Consider the formula below for evaluating the number of hash table comparisons:

where:

c = the number of comparisons

L = the load factor of the hash table

Given that a hash table of size 66,319 contains 34,754 items, how many comparisons are expected in a linear probe for an item? Use up to 2 decimal places of the load factor when computing c. Your final answer could include up to 2 decimal places, no spaces.

Question

Consider the formula below for evaluating the number of hash table comparisons:

where:

c = the number of comparisons

L = the load factor of the hash table

Given that a hash table of size 66,319 contains 34,754 items, how many comparisons are expected in a linear probe for an item?  Use up to 2 decimal places of the load factor when computing c. Your final answer could include up to 2 decimal places, no spaces.

BlackTom AI · Accepted Answer

To approach this problem, I first determine the load factor L by dividing the number of items by the hash table size. Here, L = 34,754 / 66,319, which computes to approximately 0.5180 (more precisely about 0.5180–0.5181 when using additional decimal places). Next, I recall the standard formulas for probe counts in linear probing:
- For an unsuccessful search, the expected number of probes (comparisons) is 1 / (1 − L).
- For a successful search (finding an item), the commonly cited average is (1/2) * (1 + 1 / (1 − L)).
These reflect the two main scenarios in linear probing: not finding the item, and finding the item, respectively.

Since the prompt asks for the number of comparisons for an item using linear probing, we interpret this as the typical case of locating an item (a successful search). Therefore, we use the successful-search formula: c = (1/2) * (1 + 1 / (1 − L)).

Plugging in L ≈ 0.5180:
1 − L ≈ 0.4820
1 / (1 − L) ≈ 1 / 0.4820 ≈ 2.074
Then c ≈ (1/2) * (1 + 2.074) ≈ (1/2) * 3.074 ≈ 1.537.
Rounding to two decimal places as requested, c ≈ 1.54.

If instead one were calculating the cost of an unsuccessful search, the result would be 1 / (1 − L) ≈ 2.07, which is notably higher than the approximately 1.54 for a successful search. The key conceptual distinction is between searching for an item that is present versus querying for an absent item.

Summary of steps:
- Compute L = 34,754 / 66,319 ≈ 0.5180.
- Compute 1 − L ≈ 0.4820.
- Compute 1 / (1 − L) ≈ 2.074.
- For a successful search (finding an item), use c = (1/2) * (1 + 1 / (1 − L)) ≈ 1.537 → 1.54.

Note: If the problem intended the cost of an unsuccessful search, the value would be about 2.07, but the interpretation here assumes a successful item lookup in a linear probe.

CMPSC 132 Spring 2025 Module 6.2 Checkpoint

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