CS140 Lecture notes -- Hashing

Wikipedia Links

• Hash Tables.
• Hash Functions.
• Linear Probing.
• Quadratic Probing.
• Double Hashing.

Overview

1. In the worst case their insert, delete, and find operations require n accesses. If the hash table is properly implemented and if the table does not get too full, then this worst case performance is highly unlikely.
2. They do not store the data in sorted order. However, scripting languages provide sort algorithms that efficiently sort the keys in a hash table, if ordering is required.

There are three components that are involved with performing storage and retrieval with Hash Tables:

• A hash table. This is a fixed size table that stores data of a given type. The best performance is normally achieved when the table's size is a prime number.
• A hash function: This is a function that converts a piece of data into an integer. Sometimes we call this integer a hash value. The integer should be at least as big as the hash table. When we store a value in a hash table, we compute its hash value with the hash function, take that value modulo the hash table size, and that's where we store/retrieve the data.
• A collision resolution strategy: There are times when two pieces of data have hash values that, when taken modulo the hash table size, yield the same value. That is called a collision. You need to handle collisions. We will detail four collision resolution strategies: Separate chaining, linear probing, quadratic probing, and double hashing.

An example helps to illustrate the basic concept. Let's suppose that our hash table is of size 10, and that we are hashing strings. We'll talk about hash functions later, but let's suppose that we have four strings that we want to store in our hash table: "Luther," "Rosalita", "Binky" and "Dontonio." Let's also suppose that we have a hash function that converts these strings into the following values:

• "Luther" has a hash value of 3249384281.
• "Rosalita" has a hash value of 2627953124.
• "Binky" has a hash value of 216319842.
• "Dontonio" has a hash value of 2797174031.

Similarly, "Rosalita" goes into index 4, and "Binky" into index 2. If we insert them into the hash table, the table looks as follows:

To find a string in the hash table, we calculate its hash value modulo ten, and we look at that index in the hash table. For example, if we want to see if "Luther" is in the hash table, we look in index 1.

Now, suppose we want to look for "Dontonio" in the hash table. Since its hash value is 2797174031, we look in index 1. It's not there. Now suppose we wanted to insert "Dontonio." Well, that's a problem, because "Luther" is already at index one. That is a collision.

I'm not going to fix this problem until later in these lecture notes when we discuss collision resolution strategies. However, it demonstrates why collisions are problems.

Properties of hash tables and hash functions

That's a pretty intuitively named property. In the hash table pictured above, the load factor is 0.3, because there are three strings in the hash table, and the table size is ten. We can typically quantify how well a hash table is functioning by its load factor.

The selection of a hash function is very important. You want your hash function to have two important properties:

• It should be quick to compute, often constant time, or linear in the size of the data that you are hashing.
• The hash values that it computes should be uniformly distributed from zero to one minus the hash table size. That minimizes collisions.

Interesting Hash Functions for Strings

We start with the most obvious hash function for strings -- add up the ASCII values of the characters. I'm going to call this hash function "BAD". I have this programmed in badhash.cpp:

#include <iostream>
using namespace std;

unsigned int bad_hash(string &s)
{
  int i;
  unsigned int h;
  
  h = 0;

  for (i = 0; i < s.size(); i++) {
    h += s[i];
  }
  return h;
}

main()
{
  string s;
  unsigned int h;

  while (getline(cin, s)) {
    h = bad_hash(s);
    cout << h << endl;
  }
}

Why the unsigned int? The reason is that I want to allow my hash values to go into any 32-bit integer, and I don't want negative numbers. To do that, I specify that the integer be unsigned. Knowing that the ASCII value of 'A' is 65 and that the ASCII value of 'a' is 97, we can verify that the hash function works as advertised:

UNIX> badhash
A
65
a
97
Aa
162
<CNTL-D>
UNIX> 

unsigned int djb_hash(string &s)
{
  int i;
  unsigned int h;
  
  h = 5381;

  for (i = 0; i < s.size(); i++) {
    h = (h << 5) + h + s[i];
  }
  return h;
}

Although the web page says that this is equivalent to:

where the carat operator is bitwise exclusive-or, that is not what is implemented. Instead, we take h_i-1 and "bit-shift" it, five places to the left. That is equivalent to multiplying by 32, which is 2⁵. However if any of the leftmost 5 bits of the number are set, they are "shifted away" -- they go away. We then add h_i-1 -- that is how the author performs multiplication by 33. Finally, we add s[i] rather than performing the bitwise exclusive-or (if you have not seen C++'s bitwise operations before, see these bit notes). I don't know why the author's code is different from the author's stated algorithm-- I'm just copying code from the web page.

We can double-check ourselves though:

UNIX> djbhash
a
177670                                  5381 * 33 + 97 = 177670
aA
5863175                               177670 * 33 + 65 = 5863175
aAA
193484840                            5863175 * 33 + 65 = 193484840
aAAA
2090032489
<CNTL-D>
UNIX> 

That last number takes a little work to verify. First, let's look at 193484840 in hexadecimal (you can do this in C++, but the scripting language awk supports printf, so it's easier):

UNIX> echo 193484840 | awk '{ printf "0x%08x\n", $1 }'
0x0b885828
UNIX> 

0000 1011   1000 1000   0101 1000   0010 1000

Performing the left bit-shift by 5 turns this number into:

0111 0001   0000 1011   0000 0101   0000 0000

I've tried to use the blue color to help you -- those are the original bits that remain after the bit-shift. This is 0x710b0500 in hex, which is 1896547584 in decimal (see your lab for how to do that conversion). And finally, 1896547584 + 193484840 + 65 = 2090032489. We've verified that the function works!!

One last hash function comes from [Pearson10]. We'll call it ACM. It uses a permutation table to calculate the hashes: acmhash.cpp:

static unsigned char perm_table[256] = {
  1, 87, 49, 12, 176, 178, 102, 166, 121, 193, 6, 84, 249, 230, 44, 163,
  14, 197, 213, 181, 161, 85, 218, 80, 64, 239, 24, 226, 236, 142, 38, 200,
  110, 177, 104, 103, 141, 253, 255, 50, 77, 101, 81, 18, 45, 96, 31, 222,
  25, 107, 190, 70, 86, 237, 240, 34, 72, 242, 20, 214, 244, 227, 149, 235,
  97, 234, 57, 22, 60, 250, 82, 175, 208, 5, 127, 199, 111, 62, 135, 248,
  174, 169, 211, 58, 66, 154, 106, 195, 245, 171, 17, 187, 182, 179, 0, 243,
  132, 56, 148, 75, 128, 133, 158, 100, 130, 126, 91, 13, 153, 246, 216, 219,
  119, 68, 223, 78, 83, 88, 201, 99, 122, 11, 92, 32, 136, 114, 52, 10,
  138, 30, 48, 183, 156, 35, 61, 26, 143, 74, 251, 94, 129, 162, 63, 152,
  170, 7, 115, 167, 241, 206, 3, 150, 55, 59, 151, 220, 90, 53, 23, 131,
  125, 173, 15, 238, 79, 95, 89, 16, 105, 137, 225, 224, 217, 160, 37, 123,
  118, 73, 2, 157, 46, 116, 9, 145, 134, 228, 207, 212, 202, 215, 69, 229,
  27, 188, 67, 124, 168, 252, 42, 4, 29, 108, 21, 247, 19, 205, 39, 203,
  233, 40, 186, 147, 198, 192, 155, 33, 164, 191, 98, 204, 165, 180, 117, 76,
  140, 36, 210, 172, 41, 54, 159, 8, 185, 232, 113, 196, 231, 47, 146, 120,
  51, 65, 28, 144, 254, 221, 93, 189, 194, 139, 112, 43, 71, 109, 184, 209};

unsigned int acm_hash(string &s)
{
  int i, j;
  unsigned int h;
  unsigned char byte_hash[4];
  
  for (j = 0; j < 4; j++) byte_hash[j] = 0;
  j = 0;
  for (i = 0; i < s.size(); i++) {
    byte_hash[j] = perm_table[byte_hash[j]] ^ s[i];
    j++;
    if (j == 4) j = 0;
  }
  h = 0;
  for (j = 0; j < 4; j++) h = (h << 8) | byte_hash[j]; 
  return h;
}

Now, I know this is a little confusing, but such is life. The permutation table contains a permutation of the numbers between 0 and 255. We then calculate the hashes as follows:

Otherwise: h_i = perm_table[h_i-4] ^ s[i]

Again, the carat is the bitwise exclusive-or operator. After calculating all the h_i, we only consider the last four. These are held in the array byte_hash. Specifically, byte_hash[0] holds whichever of the last four h_i has (i%4) = 0. byte_hash[1] holds whichever of the last four h_i has (i%4) = 1, and so on. We then construct h so that byte_hash[0] is the first byte, byte_hash[1] is the second byte, and so on.

Let's walk through some examples:

UNIX> acmhash
a
1610612736
aaaa
1616928864
aaaaa
3848298592
<CNTL-D>
UNIX> 

96 is 0x60 in hexadecimal. Therefore, h is equal to 0x60000000. Check it out:

UNIX> echo 1610612736 | awk '{ printf "0x%08x\n", $1 }'
0x60000000
UNIX>

UNIX> echo 1616928864 | awk '{ printf "0x%08x\n", $1 }'
0x60606060
UNIX> 

Thus, h is equal to 0xe5606060:

UNIX> echo 3848298592 | awk '{ printf "0x%08x\n", $1 }'
0xe5606060
UNIX> 

I know that was detailed and probably very difficult. I'll make sure we go over it in class.

Evaluating how good those three hash functions are

UNIX> wc names_100000.txt
  100000  216288 1599207 names_100000.txt
UNIX> head names_100000.txt
Charlie Muffin
Alexandra Map
Sophia Compensatory
Jack Havoc PhD
Joseph Tawny
Kaylee Torture
Addison Oppressor
Sophie Tercel
Caleb Troglodyte
Victoria Garish
UNIX> 

UNIX> head names_100000.txt | badhash | awk '{ print $1%200000 }'
1341
1230
1928
1190
1180
1392
1711
1255
1572
1471
UNIX> head names_100000.txt | djbhash | awk '{ print $1%200000 }'
121954
19475
49837
65483
86881
97205
141524
12204
127017
54052
UNIX> head names_100000.txt | acmhash | awk '{ print $1%200000 }'
10430
173738
65289
133744
46463
21690
76893
59060
133035
185216
UNIX> 

To evaluate even more, let's calculate all 100,000 hash values, pipe it through awk to print the numbers mod 200,000 and then pipe that through sort -un. This sorts the values numerically, stripping out duplicates. If we pipe that to wc, we can see how many distinct values are produced. The higher that number, the fewer collisions there are:

UNIX> cat names_100000.txt | badhash | awk '{ print $1%200000 }' | sort -un | wc
    2305    2305   11106
UNIX> cat names_100000.txt | djbhash | awk '{ print $1%200000 }' | sort -un | wc
   78605   78605  506578
UNIX> cat names_100000.txt | acmhash | awk '{ print $1%200000 }' | sort -un | wc
   78610   78610  506934
UNIX> 

Finally, I wrote a program that generated 100000 random numbers between 0 and 199999 to see how many distinct values that program would generate. I ran it 10 times and averaged the results. It produced an average of 78742 distinct values. Hence if I had a purely random function, which is what the random number generator is, it would only produce about 130-140 more distinct hash values then either DJB or ACM. That means that DJB and ACM are pretty darn good!

Collision Resolution Strategy Number One: Separate Chaining

The nice thing about separate chaining is that it does not place limitations on the load factor. It should be pretty clear that if the load factor of a hash table with separate chaining is x, then the average size of each vector is x. That means that to find a piece of data in a hash table, the performance is going to linearly dependent on x. We'll demonstrate that below.

The code in cc_hacker.cpp programs up a simple hash table with separate chaining. It stores names and credit card numbers in the hash table. As command line arguments, it accepts a hash table size and a file of credit card numbers/names as in cc_names_10.txt:

5034036940778753 Samantha Bustle
9614647402933149 Sydney Macaque
9520044178288547 Eli Boone
4591437720049815 Lillian Handiwork
9464976002919633 Carson Shipmen
1263239861455040 Joshua Lucid IV
1750256670725811 Aaron Flaunt
8988852954721062 Brooklyn Samantha Hoofmark
8293347969318340 Avery Parsifal
1579529982933479 Alyssa Kissing

For each line, it creates a Person instance, calculates a hash of the name using DJB and then inserts the name into the hash table. Since we're using separate chaining, the hash table is a table of Person vectors. To insert a Person, we calculate the hash of the name, take it modulo the table size and call push_back() on the vector that is there.

After reading the table, we read names on standard input, and look them up in the table.

Here's an example:

UNIX> djbhash
Sydney Macaque
1929449534
Eli Boone
3852330034
Samantha Bustle
2419233313
Jim Plank
2620064763
<CNTL-D>
UNIX> cc_hacker 10 cc_names_10.txt
Enter a name> Sydney Macaque
Found it: Table entry 4: Sydney Macaque 9614647402933149
Enter a name> Eli Boone
Found it: Table entry 4: Eli Boone 9520044178288547
Enter a name> Samantha Bustle
Found it: Table entry 3: Samantha Bustle 5034036940778753
Enter a name> Jim Plank
Not found.  Table entry 3
Enter a name> <CNTL-C>
UNIX> 

As mentioned above, the average number of entries in each "chain" is equal to the load factor. Therefore, the performance of lookup will be proportional to the load factor.

To demonstrate this, the file cc_names_100K.txt has 100,000 random names/credit cards, and cc_just_names_100K.txt has just the names, in a different order.

As we increase the table size, the program runs faster: (Redirecting standard output to /dev/null simply says to not print standard output).

UNIX> time cc_hacker 10 cc_names_100K.txt < cc_just_names_100K.txt > /dev/null
18.283u 0.234s 0:18.51 100.0%	0+0k 0+0io 0pf+0w
UNIX> time cc_hacker 100 cc_names_100K.txt < cc_just_names_100K.txt > /dev/null
2.294u 0.098s 0:02.39 99.5%	0+0k 0+0io 0pf+0w
UNIX> time cc_hacker 1000 cc_names_100K.txt < cc_just_names_100K.txt > /dev/null
0.545u 0.072s 0:00.61 100.0%	0+0k 0+1io 0pf+0w
UNIX> time cc_hacker 10000 cc_names_100K.txt < cc_just_names_100K.txt > /dev/null
0.385u 0.070s 0:00.45 100.0%	0+0k 0+0io 0pf+0w
UNIX> 

Collision Resolution Strategies 2-4: Open Addressing

You may think that open addressing is stupid -- computers have tons of memory these days, so separate chaining should work easily. However, suppose you have to work in constrained environments (e.g. embedded processors) with much tighter memory requirements. Then, being able to preallocate your hash table might be very important.

The remainder of these lecture notes are without computer code, because I want to give you the joy of programming these collision resolution strategies yourself!

With open addressing, you will generate a sequence of hash values for a given key until you find a value that corresponds to a free slot in the hash table. To be precise, you will generate a sequence of hash values, h₀, h₁, h₂, ..., and you look at each in the hash table until you find the value (when you are doing look up), or until you find an empty hash table entry (when you are inserting, or when you are looking up a value that is not there). The various h_i are defined as:

H() is the hash function, and F() is a function that defines how to resolve collisions. It is usually convenient to make sure that F(0, key) equals zero, and it is necessary for F(i, key) not to equal zero when i doesn't equal zero.

Linear Probing

UNIX> djbhash
Luther
3249384281
Rosalita
2627953124
Binky
216319842
Dontonio
2797174031
Frenchy
561643892
<CNTL-D>
UNIX> 

Dontonio hashes to 1, so we have a collision. Formally:

Therefore, we insert "Dontonio" into index 3:

Similarly, "Frenchy" hashes to 2, so it collides and ends up in index 5:

If we try to find the key "Baby-Daisy", we first find its hash value using djbhash: 2768631242. We'll have to check indices 2, 3, 4 and 5 before we find that index 6 is empty and we can conclude that "Baby-Daisy" is not in the table.

Quadratic Probing

We'll use the same example -- "Luther", "Rosalita" and "Binky" are in the hash table and we want to insert "Dontonio".

Since there is a collision, we test values of h_i until we find one that's not in the table:

Thus, "Dontonio" goes into index 5:

Now when we insert "Frenchy", it collides at h₀ = 2, but not at h₁ = 3:

If we try to find "Baby-Daisy", we'll check indices 2, 3 and 6 to determine that she is not in the table.

Quadratic Probing can have an issue that all hash table entries are not checked by the various h_i. For example, suppose we've inserted "Luther" (3249384281), "Rosalita" (2627953124), "Princess" (2493584940), "Thor" (2089609346), Waluigi (385020695) and "Dave" (2089026949) into the table:

Suppose we want to now insert "Peach", which has has a hash value of 232764550. We go through the following sequences of h_i:

That's a problem, as we can't put "Peach" into the table. You need to do a little math to help ensure that h_i covers all possible values. Having the table size be prime helps, as would different settings of c₁ and c₂ in the more general version of quadratic probing.

Double Hashing

Let's try an example -- let's have H2 be equal to acmhash modulo the table size, and if that value is equal to 0, then set it to one. We'll start with "Luther", "Rosalita" and "Binky" in the table:

Again, we want to insert "Dontonio", whose hash value is 2797174031. Since that collides, we need to calculate H2("Dontonio"), which equals 2511778359%10 = 9. Thus h₁ = (2797174031+9)%10 = 0, and we can put "Dontonio" into index zero:

Next comes "Frenchy", whose hash value is 561643892. That collides, so we calculate H2("Frenchy"), which equals 3425106287%10 = 7. Thus h₁ = (561643892+7)%10 = 9, and we can put "Frenchy" into index nine:

Suppose we want to find "Baby-Daisy". Her hash value is 2768631242, which collides with "Binky". H2("Baby-Daisy") is equal to 2774673939%10=9. Therefore, we'll check hash locations 1, 0, 9 and 8 before seeing that she is not in the hash table.

Finally, as with quadratic probing, you need to be careful with double hashing. Suppose I try to insert "Ryan" into the hash table. The DJB hash value is 2089555519 and the ACM hash value is 1400397935. Do you see why that's problematic:

You can fix the problem in a few ways. The easiest, as with quadratic probing, is to make sure that the table size is a prime number. There are other fixes, but I won't go into them (maybe in class).

Performance Comparison

The following table compares the expected number of probes required by each of the 4 techniques discussed in these notes for various operations. In the table, L represents the load factor.

Operation	Separate Chaining	Linear Probing	Quadratic Probing	Double Hashing
Insertion	1	½(1 + 1/(1-L)2)	1/(1-L) + ½	1/(1-L)
Successful Search	1 + L/2	½(1 + 1/(1-L))	(1/L)*ln(1/(1-L)) + ½	ln(1/(1-L))
Unsuccessful Search	1 + L	½(1 + 1/(1-L)2)	1/(1-L) + ½	1/(1-L)

ln is the natural logarithm.

Some additional notes of interest:

1. If memory is not an issue, then separate chaining is easily the best algorithm. Note that load factors less than 1, separate chaining is never expected to require more than 2 probes for any operation.
2. Deletions require the same number of probes as successful searches, since a delete first finds the target key.
3. The suggested load factor for separate chaining is 1, which means you should try to make the table size equal to the expected number of keys.
4. Linear probing has a problem called primary clustering, which means that keys can cluster around the initial probe location and require several searches for an insert operation to locate a free entry, or for a search operation to determine whether a key is in the table. This clustering effect is why linear probing is less efficient than quadratic probing or double hashing. For example, if L is 0.75 then 8.5 probes are expected for an insertion using linear probing and if L is 0.9, 50 probes are expected. Conversely, insertions in quadratic probing and double hashing would be expected to require 4 and 10 probes for the same respective loads.
5. A reasonable load for linear probing is considered to be 0.5. With this load factor, an insertion is expected to require 2.5 probes and a successful search is expected to require 1.5 probes.
6. Quadratic probing has a problem called secondary clustering, which means that keys can cluster around the secondary insertion points for a key. This is less of a problem than primary clustering, and in practice, only adds about ½ probe to a search or insertion.
7. Double hashing with a good second function achieves the theoretical best performance. However, on average it is only a ½ probe better than quadratic probing, and since it is more complicated than quadratic probing and the computation of the second hash function requires more time than computing i², quadratic probing is typically preferred.
8. If the hash table size is prime, then the insert operation in quadratic probing is guaranteed to find an empty location if the table is less than half full. Whether the insert operation for double hashing finds an empty location is dependent on whether the second hash function can achieve coverage of all of the available entries.

Summary