Welcome to Lesson 14!

Learning Objectives
By the end of today's lessons, you should know
  • What is the Big-O runtime of the hash table operations (insert, delete, search)?
  • What is the load factor?
  • What are some guidelines for selecting a good hashing algorithm?

  • Quiz 5 after the break
  • Collect Lab 6
  • Return Lab 5
  • Lab 7 Assigned - due one week from today
  • Midterm postponed by one class day
    • New date: Tuesday, November 22
  • No online office hours tomorrow due to Veteran's Day Holiday
    • However, I will check my email a couple of times tomorrow to answer questions
  • Discuss Project Report 3 due in one week

Review Activity
With a partner, answer the following questions...
  • Trace BST Insert on the following values: 10, 12, 15, 4, 7, 13, 9, 11
  • What would be the output when calling pre, post and in order print on the above BSTs?
  • What is the Big-O runtime of the BST operations (search, insert, delete)? How do these compare to the Big-O runtime for a linked list?
  • How is data organized in a hash table?
  • What is a hashing algorithm?
  • What is a collision?
  • What are two strategies that used to handle collisions?
    • What is the difference between them?

Hash Tables Continued

Separate Chaining
  • Another approach to collision resolution is called separate chaining.
  • Rather than limiting ourselves to storing only a single value at each index of the table, with separate chaining we store a series of 0 or more linked elements at each index.
  • In essence, we make our hash table an array of linked lists, as depicted below:
  • Now, each time that a name is inserted, and there is a collision, we simply push the new name onto the back of our list at that particular index.
  • Therefore, we could re-envision our previous names hash table like so:
    names[0] = "";
    names[1] = "";
    names[2] = "";
    names[3] = "Kun";
    names[4] = "Doma" --> "Sean";
    names[5] = "Howon" --> "Qilin";
    names[6] = "Maryna";
    names[7] = "Anthony" --> "Brandon" --> "Cameron";
    names[34] = "";
  • Now when I want to search for the name "Cameron," if his name is in the table, it must be in the list at index 7.
  • To check, simply perform a linear search through the list at names[7].
  • Later in the lesson, we will look at how to implement a version of a hash table that uses separate chaining.

Big-O Run-Time of a Hash Table 
  • For hash table operations such as search, insert and delete we get the following Big-O run-times:
  • Best case: O(1)
    • Element is alone at collision free index.
  • Worst case: O(n)
    • Worst case is a linear search, which is O(n).
    • Holds for both linear probing and chaining.
  • Average case: O(1)?
  • Average case run-time of hash table operations is a question. Assuming a perfect hash function, these operations are always O(1). 
  • However, without a perfect hash function, average run-time will be dependent upon how big the table is and how many values it is currently storing (i.e. on how full the table is).
  • As more and more values are inserted into the table, the likelihood of collision increases resulting in longer chains.
  • The speed of hash table operations is therefore dependent on something called the load factor, which is defined to be n/k, where k is the the table size (number of buckets) and n is the number of keys.
load factor = n / k
  • The larger the value for the load factor, the slower the hash table operations.
  • Therefore, average case for a hash table is sometimes considered to be O(n/k). 
  • You might be thinking: But... O(n/k) = O(n), as we do not include lower order terms and coefficients in our Big-O run-time. 
    • So, is a hash table about as efficient as a linked list then?
    • In practice, O(n/k) is still a practical improvement over O(n).
    • Moreover, if your table size is large enough, average run-time approaches constant time:
      • If I set k = 2n (table size is twice the number of keys)
      • O(n/2n) -> O(1/2) -> O(1) 

Improving Our Hashing Algorithm
  • The above hashing algorithm, which mapped a key to an index according to the length of the key, was not a good hashing algorithm.
  • What was wrong with this algorithm?
    • Too many collisions - we were unable to get an even distribution of values across the table
      • Clustering of values around buckets 2-10
    • As a related issue, some indices were unusable
      • index 0 can never hold a value as there are no names of length 0
    • Both of these problems contribute to a loss of efficiency.
  • Perhaps we can find a better hashing algorithm...
  • Let's try the following:
    • Sum the ASCII values of each character in the key
    • Scale the result by applying % table size.
int hash(string key, const int TABLE_SIZE)
    int index, sum = 0;
    for(int i = 0; i < key.length(); i++)
        sum += (int) key[i]; //summing the ASCII values for each character in the string
    index = sum % TABLE_SIZE; //dividing the summed ASCII values by 35 && storing remainder as my index
    return index;

Guidelines for Choosing a Good Hash Function
  • But, how do we know that our new algorithm is a good one?
  • Here are some guidelines for writing good hashing functions:
    1. Use all of the information provided by a key.
      • Our algorithm uses all characters in the name to create the key.
    2. Make sure the elements to insert can be spread across the entire table.
      • By applying % operator we make sure our summed ASCII values will be better distributed throughout the table.
    3. Verify that your algorithm can map similar keys to different buckets.
      • Will values like "Mark" and "Matt" be placed in the same bucket?
      • Applying the hashing algorithm, with table size of 37
      • Mark: mapped to index 25
      • Matt: mapped to index 36
    4. Make sure it uses only simple operations (such as the basic arithmetic operators) to minimize run-time.
      • When you write a hash table, you will call your hashing function often (insert, search, delete).
      • Make certain your hashing function is not computationally intensive.
  • While the hashing function is an important aspect in helping to increase the effectiveness of a hash table, other aspects also need to be considered.

Guidelines for Selecting a Table Size

  • In order to effectively minimize collisions (thereby maintaining efficiency), a good rule-of-thumb is to make your table size be twice the number of entries you are planning to insert.
  • However, more often than not, the number of entries to be inserted will not be known in advance.
  • Therefore, it may be necessary to re-size your hash table:
    • Remember the load factor: n /k (number_of_keys  divided by table_size)
    • As the load factor increases, collisions will become more common, resulting in decreased efficiency.
    • How can we decrease the load factor? We increase the table size!
    • When should we re-size the table?
      • Linear probing: Array indices are used up quickly. Re-size when load factor >= 1/2
      • Separate Chaining: Opinions vary: between 2/3 and 3/4.
    • Simplest re-sizing method:
      • Create new hash table with an array sized twice as large as the current hash table.
      • Linearly traverse the old hash table.
        • Copy each element and insert into new hash table.
        • Elements will be rehashed when inserted into new hash table.
      • Delete old hash table.

Work on Lab 7

Wrap Up
  • With your partner, answer the questions from today's learning objectives

Upcoming Assignments
  • Lab 7 due Thursday
~Have a Great Weekend!~