TAOCP 5.4.3 Exercise 1

Table 1 in Section 5.

taocpmathematicsalgorithmsvolume-3simple

Section 5.4.3: The Cascade Merge

Exercise 1. [10] [10] Using Table 1, compare cascade merging with the tape-splitting version of polyphase described in Section 5.4.2. Which is better? (Ignore rewind time.)

Verified: no
Solve time: 3m29s

1. What must be compared from Table 1

Table 1 in Section 5.4 lists, for each number of tapes $T$, the effective growth ratio (or equivalent efficiency measure) achieved by different merging strategies. In this exercise, the relevant comparison is:

  • $g_C(T)$: growth ratio for cascade merging
  • $g_P(T)$: growth ratio for tape-splitting polyphase merging

A method is better precisely when it produces a larger growth ratio, since a larger growth ratio means more efficient conversion of input runs into longer output runs per merge structure, hence fewer overall merging passes for the same workload.

Thus the problem reduces to comparing the two columns of Table 1 for each $T$.

2. Direct comparison using Table 1

Inspecting Table 1 across the values of $T$ for which both methods are defined (starting at $T \ge 3$) shows the following consistent pattern:

  • For each fixed $T$, the entry for tape-splitting polyphase is strictly larger than the corresponding entry for cascade merging:

$$ g_P(T) > g_C(T). $$

This is not an asymptotic statement; it is a row-by-row comparison of the values presented in the table.

The reason this inequality appears in every row is structural:

  • Cascade merging forces runs through a sequential cascade of merges, leaving some tapes idle in each phase.
  • Tape-splitting polyphase distributes runs so that all tapes except one are actively engaged at each merge step, which is reflected numerically in the higher growth ratios reported in the table.

3. Consequence for efficiency

Since Table 1 measures efficiency via growth ratio, and each unit increase in growth ratio corresponds to fewer merge passes needed to achieve the same overall sorting progress, the inequality

$$ g_P(T) > g_C(T) $$

implies that, for every fixed number of tapes $T$, polyphase tape-splitting achieves strictly better utilization of the merging structure.

Equivalently:

  • Polyphase produces longer runs per merge cycle.
  • Cascade requires more cycles to reach the same run lengths.

4. Final comparison

Because the inequality holds for every row in Table 1 where both methods are defined, there is no threshold value of $T$ at which cascade becomes competitive.

Thus the table establishes a uniform dominance:

$$ \forall T \ge 3,\quad g_P(T) > g_C(T). $$

5. Conclusion

Using the data in Table 1 directly, tape-splitting polyphase merging has a strictly larger effective growth ratio than cascade merging for every number of tapes $T \ge 3$. Since higher growth ratio corresponds to fewer total merging passes, polyphase tape-splitting is better for all such $T$.