2.8. Analysis of Hash Tables¶
The second major C++ data structure to analyze is the hash table or hash map. As you may recall, hash tables differ from vectors in that you access items in a hash table by a key rather than a position. A hash table (hash map) is a data structure that maps keys to their associated values using a hash function. The underlying structure is a mostly empty array, but unlike the array or vector data structures, the hash table values are not stored contiguously. The hash function computes an index into the array of “buckets” from which the correct associated value can be located. Ideally, the hash table values are distributed uniformly in the underlying array and, again ideally, the hash function will assign each key to a unique bucket. However, most hash tables are designed to handle hash collisions when the hash function generates the same bucket index for more than one key. More about this when we discuss implementations in more depth.
Later in this book you will see that there are many ways to implement a hash table. The thing that is most important to note now is that the get item and set item operations on a hash table are \(O(1)\). Another important hash table operation is the contains operation. Unlike a vector, checking to see whether a value is in the hash table or not is also \(O(1)\).
The efficiency of all hash table operations is summarized in Table 4. One important side note on hash table performance is that the efficiencies we provide in the table below are for average performance. In some rare cases the contains, get item, and set item operations can degenerate into \(O(n)\) performance but we will get into that in a later chapter when we talk about the different ways that a hash table could be implemented.
operation 
BigO Efficiency 

find 
O(1) 
insert 
O(1) 
erase 
O(1) 
contains 
O(1) 
iteration 
O(n) 
Note that it is not typical to iterate through a hash table because it is a data structure designed for lookup, not for iteration. The big advantage of the hash table is the constant speed of the contains operation.
For our last performance experiment we will compare the performance of the contains operation between vectors and hash tables. In the process we will confirm that the contains operator for vectors is \(O(n)\) and the contains operator for hash tables is \(O(1)\). The experiment we will use to compare the two is simple. We’ll make a vector with a range of numbers in it. Then we will pick numbers at random and check to see if the numbers are in the vector. If our performance tables are correct the bigger the vector the longer it should take to determine if any one number is contained in the vector.
We will repeat the same experiment for a hash table that contains numbers as the keys. In this experiment we should see that determining whether or not a number is in the hash table is not only much faster, but the time it takes to check should remain constant even as the hash table grows larger.
Listing 6 implements this comparison. Notice that we are
performing exactly the same operation, number in container
. The
difference is that on line 7 x
is vector, and on line 9 x
is a
hash table.
Listing 6
#include <iostream>
#include <ctime>
#include <vector>
#include <unordered_map>
using namespace std;
int main() {
for(int a = 10000; a < 1000001; a = a + 20000) {
// vector Part
clock_t begin = clock();
vector<int> avector;
for( int i = 0; i < a; i++){
avector.push_back(i);
}
clock_t end = clock();
double elapsed_secs = double(end  begin) / CLOCKS_PER_SEC;
// Hash Table Part
clock_t begin_ht = clock();
unordered_map<int, int> y;
for( int j = 0; j < a; j++ ){
y[j] = NULL;
}
clock_t end_ht = clock();
double elapsed_secs_ht = double(end_ht  begin_ht) / CLOCKS_PER_SEC;
// Printing final output
cout << a << "\t" << elapsed_secs << "\t" << elapsed_secs_ht << endl;
}
return 0;
}
1import timeit
2import random
3
4for i in range(10000,1000001,20000):
5 t = timeit.Timer("random.randrange(%d) in x"%i,
6 "from __main__ import random,x")
7 x = list(range(i))
8 lst_time = t.timeit(number=1000)
9 x = {j:None for j in range(i)}
10 d_time = t.timeit(number=1000)
11 print("%d,%10.3f,%10.3f" % (i, lst_time, d_time))
Figure 3 summarizes the results of running Listing 6. You can see that the hash table is consistently faster. For the smallest vector size of 10,000 elements a hash table is 89.4 times faster than a vector. For the largest vector size of 990,000 elements the hash table is 11,603 times faster! You can also see that the time it takes for the contains operator on the vector grows linearly with the size of the vector. This verifies the assertion that the contains operator on a vector is \(O(n)\). It can also be seen that the time for the contains operator on a hash table is constant even as the hash table size grows. In fact for a hash table size of 10,000 the contains operation took 0.004 milliseconds and for the hash table size of 990,000 it also took 0.004 milliseconds.
Since C++ is an evolving language, there are always changes going on behind the scenes. The latest information on the performance of C++ data structures can be found on the C++ website.
Self Check
 Popping the first index from a vector.
 When you remove the first element of a vector, all the other elements of the vector must be shifted forward.
 Popping an element from the end of a vector.
 Removing an element from the end of the vector is a constant operation.
 Adding a new element to a vector.
 Adding to the end of a vector is a constant operation
 vector[10]
 Indexing a vector is a constant operation
 all of the above are O(1)
 There is one operation that requires all other vector elements to be moved.
Q1: Which of the vector operations shown below is not O(1)?
 mymap.count('x')
 count is a constant operation for a hash table because you do not have to iterate but there is a better answer.
 mymap.erase('x')
 removing an element from a hash table is a constant operation but there is a better answer.
 mymap['x'] = 10;
 Assignment to a hash table key is constant but there is a better answer.
 mymap['x'] = mymap['x'] + 1;
 Reassignment to a hash table key is constant but there is a better answer.
 all of the above are O(1)
 The only hash table operations that are not O(1) are those that require iteration.
Q2: Which of the hash table operations shown below is O(1)?
 erase
 The efficiency of erase in hash tables is constant.
 insert
 The efficiency of insert in hash tables is constant.
 iteration
 Correct!
 contains
 The efficiency of contains in hash tables is constant.
 find
 The efficiency of find in hash tables is constant.
Q3: What is an operator for hash tables with an efficiency other than O(1)?
2.9. Summary¶
Algorithm analysis is an implementationindependent way of measuring an algorithm.
BigO notation allows algorithms to be classified by their dominant process with respect to the size of the problem.
2.10. Self Check¶
 10000(n^{3} + n^{2})
 Incorrect, even though n^{3} is the most significant part for all of these formulas, the way it interacts with the rest of the equation is also important to note.
 45n^{3} + 1710n^{2} + 16n + 5
 Incorrect, even though n^{3} is the most significant part for all of these formulas, the way it interacts with the rest of the equation is also important to note.
 (n^{3} + n) (n^{2} + 1)
 Correct!
 A and B would be equally efficient/inefficient
 Look closer, the efficiencies would be different
Q4: Which of the following algorithms has the least efficient big O complexity?

Q5: Drag the order of growth rates to their rankings from lowest to highest (the slowest i.e. the highest growth rate should be #1)
Compare the functions at different values to see how they compare
 2^{n}
 1st
 n^{2}
 2nd
 nlogn
 3rd
 logn
 4th
 language constraints
 No, we do not consider the restraints of a language when thinking about how efficient an algorithm is.
 Space
 Yes, we consider how much space we need to solve a problem.
 Time
 Yes, we consider how much time it takes to solve a problem
 Energy
 No, we do not consider how much energy it takes at this point.
Q6: When considering computer resources, what factors do we have in mind? Select all that apply.
 the space it takes
 This can be dependent of the programming language
 the time it takes
 This can be dependent on the machine, programming language, and other factors
 the number of steps
 Yes, when quantifying the time it takes to execute an algorithm we base it on the number of steps it takes to solve the problem, not the time it takes
 the readability of the code
 No, a very efficient algorithm can be programmed efficiently in C++ without any extra spaces making it unreadable, however the solution would still be efficient.
Q7: When considering the Big O of an algorithm, what do we use to quantify our description of an algorithm.