Maps, Dictionaries, Hashing

1
Maps, Dictionaries, Hashing
2
Outline and Reading

• Map ADT (9.1)
• Dictionary ADT (9.5)
• Hash Tables (9.2)
• Ordered Maps (9.3)

3
Map ADT

• The map ADT models a searchable collection of
  key-element items
• The main operations of a map are searching,
  inserting, and deleting items
• Multiple items with the same key are not allowed
• Applications
• address book
• mapping host names (e.g., cs16.net) to internet
  addresses (e.g., 128.148.34.101)

• Map ADT methods
• find(k) if M has an entry with key k, return an
  iterator p referring to this element, else,
  return special end iterator.
• put(k, v) if M has no entry with key k, then add
  entry (k, v) to M, otherwise replace the value of
  the entry with v return iterator to the
  inserted/modified entry
• erase(k) or erase(p) remove from M entry with
  key k or iterator p An error occurs if there
  is no such element.
• size(), isEmpty()

4
Map - Direct Address Table

• A direct address table is a map in which
• The keys are in the range 0,1,2,,N-1
• Stored in an array of size N - T0,N-1
• Item with key k stored in Tk
• Performance
• insertItem, find, and removeElement all take O(1)
  time
• Space - requires space O(N), independent of n,
  the number of items stored in the map
• The direct address table is not space efficient
  unless the range of the keys is close to the
  number of elements to be stored in the map, I.e.,
  unless n is close to N.

5
Dictionary ADT

• The dictionary ADT models a searchable collection
  of key-element items
• The main difference from a map is that multiple
  items with the same key are allowed
• Any data structure that supports a dictionary
  also supports a map
• Applications
• Dictionary that has multiple definitions for the
  same word

• Dictionary ADT methods
• find(k) if the dictionary has an entry with key
  k, returns an iterator p to an arbitrary element
• findAll(k) Return iterators (b,e) s.t. that all
  entries with key k are between them
• insert(k, v) insert entry (k, v) into D, return
  iterator to it
• erase(k), erase(p) remove arbitrary entry with
  key k or entry referenced by iterator p. Error
  occurs if there is no such entry
• Begin(), end() return iterator to first or just
  beyond last entry of D
• size(), isEmpty()

6
Map/Dictionary - Log File (unordered sequence
implementation)

• 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
• insert takes O(1) time since we can insert the
  new item at the beginning or at the end of the
  sequence
• find and erase take O(n) time since in the worst
  case (item is not found) we traverse the entire
  sequence to find the item with the given key
• Space - can be O(n), where n is the number of
  elements in the dictionary
• 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)

7
Map/Dictionarie implementations

• n - elements in map/Dictionary

Insert Find Space
Log File O(1) O(n) O(n)
Direct Address Table (map only) O(1) O(1) O(N)
8
Hash Tables

• Hashing
• Hash table (an array) of size N, H0,N-1
• Hash function h that maps keys to indices in H
• Issues
• Hash functions - need method to transform key to
  an index in H that will have nice properties.
• Collisions - some keys will map to the same index
  of H (otherwise we have a Direct Address Table).
  Several methods to resolve the collisions
• Chaining - put elements that hash to same
  location in a linked list
• Open addressing - if a collision occurs, have a
  method to select another location in the table

9
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 Nis 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
• When implementing a dictionary with a hash table,
  the goal is to store item (k, o) at index i
  h(k)

10
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 functionh(x) last four digits of x

11
Collisions

• Collisions occur when different elements are
  mapped to the same cell
• collisions must be resolved
• Chaining (store in list outside the table)
• Open addressing (store in another cell in the
  table)

• Example with Modulo Method
• h(k) k mod N
• If N10, then
• h(k)0 for k0,10,20,
• h(k) 1 for k1, 11, 21, etc

12
Collision Resolution with Chaining

• 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

• Chaining is simple, but requires additional
  memory outside the table

13
Exercise chaining

• Assume you have a hash table H with N9 slots
  (H0,8) and let the hash function be h(k)k mod
  N.
• Demonstrate (by picture) the insertion of the
  following keys into a hash table with collisions
  resolved by chaining.
• 5, 28, 19, 15, 20, 33, 12, 17, 10

14
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

0
1
2
3
4
5
6
7
8
9
10
11
12
15
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

18

0
1
2
3
4
5
6
7
8
9
10
11
12
16
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18

0
1
2
3
4
5
6
7
8
9
10
11
12
17
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
22

0
1
2
3
4
5
6
7
8
9
10
11
12
18
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
22

0
1
2
3
4
5
6
7
8
9
10
11
12
19
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
22

0
1
2
3
4
5
6
7
8
9
10
11
12
20
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
22

0
1
2
3
4
5
6
7
8
9
10
11
12
21
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
22

0
1
2
3
4
5
6
7
8
9
10
11
12
22
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
22

0
1
2
3
4
5
6
7
8
9
10
11
12
23
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
24
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
25
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
26
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
27
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
28
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22

0
1
2
3
4
5
6
7
8
9
10
11
12
29
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22
31

0
1
2
3
4
5
6
7
8
9
10
11
12
30
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22
31

0
1
2
3
4
5
6
7
8
9
10
11
12
31
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22
31

0
1
2
3
4
5
6
7
8
9
10
11
12
32
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22
31

0
1
2
3
4
5
6
7
8
9
10
11
12
33
Collision Resolution in Open Addressing - Linear
Probing

• 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. So the i-th cell checked is
• H(k,i) (h(k)i)mod N
• 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

41


18
44
59
32
22
31
73

0
1
2
3
4
5
6
7
8
9
10
11
12
34
Search with Linear Probing

• Consider a hash table A that uses linear probing
• find(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 find(k) i ? h(k) p ? 0 repeat c ?
Ai if c ? return Position(null) else
if c.key () k return Position(c)
else i ? (i 1) mod N p ? p 1 until
p N return Position(null)
35
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 the
  position of this item
• Else, we return a null position

• insertItem(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

36
Exercise Linear Probing

• Assume you have a hash table H with N11 slots
  (H0,10) and let the hash function be h(k)k mod
  N.
• Demonstrate (by picture) the insertion of the
  following keys into a hash table with collisions
  resolved by linear probing.
• 10, 22, 31, 4, 15, 28, 17, 88, 59

37
Open Addressing Double Hashing

• Common choice of compression map for the
  secondary hash function d2(k) q - (k mod q)
• where
• q lt N
• q is a prime
• The possible values for d2(k) are 1, 2, , q

• Double hashing uses a secondary hash function
  d(k) and handles collisions by placing an item
  in the first available cell of the seriesh(k,i)
  (h(k) id(k)) mod N for i 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

38
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

0
1
2
3
4
5
6
7
8
9
10
11
12
31

41


18
32
59
73
22
44

0
1
2
3
4
5
6
7
8
9
10
11
12
39
Exercise Double Hashing

• Assume you have a hash table H with N11 slots
  (H0,10) and let the hash functions for double
  hashing be
• h(k,i)(h(k) ih2(k))mod N
• h(k)k mod N
• h2(k)1 (k mod (N-1))
• Demonstrate (by picture) the insertion of the
  following keys into H
• 10, 22, 31, 4, 15, 28, 17, 88, 59

40
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

41
Hash Code Maps

• Memory address
• We reinterpret the memory address of the key
  object as an integer
• 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.,
  char, short, int and float on many machines)

• 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 on many
  machines)

42
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 Horners 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)

43
Compression Maps

• Division
• 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

44
Performance of Hashing

• 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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Uniform Hashing Assumption

• The probe sequence of a key k is the sequence of
  slots that will be probed when looking for k
• In open addressing, the probe sequence is h(k,0),
  h(k,1), h(k,2), h(k,3),
• Uniform Hashing Assumption Each key is equally
  likely to have any one of the N! permutations of
  0,1, 2, , N-1 as is probe sequence
• Note Linear probing and double hashing are far
  from achieving Uniform Hashing
• Linear probing N distinct probe sequences
• Double Hashing N2 distinct probe sequences

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Performance of Uniform Hashing

• Theorem Assuming uniform hashing and an
  open-address hash table with load factor a n/N
  lt 1, the expected number of probes in an
  unsuccessful search is at most 1/(1-a).
• Exercise compute the expected number of probes
  in an unsuccessful search in an open address hash
  table with a ½ , a3/4, and a 99/100.

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Maps/Dictionaries

• n elements in map/dictionary,
• Npossible keys (it could be Ngtgtn) or size of
  hash table

Insert Find Space
Log File O(1) O(n) O(n)
Direct Address Table (map only) O(1) O(1) O(N)
Hashing (chaining) O(1) O(n/N) O(nN)
Hashing (open addressing) O(1/(1-n/N)) O(1/(1-n/N)) O(N)
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Ordered Map

• In an ordered Map, we wish to perform the usual
  map operations, but also maintain an order
  relation for the keys in the dictionary.
• Naturally supports
• Look-Up Tables - store dictionary in a vector by
  non-decreasing order of the keys
• Binary Search

• Ordered Dictionary ADT
• In addition to the generic dictionary ADT, the
  ordered dictionary ADT supports the following
  functions
• closestBefore(k) return the position of an item
  with the largest key less than or equal to k
• closestAfter(k) return the position of an item
  with the smallest key greater than or equal to k

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Lookup Table

• A lookup table is a dictionary implemented by
  means of a sorted sequence
• We store the items of the dictionary in an
  array-based sequence, sorted by key
• We use an external comparator for the keys
• Performance
• find takes O(log n) time, using binary search
• insertItem takes O(n) time since in the worst
  case we have to shift n/2 items to make room for
  the new item
• removeElement take O(n) time since in the worst
  case we have to shift n/2 items to compact the
  items after the removal
• The lookup table is effective only for
  dictionaries of small size or for dictionaries on
  which searches are the most common operations,
  while insertions and removals are rarely
  performed (e.g., credit card authorizations)

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Example of Ordered Map Binary Search

• Binary search performs operation find(k) on a
  dictionary implemented by means of an array-based
  sequence, sorted by key
• similar to the high-low game
• at each step, the number of candidate items is
  halved
• terminates after a logarithmic number of steps
• Example find(7)

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52
Universal Hashing

• A family of hash functions is universal if, for
  any 0lti,jltM-1,
• Pr(h(j)h(i)) lt 1/N.
• Choose p as a prime between M and 2M.
• Randomly select 0ltaltp and 0ltbltp, and define
  h(k)(akb mod p) mod N

• Theorem The set of all functions, h, as defined
  here, is universal.

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Proof of Universality (Part 1)

• Let f(k) akb mod p
• Let g(k) k mod N
• So h(k) g(f(k)).
• f causes no collisions
• Let f(k) f(j).
• Suppose kltj. Then

• So a(j-k) is a multiple of p
• But both are less than p
• So a(j-k) 0. I.e., jk. (contradiction)
• Thus, f causes no collisions.

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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 yf(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
  hs 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.