Searching Algorithms (Edexcel IAL Maths)

Binary Search

Spec point — 1.2

What is the binary search algorithm?

• The binary search algorithm inspects an ordered list to determine if a specified item is in the list

  • If the item is in the list, the search algorithm will find and locate the item

How does the binary search algorithm work?

• The binary search algorithm works by finding a pivot that splits an ordered sub-list into two halves

  • In the first search, the sub-list is the complete original list

• The item being searched for is compared to the pivot

  • If the item is the same as the pivot then the item has been found
  • If the item comes before the pivot, the search continues in the sub-list before the pivot
  • If the item comes after the pivot, the search continues in the sub-list after the pivot

• The algorithm is complete when

  • the item has been located or
  • the item is identified as not being in the list

Which item is the pivot in the binary search algorithm?

• The pivot is the value in the middle of a sub-list of items

  • To find the position of the pivot

    • Add the positions of the first and last items in the sub-list
    • Divide the value by 2
    • Round up to the next integer value if necessary

Exam Hint

• If the items in the list are not numbered, it is a good idea to number them in ascending order as a first step

• Make a clear statement at the end of the algorithm to show that it is complete

  • If the item has been found: 'The search is complete, ..... has been found in the list'
  • If the item has not been found: '..... is not in the list'

Worked Example Below is a list of names.

1. Becky
2. Jason
3. Juan
4. Kia
5. Lorcan
6. Ryan
7. Samira
8. Selina

Use the binary search algorithm to find the following names in the list.

a) Jason

Answer:

Find the pivot

$\frac{( 1 + 8 )}{2} = 4.5 \rightarrow 5$th item

5. Lorcan

Compare 'Jason' to the pivot

Jason comes before Lorcan Reject items 5 to 10

Write down the reduced list

1. Becky 2. Jason 3. Juan 4. Kia

Find the pivot of the reduced list

$\frac{( 1 + 4 )}{2} = 2.5 \rightarrow 3$rd item

3. Juan

Compare 'Jason' to the pivot

Jason comes before Juan Reject items 3 to 4

Write down the reduced list

1. Becky 2. Jason

Find the pivot of the reduced list

$\frac{( 1 + 2 )}{2} = 1.5 \rightarrow 2$nd item

2. Jason

The name has been found so write a final statement

Final answer: The search is complete Jason has been found

b) Michelle

Answer:

Find the pivot

$\frac{( 1 + 8 )}{2} = 4.5 \rightarrow 5$th item

5. Lorcan

Compare 'Michelle' to the pivot

Michelle comes after Lorcan Reject items 1 to 5

Write down the reduced list

6. Ryan 7. Samira 8. Selina

Find the pivot of the reduced list

$\frac{( 6 + 8 )}{2} = 7$th item

7. Samira

Compare Michelle to the pivot

Michelle comes before Samira Reject items 7 to 8

Write down the reduced list

The list has been reduced to one name, which is not Michelle

6. Ryan

Reject item 6

Write a final statement

Final answer: Michelle is not in the list