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Fundamentals of Python:From First Programs Through Data
Structures
Chapter 19
Unordered Collections: Sets and Dictionaries
Fundamentals of Python: From First Programs Through Data Structures 2
Objectives
After completing this chapter, you will be able to:
• Implement a set type and a dictionary type using lists
• Explain how hashing can help a programmer achieve constant access time to unordered collections
• Explain strategies for resolving collisions during hashing, such as linear probing, quadratic probing, and bucket/chaining
Fundamentals of Python: From First Programs Through Data Structures 3
Objectives (continued)
After completing this chapter, you will be able to: (continued)
• Use a hashing strategy to implement a set type and a dictionary type
• Use a binary search tree to implement a sorted set type and a sorted dictionary type
Fundamentals of Python: From First Programs Through Data Structures 4
• A set is a collection of items in no particular order
• Most typical operations:– Return the number of items in the set– Test for the empty set (a set that contains no items)– Add an item to the set– Remove an item from the set– Test for set membership– Obtain the union of two sets– Obtain the intersection of two sets– Obtain the difference of two sets
Using Sets
Fundamentals of Python: From First Programs Through Data Structures 5
Using Sets (continued)
Fundamentals of Python: From First Programs Through Data Structures 6
The Python set Class
Fundamentals of Python: From First Programs Through Data Structures 7
The Python set Class (continued)
Fundamentals of Python: From First Programs Through Data Structures 8
A Sample Session with Sets
Fundamentals of Python: From First Programs Through Data Structures 9
A Sample Session with Sets (continued)
Fundamentals of Python: From First Programs Through Data Structures 10
Applications of Sets
• Sets have many applications in the area of data processing– Example: In database management, answer to query
that contains conjunction of two keys could be constructed from intersection of sets of items associated with those keys
Fundamentals of Python: From First Programs Through Data Structures 11
Implementations of Sets
• Arrays and lists may be used to contain the data items of a set– A linked list has the advantage of supporting
constant-time removals of items• Once they are located in the structure
• Hashing attempts to approximate random access into an array for insertions, removals, and searches
Fundamentals of Python: From First Programs Through Data Structures 12
Relationship Between Sets and Dictionaries
• A dictionary is an unordered collection of elements called entries– Each entry consists of a key and an associated
value– A dictionary’s keys must be unique, but its values
may be duplicated
• One can think of a dictionary as having a set of keys
Fundamentals of Python: From First Programs Through Data Structures 13
List Implementations of Sets and Dictionaries
• The simplest implementations of sets and dictionaries use lists
• This section presents these implementations and assesses their run-time performance
Sets
• List implementation of a set
Fundamentals of Python: From First Programs Through Data Structures 14
Fundamentals of Python: From First Programs Through Data Structures 15
Dictionaries
• Our list-based implementation of a dictionary is called ListDict– The entries in a dictionary consist of two parts, a key
and a value
• A list implementation of a dictionary behaves in many ways like a list implementation of a set
Fundamentals of Python: From First Programs Through Data Structures 16
Dictionaries (continued)
Fundamentals of Python: From First Programs Through Data Structures 17
Dictionaries (continued)
Fundamentals of Python: From First Programs Through Data Structures 18
Dictionaries (continued)
Fundamentals of Python: From First Programs Through Data Structures 19
Complexity Analysis of the List Implementations of Sets and
Dictionaries• The list implementations of sets and dictionaries
require little programmer effort– Unfortunately, they do not perform well
• Basic accessing methods must perform a linear search of the underlying list– Each basic accessing method is O(n)
Fundamentals of Python: From First Programs Through Data Structures 20
Hashing Strategies
• Key-to-address transformation or a hashing function– Acts on a given key by returning its relative position in
an array
• Hash table– An array used with a hashing strategy
• Collision– Placement of different keys at the same array index
Fundamentals of Python: From First Programs Through Data Structures 21
Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 22
Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 23
The Relationship of Collisions to Density
• Density– The number of keys relative to the length of an array
• As the density decreases, so does the probability of collisions
• Keeping a low load factor even (say, below .2) seems like a good way to avoid collisions– Cost of memory incurred by load factors below .5 is
probably prohibitive for data sets of millions of items– Even load factors below .5 cannot prevent many
collisions from occurring for some data sets
Fundamentals of Python: From First Programs Through Data Structures 24
Hashing with Non-Numeric Keys
• Try returning the sum of the ASCII values in the string
• This method has effect of producing same keys for anagrams– Strings that contain same characters, but in different
order
• First letters of many words in English are unevenly distributed– This might have the effect of weighting or biasing the
sums generated
Fundamentals of Python: From First Programs Through Data Structures 25
Hashing with Non-Numeric Keys (continued)
• One solution:– If length of string is greater than a certain threshold
• Drop first character from string before computing sum
• Can also subtract the ASCII value of the last character
• Python also includes a standard hash function for use in hashing applications– Function can receive any Python object as an
argument and returns a unique integer
Fundamentals of Python: From First Programs Through Data Structures 26
Hashing with Non-Numeric Keys (continued)
Fundamentals of Python: From First Programs Through Data Structures 27
Linear Probing
• Linear probing– Simplest way to resolve a collision– Search array, starting from collision spot, for the first
available position
• At the start of an insertion, the hashing function is run to compute the home index of the item– If cell at home index is not available, move index to
the right to probe for an available cell– When search reaches last position of array, probing
wraps around to continue from the first position
Linear Probing (continued)
• For retrievals, stop probing process when current array cell is empty or it contains the target item– If target item is found, its cell is set to DELETED
Fundamentals of Python: From First Programs Through Data Structures 28
Fundamentals of Python: From First Programs Through Data Structures 29
Linear Probing (continued)
• Problem: After several insertions/removals, item is farther away from its home index than needs to be– Increasing the average overall access time
• Two ways to deal with this problem:– After a removal, shift items on the cell’s right over to
the cell’s left until an empty cell, a currently occupied cell, or the home indexes for each item are reached
– Regularly rehash the table (e.g., if load factor is .5)
• Clustering: Occurs when items causing a collision are relocated to the same region within the array
Fundamentals of Python: From First Programs Through Data Structures 30
Linear Probing (continued)
Quadratic Probing
• To avoid clustering associated with linear probing, we can advance the search for an empty position a considerable distance from the collision point– Quadratic probing: Increments the home index by
the square of a distance on each attempt
• Problem: By jumping over some cells, one or more of them might be missed– Can lead to some wasted space
Fundamentals of Python: From First Programs Through Data Structures 31
Quadratic Probing (continued)
• Here is the code for insertions, updated to use quadratic probing:
Fundamentals of Python: From First Programs Through Data Structures 32
Chaining
• Items are stored in an array of linked lists (chains) – Each item’s key locates the bucket (index) of the
chain in which the item resides or is to be inserted
• Retrieval and removal each perform these steps:– Compute the item’s home index in the array– Search the linked list at that index for the item
• To insert an item:– Compute the item’s home index in the array– If cell is empty, create a node with item and assign
the node to cell; else (collision), insert item in chain
Fundamentals of Python: From First Programs Through Data Structures 33
Fundamentals of Python: From First Programs Through Data Structures 34
Chaining (continued)
Complexity Analysis
• Linear probing: Complexity depends on load factor (D) and tendency of items to cluster– Worst case (method traverses entire array before
locating item’s position): behavior is linear– Average behavior in searching for an item that
cannot be found is (1/2) [1 + 1/(1 – D)2]
• Quadratic probing: Tends to mitigate clustering– Average search complexity is 1 – loge(1 – D) – (D /
2) for the successful case and 1 / (1 – D) – D – loge(1 – D) for the unsuccessful case
Fundamentals of Python: From First Programs Through Data Structures 35
Complexity Analysis (continued)
• Chaining:– Locating an item consists of two parts:
• Computing home index constant time behavior• Searching linked list upon a collision linear
– Worst case (all items that have collided with each other are in one chain, which is a linked list): O(n)
– If lists are evenly distributed in array and array is fairly large, the second part can be close to constant
– Best case (a chain of length 1 occupies each array cell): O(1)
Fundamentals of Python: From First Programs Through Data Structures 36
Case Study: Profiling Hashing Strategies
• Request:– Write a program that allows a programmer to profile
different hashing strategies
• Analysis:– Should allow to gather statistics on number of
collisions caused by the hashing strategies– Other useful information:
• Hash table’s load factor
• Number of probes needed to resolve collisions during linear or quadratic probing
Fundamentals of Python: From First Programs Through Data Structures 37
Fundamentals of Python: From First Programs Through Data Structures 38
Case Study: Profiling Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 39
Case Study: Profiling Hashing Strategies (continued)
Case Study: Profiling Hashing Strategies (continued)
• Analysis (continued):– Here are the profiler’s results:
• Design:– Profiler class requires instance variables to track
a table, number of collisions, and number of probesFundamentals of Python: From First Programs Through Data Structures 40
Case Study: Profiling Hashing Strategies (continued)
• Implementation:
Fundamentals of Python: From First Programs Through Data Structures 41
Fundamentals of Python: From First Programs Through Data Structures 42
Case Study: Profiling Hashing Strategies (continued)
Hashing Implementation of Dictionaries
• HashDict uses the bucket/chaining strategy– To manage the array, declare three instance
variables: _table, _size, and _capacity
Fundamentals of Python: From First Programs Through Data Structures 43
Hashing Implementation of Sets
• The design of the methods for HashSet is also the same as the methods in HashDict, except:– __contains__ searches for an item (not key)– add inserts item only if it is not already in the set– A single iterator method is included instead of
separate methods that return keys and values
Fundamentals of Python: From First Programs Through Data Structures 44
Sorted Sets and Dictionaries
• Each item added to a sorted set must be comparable with its other items– Same applies for keys added to a sorted dictionary
• The iterator for each type of collection guarantees its users access to items or keys in sorted order
• Implementation alternatives:– List-based: must maintain a sorted list of the items– Hashing implementation: not feasible– Binary search tree implementation: generally provide
logarithmic access to data items
Fundamentals of Python: From First Programs Through Data Structures 45
Sorted Sets and Dictionaries (continued)
Fundamentals of Python: From First Programs Through Data Structures 46
Summary
• A set is an unordered collection of items– Each item is unique– List-based implementation linear-time access– Hashing implementation constant-time access
• Items in a sorted set can be visited in sorted order– A tree-based implementation of a sorted set
supports logarithmic-time access
• A dictionary is an unordered collection of entries, where each entry consists of a key and a value– Each key is unique; its values may be duplicated
Fundamentals of Python: From First Programs Through Data Structures 47
Summary (continued)
• A sorted dictionary imposes an ordering by comparison on its keys
• Implementations of both types of dictionaries are similar to those of sets
• Hashing: Technique for locating an item in constant time– Techniques to resolve collisions: linear collision
processing, quadratic collision processing, chaining– The run-time and memory aspects involve the load
factor of the array
Fundamentals of Python: From First Programs Through Data Structures 48