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17.20. Quick Hashing¶

17.20.1. A Quick Introduction to Hashing¶

17.20.1.1. Hashing (1)¶

Hashing: The process of mapping a key value to a position in a table.

A hash function maps key values to positions. It is denoted by $h$.

A hash table is an array that holds the records. It is denoted by HT.

HT has $M$ slots, indexed form 0 to $M-1$.

17.20.1.2. Hashing (2)¶

For any value $K$ in the key range and some hash function $h$, $h(K) = i$, $0 <= i < M$, such that key(HT[i]) $= K$.

Hashing is appropriate only for sets (no duplicates).

Good for both in-memory and disk-based applications.

Answers the question “What record, if any, has key value K?”

17.20.1.3. Collisions¶

• Given: hash function h with keys $k_1$ and $k_2$. $\beta$ is a slot in the hash table.

• If $\mathbf{h}(k_1) = \beta = \mathbf{h}(k_2)$, then $k_1$ and $k_2$ have a collision at $\beta$ under h.

• Search for the record with key $K$:
1. Compute the table location $\mathbf{h}(K)$.

2. Starting with slot $\mathbf{h}(K)$, locate the record containing key $K$ using (if necessary) a collision resolution policy.

17.20.1.4. Closed Hashing¶

• Closed hashing stores all records directly in the hash table.

• Each record $i$ has a home position $\mathbf{h}(k_i)$.

• If another record occupies the home position for $i$, then another slot must be found to store $i$.

• The new slot is found by a collision resolution policy.

• Search must follow the same policy to find records not in their home slots.

17.20.1.5. Collision Resolution¶

• During insertion, the goal of collision resolution is to find a free slot in the table.

• Probe sequence: The series of slots visited during insert/search by following a collision resolution policy.

• Let $\beta_0 = \mathbf{h}(K)$. Let $(\beta_0, \beta_1, ...)$ be the series of slots making up the probe sequence.

17.20.1.6. Insertion¶

// Insert e into hash table HT
void hashInsert(const Key& k, const Elem& e) {
int home;                     // Home position for e
int pos = home = h(k);        // Init probe sequence
for (int i=1; EMPTYKEY != (HT[pos]).key(); i++) {
pos = (home + p(k, i)) % M; // probe
if (k == HT[pos].key()) {
println("Duplicates not allowed");
return;
}
}
HT[pos] = e;
}


17.20.1.8. Probe Function¶

• Look carefully at the probe function p():

pos = (home + p(k, i)) % M; // probe

• Each time p() is called, it generates a value to be added to the home position to generate the new slot to be examined.

• $p()$ is a function both of the element’s key value, and of the number of steps taken along the probe sequence. Not all probe functions use both parameters.

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17.20.1.10. Deletion¶

• Deleting a record must not hinder later searches.

• We do not want to make positions in the hash table unusable because of deletion.

• Both of these problems can be resolved by placing a special mark in place of the deleted record, called a tombstone.

• A tombstone will not stop a search, but that slot can be used for future insertions.

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