not exactly equal.

5. Conclusion

The claim is false. Even in the absence of any color assignments, Algorithm C retains additional bookkeeping and conditional checks that are not present in Algorithm X. These contribute extra constant-time work per relevant step, so the two algorithms do not have exactly the same running time, even though they explore the same search tree.

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