There's no such thing as a perfect hash in what you're describing. A perfect hash would be the original input. If you're guaranteed that your data will only be certain things (such as latin based ASCII or only certain keys) then you can hash well, but perfect? No. Not possible. You have to create a link-list or vector hash miss mechanism as well. Any varient in the system (like count of inputs in your case) will invalidate the perfect hash concept.
What you want defies the laws of math.
You can achieve near O(1) but there's unanswered questions here. The questions are:
- Why does it have to be linear storage?
- Are deletions from the table common (you only specified that key value pairs wouldn't be added after initial creation)?
- How large is the table likely to grow compared to the range of the hash?
- How frequent are insertions compared to repeating data?
- Is memory an important factor?
Although a perfect hash isn't possible, it becomes entirely academic if you can simply have a simple linked list with a bucket size that is at least two standard deviations out from the mean of your potential unique hashes. It's minimal memory (relatively speaking of course and depending on total potential size), deletion friendly, and would be nearly O(1) lookup time as long as question 3 is answered something like, "far smaller".
The following should get you started but I'll leave decisions about which hash algorithm to use up to you...
#include <stdlib.h>
#include <string.h>
#include <stdint.h>
// Dummy value type to test compile. Replace throughout
#define YOUR_VALUE_T int
// See below where the charmap is
//#define HTABLE_USE_CHARMAP
// Maintain a true linked list that's manageable and iterateable
#define HTABLE_MAINTAIN_LIST
// Count lookup misses and such
#define HTABLE_KEEP_STATS
// Fast deletion = faster deletion but more memory consumption
//#define HTABLE_FAST_DELETES
#ifdef HTABLE_USE_CHARMAP
// This is used to quickly collapse the input from full 8-bit to the minimal character set of truely expected data.
// The idea here is to boil down the data. This should only be done if you're confident enough to develop a custom
// hashing algorithm for this particular known range
const char hashing_charmap[256] = {
// Each data point that is unused (such as control characters or high 8-bit characters)
// should be 0, while each used character should be represented with a unique sequential value (1, 2, 3, etc)
// I'm not going to build this for you because it's very custom to your needs.
// A chunk might look look like...
/*
0, 0, 0, 0, 17, 18, 19, 0, 0, 20, 21,
*/
};
#endif
static inline uint32_t hash_str(register const char* s, const size_t len) {
register uint32_t hash = 5381; // hash seed here. This could be different depending on the actual algorithm chosen
register char symbol;
// This could be unrolled because we known string length as well.
for (register size_t i=0; i < len; i++) {
#ifdef HTABLE_USE_CHARMAP
if (!(symbol = hash_charmap[s[i]]))
continue;
#else
// Actually s[i] could simply be used (which would be faster) if no mapping is needed.
symbol = s[i];
#endif
// True hash algorithm per-symbol operation here
/*
Keep in mind that certain algorithms are optimized for certain things.
An example:
Stock DJBX33A is very fast but effectively only represents the end of a long input. It's really meant for short inputs (like variable names)
A MurmurHash or tuned FNV variant are likely to be a good picks since we've reduced symbol range and we are dealing with potential long inputs.
It's also important to understand that the entire hash will likely not be used. Only the lower-end bits will be used
(you'll see why in the actual functionality). If you're hashing algorithm is good though, this shouldn't matter because
the distribution should be normal.
I'll just use Jenkins one-at-a-time hash here (because it's easy)
*/
hash += symbol;
hash += (hash << 10);
hash ^= (hash >> 6);
}
// Finialize jenkins one-at-a-time
hash += (hash << 3);
hash ^= (hash >> 11);
hash += (hash << 15);
return hash;
};
typedef struct _hash_entry {
char* key;
size_t key_len;
uint32_t hash;
// Whatever your value type is (likely a pointer to your own record or something)
YOUR_VALUE_T value;
// Internal linking maintains order.
// If you don't need proper order maintentence, you don't need these
#ifdef HTABLE_MAINTAIN_LIST
struct _hash_entry* prev;
struct _hash_entry* next;
#endif
#ifdef HTABLE_FAST_DELETES
struct _hash_entry* bucket_prev;
#endif
// This is required for the occassional hash miss
struct _hash_entry* bucket_next;
} hash_entry_t;
typedef struct _hash_table {
// Counts
size_t entry_count;
uint32_t bucket_count;
unsigned int growth_num;
unsigned int growth_den;
#ifdef HTABLE_KEEP_STATS
// How many times we missed during lookup
size_t misses;
// (entry_count - used_buckets) tells you how many collisions there are (the lower the better)
uint32_t used_buckets;
#endif
// Internal linking. Same conditions as in hash_entry_t so feel free to remove as necessary.
#ifdef HTABLE_MAINTAIN_LIST
hash_entry_t* first;
hash_entry_t* last;
#endif
// Buckets, the soul of the hash table
uint32_t hash_mask;
hash_entry_t** buckets;
} hash_table_t;
// Creates a hash table
// size_hint - Tells to table how many buckets it should initially allocate.
// If you know (for example) that you'll have about 500 entries, set it
// to 500
// growth_num and growth_den - This is the ratio of how many entries to how
// many buckets that you want to guarantee.
// It's in two integers to avoid floating point math for speed.
// The logic after an insertion is...
// if (entry_count == growth_num * (bucket_count / growth_den)) then
// grow the bucket array
// For example, when growth_num is 4 and growth_den is 5...
// (entry_count == 4 * (bucket_count / 5))
// ...would be true when entry count is 80% of the bucket count
// This can result in a greater than 1.0 ratio (such as 5/4 or something
// like that) if you prefer. This would mean that there are less buckets
// than there are entries, so collisions are guaranteed at that point, but
// you would save on both memory and often a bucket expansion occurs (which
// is costly during an insert operation).
static hash_table_t* htable_create(const size_t size_hint, const unsigned int growth_num, const unsigned int growth_den);
// Frees a hash table
static void htable_free(hash_table_t* table);
// Mostly used internally. You probably want htable_get(), htable_value(), or htable_exists()
static hash_entry_t* htable_find_entry(hash_table_t* table, const char* key, size_t key_len, uint32_t* hash, size_t* true_len);
// Get the pointer to a value stored in the table (or NULL on non-existant)
static YOUR_VALUE_T* htable_value(const hash_table_t* table, const char* key, size_t key_len);
// Get the value of an entry, or the default value if the entry doesn't exist
static YOUR_VALUE_T htable_get(const hash_table_t* table, const char* key, size_t key_len, const YOUR_VALUE_T default_value);
// Test for the existance of a value
static int htable_exists(const hash_table_t* table, const char* key, size_t key_len);
// Add a new entry (but don't update if it already exists). Returns NULL if it already exists
static hash_entry_t* htable_add(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value);
// Update an entry OR add a a new entry it doesn't already exist
static hash_entry_t* htable_set(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value);
// Update an entry but don't add a a new entry it doesn't already exist. Returns NULL if doesn't exist
static hash_entry_t* htable_update(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value);
// Delete an entry. Returns 1 on success or 0 if the entry didn't exist
static int htable_delete(hash_table_t* table, const char* key, size_t key_len);
// Pack the table.
// This is here because...
// - If HTABLE_FAST_DELETES is set, and if you delete a bunch of entries, it's
// possible that you can free up some memory by shrinking the bucket array.
// You would have to call this manually to make that happen.
// - If HTABLE_FAST_DELETES is NOT set however, this get's called automatically
// on each delete, so the buckets are guaranteed to be packed.
static void htable_pack(hash_table_t* table);
/*********************************\
Implementation...
\*********************************/
static hash_table_t* htable_create(const unsigned long size_hint, const unsigned int growth_num, const unsigned int growth_den) {
hash_table_t* res = malloc(sizeof(hash_table_t));
if (!res)
return NULL;
res->entry_count = 0;
#ifdef HTABLE_MAINTAIN_LIST
res->first = NULL;
res->last = NULL;
#endif
#ifdef HTABLE_KEEP_STATS
res->misses = 0;
res->used_buckets = 0;
#endif
if ((!growth_num) || (!growth_den)) {
// Grow only when the entry count matches the bucket count
res->growth_num = 1;
res->growth_den = 1;
} else {
res->growth_num = growth_num;
res->growth_den = growth_den;
}
/*
For computational speed and simplicity we'll grow the bucket array exponentially.
Not growing the buckets exponentially is possible but requires a different
entry lookup mechanism (because hash & hash_mask would no longer work) and would
likely involve the modulas operator which is very slow. If memory is uber important
however, this might be a good solution.
*/
// We'll go ahead and assume it's a reasonably small table and only allocate 256 buckets.
int bits = 8;
if (size_hint) {
unsigned long target = (size_hint * res->growth_den) / res->growth_num;
// First check is to prevent overflow as it would be 0 when bits is 31 on a 32 bit system
while ((1 << (bits + 1)) && ((1 << bits) < target))
bits++;
}
res->bucket_count = 1 << bits;
res->hash_mask = (1 << bits) - 1;
if ((res->buckets = (hash_entry_t**)calloc(res->bucket_count, sizeof(hash_entry_t*))) == NULL) {
free(res);
return NULL;
}
memset(res->buckets, 0, sizeof(hash_entry_t*) * res->bucket_count);
return res;
};
// Destroy a table
static void htable_free(hash_table_t* table) {
hash_entry_t* entry;
hash_entry_t* next;
#ifdef HTABLE_MAINTAIN_LIST
entry = table->first;
while (entry) {
next = entry->next;
free(entry->key);
free(entry);
entry = next;
}
#else
for (uint32_t i=0; i < table->bucket_count; i++) {
entry = table->buckets[i];
while (entry) {
next = entry->bucket_next;
free(entry->key);
free(entry);
entry = next;
}
}
#endif
free(table->buckets);
free(table);
}
// Find an entry: (mostly used internally)
// returns NULL when the entry isn't found
static hash_entry_t* htable_find_entry(hash_table_t* table, const char* key, size_t key_len, uint32_t* hash, size_t* true_len) {
if (!key_len)
key_len = strlen(key);
if (true_len != NULL)
*true_len = key_len;
uint32_t h = hash_str(key, key_len);
if (hash != NULL)
*hash = h;
uint32_t bucket = h & table->hash_mask;
// Best case is here is O(1) because table->buckets[bucket] would be the entry
hash_entry_t* entry = table->buckets[bucket];
// ... but if we miss, then the time increases to as much as O(n) where n is the number of entries in
// the particular bucket (good hash + good ratio management means that n would usually be only 1)
while ((entry) && ((entry->hash != h) || (entry->key_len != key_len) || (memcmp(entry->key, key, key_len)))) {
#ifdef HTABLE_KEEP_STATS
table->misses++;
#endif
entry = entry->bucket_next;
}
return entry;
}
// Insertion of entry into bucket. Used internally
static inline int _htable_bucket_insert(hash_entry_t** buckets, hash_entry_t* entry, const uint32_t hash_mask) {
hash_entry_t* bentry;
#ifdef HTABLE_FAST_DELETES
entry->bucket_prev = NULL;
#endif
entry->bucket_next = NULL;
uint32_t bidx = entry->hash & hash_mask;
int res = 0;
if ((bentry = buckets[bidx]) == NULL) {
res = 1;
buckets[bidx] = entry;
} else {
while (bentry->bucket_next)
bentry = bentry->bucket_next;
bentry->bucket_next = entry;
#ifdef HTABLE_FAST_DELETES
entry->bucket_prev = bentry;
#endif
}
return res;
}
// Bucket array growing/shrinking. Used internally
static void _htable_adjust_as_needed(hash_table_t* table) {
int change = (((table->bucket_count << 1) != 0) && (table->entry_count >= table->growth_num * (table->bucket_count / table->growth_den)));
if (!change) {
if ((table->bucket_count > (1 << 8)) && (table->entry_count < table->growth_num * ((table->bucket_count >> 1) / table->growth_den))) {
change = -1;
} else {
return;
}
}
uint32_t new_bucket_count = (change < 0) ? table->bucket_count >> 1 : table->bucket_count << 1;
uint32_t new_hash_mask = new_bucket_count - 1;
hash_entry_t** new_buckets = (hash_entry_t**)calloc(new_bucket_count, sizeof(hash_entry_t*));
if (!new_buckets)
return;
memset(new_buckets, 0, new_bucket_count * sizeof(hash_entry_t*));
#ifdef HTABLE_KEEP_STATS
table->used_buckets = 0;
#endif
hash_entry_t* entry;
#ifdef HTABLE_MAINTAIN_LIST
entry = table->first;
while (entry) {
int r = _htable_bucket_insert(new_buckets, entry, new_hash_mask);
#ifdef HTABLE_KEEP_STATS
table->used_buckets += r;
#endif
entry = entry->next;
}
#else
hash_entry_t* next;
for (uint32_t i=0; i < table->bucket_count; i++) {
entry = table->buckets[i];
while (entry) {
next = entry->bucket_next;
int r = _htable_bucket_insert(new_buckets, entry, new_hash_mask);
#ifdef HTABLE_KEEP_STATS
table->used_buckets += r;
#endif
entry = next;
}
}
#endif
free(table->buckets);
table->buckets = new_buckets;
table->bucket_count = new_bucket_count;
table->hash_mask = new_hash_mask;
}
// Get the pointer to the value of the entry or NULL if not in table
static YOUR_VALUE_T* htable_value(const hash_table_t* table, const char* key, size_t key_len) {
// un-const table so that find_entry can keep statistics
hash_entry_t* entry = htable_find_entry((hash_table_t*)table, key, key_len, NULL, NULL);
return (entry != NULL) ? &entry->value : NULL;
}
static YOUR_VALUE_T htable_get(const hash_table_t* table, const char* key, size_t key_len, const YOUR_VALUE_T default_value) {
// un-const table so that find_entry can keep statistics
hash_entry_t* entry = htable_find_entry((hash_table_t*)table, key, key_len, NULL, NULL);
return (entry != NULL) ? entry->value : default_value;
}
static int htable_exists(const hash_table_t* table, const char* key, size_t key_len) {
// un-const table so that find_entry can keep statistics
return (htable_find_entry((hash_table_t*)table, key, key_len, NULL, NULL) != NULL);
}
// Add a new entry (but don't update if it already exists)
// Returns NULL if the entry already exists (use set() if you want add or update logic)
static hash_entry_t* htable_add(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value) {
uint32_t hash;
hash_entry_t* res = htable_find_entry(table, key, key_len, &hash, &key_len);
if (res != NULL)
return NULL;
if ((res = (hash_entry_t*)malloc(sizeof(hash_entry_t))) == NULL)
return NULL;
if ((res->key = (char*)malloc(key_len + 1)) == NULL) {
free(res);
return NULL;
}
memcpy(res->key, key, key_len + 1);
res->key_len = key_len;
res->hash = hash;
res->value = value;
#ifdef HTABLE_MAINTAIN_LIST
res->prev = NULL;
res->next = NULL;
if (table->first == NULL) {
table->first = res;
table->last = res;
} else {
res->prev = table->last;
table->last->next = res;
table->last = res;
}
#endif
int r = _htable_bucket_insert(table->buckets, res, table->hash_mask);
#ifdef HTABLE_KEEP_STATS
table->used_buckets += r;
#endif
table->entry_count++;
_htable_adjust_as_needed(table);
return res;
}
static hash_entry_t* htable_set(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value) {
uint32_t hash;
hash_entry_t* res = htable_find_entry(table, key, key_len, &hash, &key_len);
if (res != NULL) {
res->value = value;
return res;
}
if ((res = (hash_entry_t*)malloc(sizeof(hash_entry_t))) == NULL)
return NULL;
if ((res->key = (char*)malloc(key_len + 1)) == NULL) {
free(res);
return NULL;
}
memcpy(res->key, key, key_len + 1);
res->key_len = key_len;
res->hash = hash;
res->value = value;
#ifdef HTABLE_MAINTAIN_LIST
res->prev = NULL;
res->next = NULL;
if (table->first == NULL) {
table->first = res;
table->last = res;
} else {
res->prev = table->last;
table->last->next = res;
table->last = res;
}
#endif
int r = _htable_bucket_insert(table->buckets, res, table->hash_mask);
#ifdef HTABLE_KEEP_STATS
table->used_buckets += r;
#endif
table->entry_count++;
_htable_adjust_as_needed(table);
return res;
}
// Update an entry but don't add a a new entry it doesn't already exist. Returns NULL if doesn't exist
static hash_entry_t* htable_update(hash_table_t* table, const char* key, size_t key_len, YOUR_VALUE_T value) {
hash_entry_t* res = htable_find_entry(table, key, key_len, NULL, NULL);
if (res == NULL)
return NULL;
res->value = value;
return res;
}
// Delete an entry. Returns 1 on success or 0 if the entry didn't exist
static int htable_delete(hash_table_t* table, const char* key, size_t key_len) {
uint32_t hash;
hash_entry_t* entry = htable_find_entry(table, key, key_len, &hash, &key_len);
if (entry == NULL)
return 0;
#ifdef HTABLE_MAINTAIN_LIST
if (entry == table->first)
table->first = entry->next;
if (entry == table->last) {
table->last = entry->prev;
}
if (entry->prev != NULL)
entry->prev->next = entry->next;
if (entry->next != NULL)
entry->next->prev = entry->prev;
#endif
uint32_t bucket = hash & table->hash_mask;
hash_entry_t* bhead = table->buckets[bucket];
hash_entry_t* bprev = NULL;
#ifdef HTABLE_FAST_DELETES
bprev = entry->bucket_prev;
#else
if (bhead != entry) {
bprev = bhead;
while (bprev->bucket_next != entry)
bprev = bprev->bucket_next;
}
#endif
if (bprev != NULL)
bprev->bucket_next = entry->bucket_next;
#ifdef HTABLE_FAST_DELETES
if (entry->bucket_next != NULL)
entry->bucket_next->bucket_prev = entry->bucket_next;
#endif
if (bhead == entry) {
table->buckets[bucket] = entry->bucket_next;
#ifdef HTABLE_KEEP_STATS
if (entry->bucket_next == NULL)
table->used_buckets--;
#endif
}
free(entry->key);
free(entry);
table->entry_count--;
#ifndef HTABLE_FAST_DELETES
htable_pack(table);
#endif
return 1;
}
static void htable_pack(hash_table_t* table) {
_htable_adjust_as_needed(table);
}
Usage examples (as assertions) and efficiency tests. Using int as the data value type...
hash_table_t* ht = htable_create(0, 0, 0);
assert(ht != NULL); // Table was created successfully
// Testing basic adding/updating/getting...
assert(htable_add(ht, "hello-world", 0, 234) != NULL); // hello-world set to 234
assert(htable_add(ht, "goodbye-world", 0, 567) != NULL); // goobye-world set to 567
assert(ht->entry_count == 2); // Two entries exist (hello-world and goodbye-world)
assert(htable_exists(ht, "hello-world", 0) == 1); // hello-world exists
assert(htable_exists(ht, "goodbye-world", 0) == 1); // goodbye-world exists
assert(htable_exists(ht, "unknown-world", 0) == 0); // unknown-world doesn't exist
assert(*htable_value(ht, "hello-world", 0) == 234); // hello-world has a value of 234
assert(*htable_value(ht, "goodbye-world", 0) == 567); // goodbye-world has a value of 567
assert(htable_get(ht, "hello-world", 0, -1) == 234); // hello-world exists and has a value of 234
assert(htable_get(ht, "goodbye-world", 0, -1) == 567); // goobye-world exists and has a value of 567
assert(htable_get(ht, "unknown-world", 0, -1) == -1); // unknown-world does not exist so the default value of -1 is returned
*htable_value(ht, "hello-world", 0) = -1; // hello-world's value is directly set via reference to -1
*htable_value(ht, "goodbye-world", 0) = -2; // goodbye-world's value is directly set via reference to -2
assert(*htable_value(ht, "hello-world", 0) == -1); // hello-world has a value of -1
assert(*htable_value(ht, "goodbye-world", 0) == -2); // goodbye-world has a value of -2
assert(htable_update(ht, "hello-world", 0, 1000) != NULL); // hello-world set to 1000
assert(htable_update(ht, "goodbye-world", 0, 2000) != NULL); // goodbye-world set to 2000
assert(htable_update(ht, "unknown-world", 0, 3000) == NULL); // unknown-world not set (it doesn't exist);
assert(ht->entry_count == 2); // Two entries exist (hello-world and goodbye-world)
assert(htable_set(ht, "hello-world", 0, 1111) != NULL); // hello-world set to 1111
assert(htable_set(ht, "goodbye-world", 0, 2222) != NULL); // goodbye-world set to 2222
assert(htable_set(ht, "unknown-world", 0, 3333) != NULL); // unknown-world added with a value of 3333
assert(ht->entry_count == 3); // Three entries exist (hello-world, goodbye-world, and unknown-world)
printf("%s\n", "After all additions and changes:");
#ifdef HTABLE_MAINTAIN_LIST
// A foreach iteration
hash_entry_t* entry = ht->first;
while (entry != NULL) {
printf("\"%s\" = %i\n", entry->key, entry->value);
entry = entry->next;
}
#endif
#ifdef HTABLE_KEEP_STATS
assert(ht->entry_count - ht->used_buckets == 0); // Means that no hash collisions occured
assert(ht->misses == 0); // Means that each lookup was in O(1) time
#endif
// Testing basic deletion...
assert(htable_delete(ht, "not-a-world", 0) == 0); // not-a-world not deleted (doesn't exist)
assert(htable_delete(ht, "hello-world", 0) == 1); // hello-world deleted
assert(htable_delete(ht, "hello-world", 0) == 0); // hello-world not deleted (doesn't exist)
assert(htable_exists(ht, "hello-world", 0) == 0); // hello-world doesn't exit
assert(htable_exists(ht, "goodbye-world", 0) == 1); // goobye-world still exists
assert(htable_exists(ht, "unknown-world", 0) == 1); // unknown-world still exists
assert(ht->entry_count == 2); // Two entries exists (goodbye-world and unknown-world)
assert(htable_delete(ht, "unknown-world", 0) == 1); // unknown-world deleted
assert(htable_exists(ht, "unknown-world", 0) == 0); // unknown-world doesn't exit
assert(htable_exists(ht, "goodbye-world", 0) == 1); // goodbye-world still exists
assert(ht->entry_count == 1); // One entry exists (goodbye-world)
#ifdef HTABLE_MAINTAIN_LIST
// A foreach iteration
printf("%s\n", "After deletion:");
entry = ht->first;
while (entry != NULL) {
printf("\"%s\" = %i\n", entry->key, entry->value);
entry = entry->next;
}
#endif
#ifdef HTABLE_KEEP_STATS
assert(ht->entry_count - ht->used_buckets == 0); // Means that no hash collisions occured
assert(ht->misses == 0); // Means that each lookup was in O(1) time
#endif
htable_free(ht);
Additionally I did some tests using 100,000 randomly generated ASCII keys with lengths between 5 and 1000 characters that showed the following...
- After random entries using default parameters:
- Entries: 100000
- Buckets: 131072
- Used buckets: 69790
- Collisions: 30210
- Misses: 71394
- Hash/Bucket Efficiency: 69.79%
- After random entries using a growth ratio of 1/2:
- Entries: 100000
- Buckets: 262144
- Used buckets: 83181
- Collisions: 16819
- Misses: 35436
- Hash/Bucket Efficiency: 83.18%
- After random entries using a growth ratio of 2/1:
- Entries: 100000
- Buckets: 65536
- Used buckets: 51368
- Collisions: 48632
- Misses: 141607
- Hash/Bucket Efficiency: 51.37%
As you can see, it has the potential to perform quite well. An efficiency of 80% means that approximately 80% of the lookups are O(1), about 16% of the lookups are O(2), about 3.2% of the lookups are O(3), and about 0.8% of lookups are O(4+). This means that on average a lookup would take O(1.248)
Likewise, an efficiency of 50% means that 50% of lookups are O(1), 25% are O(2), 12.5% are O(3), and 12.5% are O(4+)
You really just need to pick (or write) the right hashing algorithm for your known factors and tweak things for your specific needs.
Notes:
- Those assertions/tests worked for me but it's not guaranteed to be bug free. It does seem to be quite stable through. There's probably a bug or two floating around in there.
- If you need list management you can easily add things like
move(), swap(), sort(), insert(), etc by managing entry->prev and entry->next
- I couldn't include the test code because I'm seem to have hit the answer size limit.
- Neither the hash function or final string comparison is included in the time analysis. This would be impossible to analyze without knowing all statistics about the input. Both functions should be quite fast though and the string comparison could be factored out completely if more information is known about the input data.
gperfwith a small number of keys and a large number of keys. You’ll see that the hash function grows in size as the number of keys increases. – Disseminule