1. Hash Tables
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Hash Tables  1 2  3  4
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2. Outline and Reading Hash functions and hash tables (§8.3)
Hash function details
Hash code map (§8.3.3)
Compression map (§8.3.4)
Collision handling (§8.3.5)
Chaining
Linear probing
Double hashing
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3. Hashing:
A method for directly referencing items in a dictionary by doing arithmetic transformations on keys into dictionary addresses.
A hush function is perfect if there is no key collision, that is, two keys hash to the same hash value.
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4. An Example: suppose: MagicNumber = 15 int h(String s) {
return ((s[0] + s[1])% MagicNumber); }
suppose: planet solarSystem[MagicNumber];
class planet {
String name;
int numMoons;
double sunDistance; ….
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5. Suppose: Where are they located
solarSystem[h(“Mercury”)] = new planet(“Mercury”, 0, 36.0);
solarSystem[h(“Venus”)] = new planet(“Venus”, 0, 67.27);
solarSystem[h(“Earth”)] = new planet(“Earth”, 1, 93.0);
solarSystem[h(“Mars”)] = new planet(“Mars”, 2, );
solarSystem[h(“Jupiter”)] = new planet(“Jupiter”, 16, );
solarSystem[h(“Saturn”)] = new planet(“Saturn”, 12, );
solarSystem[h(“Uranus”)] = new planet(“Uranus”, 5, );
solarSystem[h(“Neptune”)] = new planet(“Neptune”, 2, 2795);
solarSystem[h(“Pluto”)] = new planet(“Pluto”, 1, 3675);
Where are they located
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6. “Ju” in ASCII are 74 and 117, 74 + 117 = 191; 191 % 15 = 11;
h(“Mercury”) = 13
h(“Venus”) = 7
h(“Earth”) = 1
h(“Mars”) = 9
h(“Jupiter”) = 11
h(“Saturn”) = 0
h(“Uranus”) = 4
h(“Neptune”) = 14
h(“Pluto”) = 8
Thus, our search function is simply:
planet search(String s){ return solarSystem[h(s)]; }
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7. Hash Functions and Hash Tables
A hash function h maps keys of a given type to integers in a fixed interval [0, N - 1]
Example: h(x) = x mod N is a hash function for integer keys
The integer h(x) is called the hash value of key x
The goal of a hash function is to uniformly disperse keys in the range [0, N - 1]
A hash table for a given key type consists of
Hash function h
Array (called table) of size N
When implementing a dictionary with a hash table, the goal is to store item (k, o) at index i = h(k)
A collision occurs when two keys in the dictionary have the same hash value
Collision handing schemes:
Chaining: colliding items are stored in a sequence
Open addressing: the colliding item is placed in a different cell of the table
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8. Example
We design a hash table for a dictionary storing items (SSN, Name), where SSN (social security number) is a nine-digit positive integer
Our hash table uses an array of size N = 10,000 and the hash function h(x) = last four digits of x
We use chaining to handle collisions  1 2  3  4 …
9997 
9998
9999 
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9. Hash Functions
A hash function is usually specified as the composition of two functions:
Hash code map: h1: keys  integers
Compression map: h2: integers  [0, N - 1]
The hash code map is applied first, and the compression map is applied next on the result, i.e., h(x) = h2(h1(x))
The goal of the hash function is to “disperse” the keys in an apparently random way
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10. Hash Code Maps Memory address: Integer cast: Component sum:
We reinterpret the memory address of the key object as an integer (default hash code of all Java objects)
Good in general, except for numeric and string keys
Integer cast:
We reinterpret the bits of the key as an integer
Suitable for keys of length less than or equal to the number of bits of the integer type (e.g., byte, short, int and float in Java)
Component sum:
We partition the bits of the key into components of fixed length (e.g., 16 or 32 bits) and we sum the components (ignoring overflows)
Suitable for numeric keys of fixed length greater than or equal to the number of bits of the integer type (e.g., long and double in Java)
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11. Hash Code Maps (cont.) Polynomial accumulation:
We partition the bits of the key into a sequence of components of fixed length (e.g., 8, 16 or 32 bits) a0 a1 … an-1
We evaluate the polynomial
p(z) = a0 + a1 z + a2 z2 + … … + an-1zn-1
at a fixed value z, ignoring overflows
Especially suitable for strings (e.g., the choice z = 33 gives at most 6 collisions on a set of 50,000 English words)
Polynomial p(z) can be evaluated in O(n) time using Horner’s rule:
The following polynomials are successively computed, each from the previous one in O(1) time
p0(z) = an-1
pi (z) = an-i-1 + zpi-1(z) (i = 1, 2, …, n -1)
We have p(z) = pn-1(z)
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12. Compression Maps Division: Multiply, Add and Divide (MAD):
h2 (y) = y mod N
The size N of the hash table is usually chosen to be a prime
The reason has to do with number theory and is beyond the scope of this course
Multiply, Add and Divide (MAD):
h2 (y) = (ay + b) mod N
a and b are nonnegative integers such that a mod N  0
Otherwise, every integer would map to the same value b
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13. Linear Probing Example:
Linear probing handles collisions by placing the colliding item in the next (circularly) available table cell
Each table cell inspected is referred to as a “probe”
Colliding items lump together, causing future collisions to cause a longer sequence of probes
Example:
h(x) = x mod 13
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73 1 2 3 4 5 6 7 8 9 10 11 12
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14. Search with Linear Probing
Consider a hash table A that uses linear probing
findElement(k)
We start at cell h(k)
We probe consecutive locations until one of the following occurs
An item with key k is found, or
An empty cell is found, or
N cells have been unsuccessfully probed
Algorithm findElement(k)
i  h(k)
p  0
repeat
c  A[i]
if c = 
return NO_SUCH_KEY
else if c.key () = k
return c.element()
else
i  (i + 1) mod N
p  p + 1
until p = N
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15. Updates with Linear Probing
To handle insertions and deletions, we introduce a special object, called AVAILABLE, which replaces deleted elements
removeElement(k)
We search for an item with key k
If such an item (k, o) is found, we replace it with the special item AVAILABLE and we return element o
Else, we return NO_SUCH_KEY
insert Item(k, o)
We throw an exception if the table is full
We start at cell h(k)
We probe consecutive cells until one of the following occurs
A cell i is found that is either empty or stores AVAILABLE, or
N cells have been unsuccessfully probed
We store item (k, o) in cell i
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16. Double Hashing
Double hashing uses a secondary hash function d(k) and handles collisions by placing an item in the first available cell of the series (i + jd(k)) mod N for j = 0, 1, … , N - 1
The secondary hash function d(k) cannot have zero values
The table size N must be a prime to allow probing of all the cells
Common choice of compression map for the secondary hash function: d2(k) = q - k mod q
where
q < N
q is a prime
The possible values for d2(k) are 1, 2, … , q
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17. Example of Double Hashing
Consider a hash table storing integer keys that handles collision with double hashing
N = 13
h(k) = k mod 13
d(k) = 7 - k mod 7
Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order 1 2 3 4 5 6 7 8 9 10 11 12 31 41 18 32 59 73 22 44 1 2 3 4 5 6 7 8 9 10 11 12
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18. Performance:
Let N be the number of slots of a hash table, n be the number of items in the table, we define load factor as:
 = n/N
If the hash function randomly distributes keys through the table, then the expected length of a successful search path is:
lengthsucc = ½(1 + 1/(1- ))
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19. Performance:
The expected length of an unsuccessful search is approximately:
lengthunsucc = ½( 1 + 1/(1 - )2)
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20. Problems with Probing:
The size of the hash table must be fixed in advance.
The search costs increase dramatically as the table becomes nearly full.
Need a special object, called AVAILABLE, to implement “delete” operation.
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21. Collision resolution using Linked Lists:
Dynamically allocate space.
Easy to insert/delete an item
Need a link for each node in the hash table.
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22. Performance:
Let N be the size of the hash table, n the number of items in the table’s linked lists, if all input sequences are equally likely and the hash function randomly distributes keys over the table, the expected length of a linked list is n/N.
lengthsucc = ½(n/N)
lengthunsucc = n/N
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23. Conclusion:
In the worst case, searches, insertions and removals on a hash table take O(n) time
The worst case occurs when all the keys inserted into the dictionary collide
The load factor a = n/N affects the performance of a hash table
Assuming that the hash values are like random numbers, it can be shown that the expected number of probes for an insertion with open addressing is 1 / (1 - a)
The expected running time of all the dictionary ADT operations in a hash table is O(1)
In practice, hashing is very fast provided the load factor is not close to 100%
Applications of hash tables:
small databases
compilers
browser caches
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24. Locators 3 a 1 g 4 e 11/22/2018 3:15 AM Hash Tables Hash Tables

25. Outline and Reading Locators (§8.7) Locator-based methods
Implementation
Positions vs. Locators
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26. Locators Application example:
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Locators
A locators identifies and tracks a (key, element) item within a data structure
A locator sticks with a specific item, even if that element changes its position in the data structure
Intuitive notion:
claim check
reservation number
Methods of the locator ADT:
key(): returns the key of the item associated with the locator
element(): returns the element of the item associated with the locator
Application example:
Orders to purchase and sell a given stock are stored in two priority queues (sell orders and buy orders)
the key of an order is the price
the element is the number of shares
When an order is placed, a locator to it is returned
Given a locator, an order can be canceled or modified
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27. Locator-based Methods
Locator-based priority queue methods:
insert(k, o): inserts the item (k, o) and returns a locator for it
min(): returns the locator of an item with smallest key
remove(l): remove the item with locator l
replaceKey(l, k): replaces the key of the item with locator l
replaceElement(l, o): replaces with o the element of the item with locator l
locators(): returns an iterator over the locators of the items in the priority queue
Locator-based dictionary methods:
insert(k, o): inserts the item (k, o) and returns its locator
find(k): if the dictionary contains an item with key k, returns its locator, else return the special locator NO_SUCH_KEY
remove(l): removes the item with locator l and returns its element
locators(), replaceKey(l, k), replaceElement(l, o)
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28. Implementation The locator is an object storing
key
element
position (or rank) of the item in the underlying structure
In turn, the position (or array cell) stores the locator
Example:
binary search tree with locators 6 d 3 a 9 b 1 g 4 e 8 c
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29. Positions vs. Locators Position
represents a “place” in a data structure
related to other positions in the data structure (e.g., previous/next or parent/child)
implemented as a node or an array cell
Position-based ADTs (e.g., sequence and tree) are fundamental data storage schemes
Locator
identifies and tracks a (key, element) item
unrelated to other locators in the data structure
implemented as an object storing the item and its position in the underlying structure
Key-based ADTs (e.g., priority queue and dictionary) can be augmented with locator-based methods
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