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# 17.10. Hashing¶

## 17.10.1. Hashing¶

### 17.10.1.1. 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 $M-1$.

### 17.10.1.2. 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?”

### 17.10.1.3. Simple Examples¶

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• More reasonable example:
• Store about 1000 records with keys in range 0 to 16,383.

• Impractical to keep a hash table with 16,384 slots.

• We must devise a hash function to map the key range to a smaller table.

### 17.10.1.4. Collisions (1)¶

• 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.

### 17.10.1.5. Collisions (2)¶

• Collisions are inevitable in most applications.
• Example: Among 23 people, some pair is likely to share a birthday.

### 17.10.1.6. Hash Functions (1)¶

• A hash function MUST return a value within the hash table range.

• To be practical, a hash function SHOULD evenly distribute the records stored among the hash table slots.

• Ideally, the hash function should distribute records with equal probability to all hash table slots. In practice, success depends on distribution of actual records stored.

### 17.10.1.7. Hash Functions (2)¶

• If we know nothing about the incoming key distribution, evenly distribute the key range over the hash table slots while avoiding obvious opportunities for clustering.

• If we have knowledge of the incoming distribution, use a distribution-dependent hash function.

### 17.10.1.8. Simple Mod Function¶

int h(int x) {
return x % 16;
}

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### 17.10.1.12. Strings Function: Character Adding¶

int sascii(String x, int M) {
char ch[];
ch = x.toCharArray();
int xlength = x.length();

int i, sum;
for (sum=0, i=0; i < x.length(); i++)
sum += ch[i];
return sum % M;
}


### 17.10.1.13. String Folding¶

// Use folding on a string, summed 4 bytes at a time
int sfold(String s, int M) {
long sum = 0, mul = 1;
for (int i = 0; i < s.length(); i++) {
mul = (i % 4 == 0) ? 1 : mul * 256;
sum += s.charAt(i) * mul;
}
return (int)(Math.abs(sum) % M);
}


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### 17.10.1.17. 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.

### 17.10.1.18. 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.

### 17.10.1.19. 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;
}


### 17.10.1.21. 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.

### 17.10.1.22. Linear Probing (1)¶

• Use the following probe function:

p(K, i) = i;

• Linear probing simply goes to the next slot in the table.

• Past bottom, wrap around to the top.

• To avoid infinite loop, one slot in the table must always be empty.

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### 17.10.1.24. Problem with Linear Probing¶

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• The primary goal of a collision resolution mechanism:
• Give each empty slot of the table an equal probability of receiving the next record.

### 17.10.1.25. Linear Probing by Steps (1)¶

• Instead of going to the next slot, skip by some constant c.
• Warning: Pick M and c carefully.

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• This effectively splits the key range, and the hash table, into two halves. This leads to reduced performance.

### 17.10.1.26. Linear Probing by Steps (2)¶

• The probe sequence SHOULD cycle through all slots of the table.
• Pick $c$ to be relatively prime to $M$.

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### 17.10.1.32. Analysis of Closed Hashing¶

The load factor is $\alpha = N/M$ where $N$ is the number of records currently in the table.

### 17.10.1.33. 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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### 17.10.1.35. Tombstones (2)¶

• Unfortunately, tombstones add to the average path length.

• Solutions:
1. Local reorganizations to try to shorten the average path length.

2. Periodically rehash the table (by order of most frequently accessed record).