1. Chapter 2, Sections 4 and 5 Priority Queues Heaps Dictionaries
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Chapter 2, Sections 4 and 5
Priority Queues
Heaps
Dictionaries
Hash Tables
Spring 2003
CS 315

2. Priority Queues
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Priority Queues
Spring 2003
CS 315

3. Outline and Reading PriorityQueue ADT (§2.4.1)
Total order relation (§2.4.1)
Comparator ADT (§2.4.1)
Sorting with a priority queue (§2.4.2)
Selection-sort (§2.4.2)
Insertion-sort (§2.4.2)
Spring 2003
CS 315

4. Priority Queue ADT A priority queue stores a collection of items
An item is a pair (key, element)
Main methods of the Priority Queue ADT
insertItem(k, o) inserts an item with key k and element o
removeMin() removes the item with smallest key and returns its element
Additional methods
minKey(k, o) returns, but does not remove, the smallest key of an item
minElement() returns, but does not remove, the element of an item with smallest key
size(), isEmpty()
Applications:
Standby flyers
Auctions
Stock market
Spring 2003
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5. Total Order Relation
Keys in a priority queue can be arbitrary objects on which an order is defined
Two distinct items in a priority queue can have the same key
Mathematical concept of total order relation 
Reflexive property: x  x
Antisymmetric property: x  y  y  x  x = y
Transitive property: x  y  y  z  x  z
Spring 2003
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6. Comparator ADT
A comparator encapsulates the action of comparing two objects according to a given total order relation
A generic priority queue uses an auxiliary comparator
The comparator is external to the keys being compared
When the priority queue needs to compare two keys, it uses its comparator
Methods of the Comparator ADT, all with Boolean return type
isLessThan(x, y)
isLessThanOrEqualTo(x,y)
isEqualTo(x,y)
isGreaterThan(x, y)
isGreaterThanOrEqualTo(x,y)
isComparable(x)
Spring 2003
CS 315

7. Sorting with a Priority Queue
We can use a priority queue to sort a set of comparable elements
Insert the elements one by one with a series of insertItem(e, e) operations
Remove the elements in sorted order with a series of removeMin() operations
The running time of this sorting method depends on the priority queue implementation
Algorithm PQ-Sort(S, C)
Input sequence S, comparator C for the elements of S
Output sequence S sorted in increasing order according to C
P  priority queue with comparator C
while S.isEmpty ()
e  S.remove (S. first ())
P.insertItem(e, e)
while P.isEmpty()
e  P.removeMin()
S.insertLast(e)
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8. Sequence-based Priority Queue
Implementation with an unsorted sequence
Store the items of the priority queue in a list-based sequence, in arbitrary order
Performance:
insertItem takes O(1) time since we can insert the item at the beginning or end of the sequence
removeMin, minKey and minElement take O(n) time since we have to traverse the entire sequence to find the smallest key
Implementation with a sorted sequence
Store the items of the priority queue in a sequence, sorted by key
Performance:
insertItem takes O(n) time since we have to find the place where to insert the item
removeMin, minKey and minElement take O(1) time since the smallest key is at the beginning of the sequence
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9. Selection-Sort
Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence
Running time of Selection-sort:
Inserting the elements into the priority queue with n insertItem operations takes O(n) time
Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to …+ n
Selection-sort runs in O(n2) time
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10. Insertion-Sort
Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence
Running time of Insertion-sort:
Inserting the elements into the priority queue with n insertItem operations takes time proportional to …+ n
Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time
Insertion-sort runs in O(n2) time
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11. In-place Insertion-sort
Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place
A portion of the input sequence itself serves as the priority queue
For in-place insertion-sort
We keep sorted the initial portion of the sequence
We can use swapElements instead of modifying the sequence 5 4 2 3 1 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5
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CS 315

12. Heaps and Priority Queues
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Heaps and Priority Queues 2 6 5 7 9
Spring 2003
CS 315

13. Priority Queue ADT (§ 2.4.1)
A priority queue stores a collection of items
An item is a pair (key, element)
Main methods of the Priority Queue ADT
insertItem(k, o) inserts an item with key k and element o
removeMin() removes the item with smallest key and returns its element
Additional methods
minKey(k, o) returns, but does not remove, the smallest key of an item
minElement() returns, but does not remove, the element of an item with smallest key
size(), isEmpty()
Applications:
Standby flyers
Auctions
Stock market
Spring 2003
CS 315

14. Total Order Relation
Keys in a priority queue can be arbitrary objects on which an order is defined
Two distinct items in a priority queue can have the same key
Mathematical concept of total order relation 
Reflexive property: x  x
Antisymmetric property: x  y  y  x  x = y
Transitive property: x  y  y  z  x  z
Spring 2003
CS 315

15. Comparator ADT (§ 2.4.1)
A comparator encapsulates the action of comparing two objects according to a given total order relation
A generic priority queue uses an auxiliary comparator
The comparator is external to the keys being compared
When the priority queue needs to compare two keys, it uses its comparator
Methods of the Comparator ADT, all with Boolean return type
isLessThan(x, y)
isLessThanOrEqualTo(x,y)
isEqualTo(x,y)
isGreaterThan(x, y)
isGreaterThanOrEqualTo(x,y)
isComparable(x)
Spring 2003
CS 315

16. Sorting with a Priority Queue (§ 2.4.2)
We can use a priority queue to sort a set of comparable elements
Insert the elements one by one with a series of insertItem(e, e) operations
Remove the elements in sorted order with a series of removeMin() operations
The running time of this sorting method depends on the priority queue implementation
Algorithm PQ-Sort(S, C)
Input sequence S, comparator C for the elements of S
Output sequence S sorted in increasing order according to C
P  priority queue with comparator C
while S.isEmpty ()
e  S.remove (S. first ())
P.insertItem(e, e)
while P.isEmpty()
e  P.removeMin()
S.insertLast(e)
Spring 2003
CS 315

17. Sequence-based Priority Queue
Implementation with an unsorted list
Performance:
insertItem takes O(1) time since we can insert the item at the beginning or end of the sequence
removeMin, minKey and minElement take O(n) time since we have to traverse the entire sequence to find the smallest key
Implementation with a sorted list
Performance:
insertItem takes O(n) time since we have to find the place where to insert the item
removeMin, minKey and minElement take O(1) time since the smallest key is at the beginning of the sequence 4 5 2 3 1 1 2 3 4 5
Spring 2003
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18. Selection-Sort
Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence
Running time of Selection-sort:
Inserting the elements into the priority queue with n insertItem operations takes O(n) time
Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to …+ n
Selection-sort runs in O(n2) time 4 5 2 3 1
Spring 2003
CS 315

19. Insertion-Sort
Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence
Running time of Insertion-sort:
Inserting the elements into the priority queue with n insertItem operations takes time proportional to …+ n
Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time
Insertion-sort runs in O(n2) time 1 2 3 4 5
Spring 2003
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20. Priority Queues
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What is a heap (§2.4.3)
A heap is a binary tree storing keys at its internal nodes and satisfying the following properties:
Heap-Order: for every internal node v other than the root, key(v)  key(parent(v))
Complete Binary Tree: let h be the height of the heap
for i = 0, … , h - 1, there are 2i nodes of depth i
at depth h - 1, the internal nodes are to the left of the external nodes
The last node of a heap is the rightmost internal node of depth h - 1 2 5 6 9 7
last node
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21. Height of a Heap (§2.4.3)
Theorem: A heap storing n keys has height O(log n)
Proof: (we apply the complete binary tree property)
Let h be the height of a heap storing n keys
Since there are 2i keys at depth i = 0, … , h - 2 and at least one key at depth h - 1, we have n  … + 2h
Thus, n  2h-1 , i.e., h  log n + 1
depth
keys 1 1 2
h-2
2h-2
h-1 1
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22. Heaps and Priority Queues
We can use a heap to implement a priority queue
We store a (key, element) item at each internal node
We keep track of the position of the last node
For simplicity, we show only the keys in the pictures
(2, Sue)
(5, Pat)
(6, Mark)
(9, Jeff)
(7, Anna)
Spring 2003
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23. Insertion into a Heap (§2.4.3)
Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap
The insertion algorithm consists of three steps
Find the insertion node z (the new last node)
Store k at z and expand z into an internal node
Restore the heap-order property (discussed next) 2 6 5 7 9 z
insertion node 2 5 6 z 9 7 1
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24. Upheap
After the insertion of a new key k, the heap-order property may be violated
Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6
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25. Removal from a Heap (§2.4.3)
Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap
The removal algorithm consists of three steps
Replace the root key with the key of the last node w
Compress w and its children into a leaf
Restore the heap-order property (discussed next) 2 6 5 7 9 w
last node 7 5 6 w 9
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26. Downheap
After replacing the root key with the key k of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 7 9 w 5 6 w 9
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27. Updating the Last Node
The insertion node can be found by traversing a path of O(log n) nodes
Go up until a left child or the root is reached
If a left child is reached, go to the right child
Go down left until a leaf is reached
Similar algorithm for updating the last node after a removal
Spring 2003
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28. Heap-Sort (§2.4.4)
Consider a priority queue with n items implemented by means of a heap
the space used is O(n)
methods insertItem and removeMin take O(log n) time
methods size, isEmpty, minKey, and minElement take time O(1) time
Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time
The resulting algorithm is called heap-sort
Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort
Spring 2003
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29. Vector-based Heap Implementation (§2.4.3)
We can represent a heap with n keys by means of a vector of length n + 1
For the node at rank i
the left child is at rank 2i
the right child is at rank 2i + 1
Links between nodes are not explicitly stored
The leaves are not represented
The cell of at rank 0 is not used
Operation insertItem corresponds to inserting at rank n + 1
Operation removeMin corresponds to removing at rank n
Yields in-place heap-sort 2 6 5 7 9 2 5 6 9 7 1 3 4
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30. Merging Two Heaps We are given two two heaps and a key k3 5 8 2 6 4
We are given two two heaps and a key k
We create a new heap with the root node storing k and with the two heaps as subtrees
We perform downheap to restore the heap-order property 7 3 2 8 5 4 6 2 3 4 8 5 7 6
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31. Bottom-up Heap Construction (§2.4.3)
We can construct a heap storing n given keys in using a bottom-up construction with log n phases
In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys
2i -1
2i+1-1
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32. Example 16 15 4 12 6 7 23 20 25 5 11 27 16 15 4 12 6 7 23 20
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33. Example (contd.) 25 5 11 27 16 15 4 12 6 9 23 20 15 4 6 23 16 25 5 12 11 9 27 20
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34. Example (contd.) 7 8 15 4 6 23 16 25 5 12 11 9 27 20 4 6 15 5 8 23 16 25 7 12 11 9 27 20
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35. Example (end) 10 4 6 15 5 8 23 16 25 7 12 11 9 27 20 4 5 6 15 7 8 23 16 25 10 12 11 9 27 20
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36. Analysis
We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path)
Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n)
Thus, bottom-up heap construction runs in O(n) time
Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort
Spring 2003
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37. Dictionaries and Hash Tables
Priority Queues
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Dictionaries and Hash Tables  1 2 3  4
Spring 2003
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38. Dictionary ADT (§2.5.1)
The dictionary ADT models a searchable collection of key-element items
The main operations of a dictionary are searching, inserting, and deleting items
Multiple items with the same key are allowed
Applications:
address book
credit card authorization
mapping host names (e.g., cs16.net) to internet addresses (e.g., )
Dictionary ADT methods:
findElement(k): if the dictionary has an item with key k, returns its element, else, returns the special element NO_SUCH_KEY
insertItem(k, o): inserts item (k, o) into the dictionary
removeElement(k): if the dictionary has an item with key k, removes it from the dictionary and returns its element, else returns the special element NO_SUCH_KEY
size(), isEmpty()
keys(), Elements()
Spring 2003
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39. Log File (§2.5.1)
A log file is a dictionary implemented by means of an unsorted sequence
We store the items of the dictionary in a sequence (based on a doubly-linked lists or a circular array), in arbitrary order
Performance:
insertItem takes O(1) time since we can insert the new item at the beginning or at the end of the sequence
findElement and removeElement take O(n) time since in the worst case (the item is not found) we traverse the entire sequence to look for an item with the given key
The log file is effective only for dictionaries of small size or for dictionaries on which insertions are the most common operations, while searches and removals are rarely performed (e.g., historical record of logins to a workstation)
Spring 2003
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40. Hash Functions and Hash Tables (§2.5.2)
The basic idea: searching through the entries in a log file can be time consuming. Is there a better way?
Hash codes, hash functions, and hash tables are one solution.
The idea: place dictionary entries in “buckets” (based on keys) in such as way that the entries of the dictionary are relatively evenly spread throughout the buckets.
To access an entry, use the key to find the correct bucket. Presumably searching the bucket is quicker than searching through an entire log file.
Spring 2003
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41. Hash Functions and Hash Tables (§2.5.2)
What does this entail?
Need an efficient way to map keys “evenly” to buckets.
The method: First, map keys to integers. The function that does this is called a hash code, and the integer output of such a mapping can be called the hash code for the original key
Want to avoid having more than one key mapped to same integer (called a collision)
Next, use another mapping, called a compression map, to map the hash codes to buckets.
The hash codes may vary considerably. That is, they may have a wide range of values, the use of the term compression.
The composition of the two operations, that is, the map from keys to buckets, is called a hash function.
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42. Hash Functions and Hash Tables (§2.5.2)
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
A hash table for a given key type consists of
Hash function h
Array (called table) of size N
Cells in this array are often called buckets, and the array a bucket array
When implementing a dictionary with a hash table, the goal is to store item (k, o) at index i = h(k)
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43. 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  1 2 3 4
9997
9998
9999 …
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44. Again…
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
Spring 2003
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45. Hash Code Maps (§2.5.3) 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)
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)
For keys longer than this, lots of collisions
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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46. 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
For sake of speed, some implementations only apply polynomial has to fraction of the characters in a long string
E.g. every eight characters, etc.
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47. Compression Maps Division: Here’s an example of what can happen:
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, but see right column…
Even prime N will generate collisions under some recurring numerical patterns
Here’s an example of what can happen:
Consider the following hash codes (200,205,210,…,600)
A total of 80 codes
Using N = 100, each code collides with three others
Using N = 101 produces no collisions
Spring 2003
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48. Compression Maps 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
The general rule: A well chosen hash function should guarantee that the probability of two different keys being hashed to the same bucket is no more than 1/N.
Spring 2003
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49. Collision Handling
Collisions occur when different elements are mapped to the same cell
Chaining: let each cell in the table point to a linked list of elements that map there  1 2 3 4
Chaining is simple, but requires additional memory outside the table
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50. Linear Probing (§2.5.5) Example: h(x) = x mod 13
Open addressing: the colliding item is placed in a different cell of the table
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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51. 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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52. 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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53. ( )2
Quadratic Probing
Quadratic probing involves iteratively trying buckets
This complicates removal operation, but avoid kind of clustering seen with linear probing
Still get secondary clustering, especially if N is not prime (can get it even if it IS prime)
Also, can fail to find an open bucket even when one is present
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54. Priority Queues
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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 A[(h(k) + 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 (Why?)
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
The key here is that for any integer i, you want to be sure that (i + jd(k) )mod N can probe all cells.
If N is not prime this may not happen. For example, consider N = 12, i= 3, and d(k) = 4. Then
i + jd(k) mod N is 3,7,11,3,7,11,3, etc. So many cells in this case are not probed.
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55. 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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56. Performance of Hashing
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
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 (load factor is expected size of each bucket)
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57. Universal Hashing
A family of hash functions is universal if, for any 0<i,j<M-1, Pr(h(j)=h(k)) < 1/N.
Choose p as a prime between M and 2M.
Randomly select 0<a<p and 0<b<p, and define h(k)=(ak+b mod p) mod N
Theorem: The set of all functions, h, as defined here, is universal.
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58. Proof of Universality (Part 1)
Let f(k) = ak+b mod p
Let g(k) = k mod N
So h(k) = g(f(k)).
f causes no collisions:
Let f(k) = f(j).
Suppose k<j. Then
So a(j-k) is a multiple of p
But both are less than p
So a(j-k) = 0. I.e., j=k. (contradiction)
Thus, f causes no collisions.
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59. Proof of Universality (Part 2)
If f causes no collisions, only g can make h cause collisions.
Fix a number x. Of the p integers y=f(k), different from x, the number such that g(y)=g(x) is at most
Since there are p choices for x, the number of h’s that will cause a collision between j and k is at most
There are p(p-1) functions h. So probability of collision is at most
Therefore, the set of possible h functions is universal.
Spring 2003
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