Time estimate: 90 min.

# 7.5. Searching Algorithms¶

Computers store vast amounts of data. One of the strengths of computers is their ability to find things quickly. This ability is called searching. For the AP CSA exam you will need to know both linear (sequential) search and binary search algorithms.

The following video is also on YouTube at https://youtu.be/DHLCXXX1OtE. It introduces the concept of searching including sequential search and binary search.

• Sequential or Linear search typically starts at the first element in an array or ArrayList and looks through all the items one by one until it either finds the desired value and then it returns the index it found the value at or if it searches the entire array or list without finding the value it returns -1.

• Binary search can only be used on data that has been sorted or stored in order. It checks the middle of the data to see if that middle value is less than, equal, or greater than the desired value and then based on the results of that it narrows the search. It cuts the search space in half each time.

If binary search requires the values in an array or list to be sorted, how can you do that? There are many sorting algorithms which are covered in the next lesson.

## 7.5.3. Runtimes¶

How do we choose between two algorithms that solve the same problem? They usually have different characteristics and runtimes which measures how fast they run. For the searching problem, it depends on your data.

Binary search is much faster than linear search, especially on large data sets, but it can only be used on sorted data. Often with runtimes, computer scientist think about the worst case behavior. With searching, the worst case is usually if you cannot find the item. With linear search, you would have to go through the whole array before realizing that it is not there, but binary search is much faster even in this case because it eliminates half the data set in each step. We can measure an informal runtime by just counting the number of steps.

Here is a table that compares the worst case runtime of each search algorithm given an array of n elements. The runtime here is measured as the number of times the loop runs in each algorithm or the number of elements we need to check in the worst case when we don’t find the item we are looking for. Notice that with linear search, the worst case runtime is the size of the array n, because it has to look through the whole array. For the binary search runtime, we can calculate the number of times you can divide n in half until you get to 1. So, for example 8 elements can be divided in half to narrow down to 4 elements, which can be further divided in half to narrow down to 2 elements, which can be further divided in half to get down to 1 element, and then if that is wrong, to 0 elements, so that is 4 divisions or guesses to get the answer (8->4->2->1->0). In the table below, every time we double the size of N, we need at most one more guess or comparison with binary search. It’s much faster than linear search!

N

Linear Search

Binary Search

2

2 comparisons

2 comparisons

4

4

3

8

8

4

16

16

5

100

100

7

Runtimes can be described with mathematical functions. For an array of size n, linear search runtime is a linear function, and binary search runtime is a function of log base 2 of n (or log n + 1 comparisons). This is called the big-O runtime function in computer science, for example O(log n) vs. O(n). You can compare the growth of functions like n and log2n as n, the data size, grows and see that binary search runs much faster for any n. You don’t need to know the log n runtime growth function for the AP exam, but you should be able to calculate how many steps binary search takes for a given n by counting how many times you can divide it in half. Or you can start at 1 and keep a count of how many times you can double it with the powers of two (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, etc.) until you reach a number that is slightly above n.

## 7.5.4. Programming Challenge : Search Runtimes¶

Let’s go back to the spellchecker that we created in Unit 6. Here is a version of the spellchecker below that reads the dictionary file into an ArrayList. The advantage of using an ArrayList instead of an array for the dictionary is that we do not need to know or declare the size of the dictionary in advance.

In Unit 6, we used linear search to find a word in the dictionary. However, the dictionary file is actually in alphabetical order. We could have used a much faster binary search algorithm! Let’s see how much faster we can make it.

Write a linear search method and a binary search method to search for a given word in the dictionary using the code in this lesson as a guide. You will need to use size and get(i) instead of [] to get an element in the ArrayList dictionary at index i. You will need to use the equals and compareTo methods to compare Strings. Have the methods return a count of how many words they had to check before finding the word or returning.

This spellchecker uses an ArrayList for the dictionary. Write a linearSearch(word) and a binarySearch(word) method. Use get(i), size(), equals, and compareTo. Return a count of the number of words checked.

Run your code with the following test cases and record the runtime for each word in this Google document (do File/Make a Copy) also seen below to record your answers.

What do you notice? Which one was faster in general? Were there some cases where each was faster? How fast were they with misspelled words? Record your answers in the window below.

## 7.5.5. Summary¶

• There are standard algorithms for searching.

• Sequential/linear search algorithms check each element in order until the desired value is found or all elements in the array or ArrayList have been checked.

• The binary search algorithm starts at the middle of a sorted array or ArrayList and eliminates half of the array or ArrayList in each iteration until the desired value is found or all elements have been eliminated.

• Data must be in sorted order to use the binary search algorithm. This algorithm will be covered more in Unit 10.

• Informal run-time comparisons of program code segments can be made using statement execution counts.