1. Fundamentals of Python: From First Programs Through Data Structures
Chapter 19
Unordered Collections: Sets and Dictionaries

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. Using Sets 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
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
Fundamentals of Python: From First Programs Through Data Structures

14. Sets List implementation of a set
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
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
Fundamentals of Python: From First Programs Through Data Structures

28. 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

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)
Fundamentals of Python: From First Programs Through Data Structures

31. 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

32. Quadratic Probing (continued)
Here is the code for insertions, updated to use quadratic probing:
Fundamentals of Python: From First Programs Through Data Structures 32

33. 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:
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

34. Chaining (continued)
Fundamentals of Python: From First Programs Through Data Structures 34

35. 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

36. 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

37. 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

38. Case Study: Profiling Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 38

39. Case Study: Profiling Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 39

40. 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 probes
Fundamentals of Python: From First Programs Through Data Structures 40

41. Case Study: Profiling Hashing Strategies (continued)
Implementation:
Fundamentals of Python: From First Programs Through Data Structures 41

42. Case Study: Profiling Hashing Strategies (continued)
Fundamentals of Python: From First Programs Through Data Structures 42

43. 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

44. 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

45. 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

46. Sorted Sets and Dictionaries (continued)
Fundamentals of Python: From First Programs Through Data Structures 46

47. 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

48. 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