Searching algorithms (linear, binary) in Intro to Computing - Full Explanation

Introduction

Finding a specific item in a list can be tricky if the list is long. Searching algorithms help us quickly locate what we want without checking every single item blindly.

Explanation

Linear Search

Linear search checks each item in the list one by one from the start until it finds the target or reaches the end. It works on any list, whether sorted or not, but can be slow for large lists because it might check every item.

Linear search looks through each item in order until it finds the target or finishes the list.

Binary Search

Binary search works only on sorted lists. It starts by looking at the middle item and compares it to the target. If the target is smaller, it repeats the search on the left half; if larger, on the right half. This splitting continues until the target is found or the search area is empty.

Binary search quickly narrows down the search by repeatedly dividing the sorted list in half.

Real World Analogy

Imagine looking for a word in a dictionary. You don't check every page from the start. Instead, you open near the middle, see if the word is before or after, then open halfway in that section, repeating until you find the word.

Linear Search → Flipping through every page of a book from start to finish to find a word.

Binary Search → Opening a dictionary near the middle and narrowing down the pages by halves to find a word quickly.

Diagram

Diagram

List: [2, 4, 6, 8, 10, 12, 14]

Linear Search:
Start → 2 → 4 → 6 → 8 → ... until target found

Binary Search:
Step 1: Check middle (8)
Step 2: Target < 8? Search left half [2,4,6]
Step 3: Check middle (4)
Step 4: Target > 4? Search right half [6]
Step 5: Check 6 → Found or not

This diagram shows how linear search checks items one by one, while binary search splits the list and checks the middle repeatedly.

Key Facts

Linear Search → A search method that checks each item in order until the target is found or the list ends.

Binary Search → A search method that repeatedly divides a sorted list in half to find the target quickly.

Sorted List → A list where items are arranged in order, such as numbers from smallest to largest.

Search Efficiency → Binary search is faster than linear search on large sorted lists because it skips half the items each step.

Common Confusions

Binary search can be used on any list.

Binary search can be used on any list. Binary search <strong>only works on sorted lists</strong>; using it on unsorted lists will not find the correct result.

Linear search is always slow.

Linear search is always slow. Linear search can be fast for small or unsorted lists, but it becomes slow as the list grows larger.

Summary

Linear search checks each item one by one and works on any list but can be slow for large lists.

Binary search works only on sorted lists and finds items faster by dividing the list in half repeatedly.

Choosing the right search method depends on whether the list is sorted and how large it is.

Practice

1. Which of the following is true about linear search?

easy

A. It checks each item one by one until it finds the target.

B. It requires the list to be sorted before searching.

C. It splits the list into halves to find the target quickly.

D. It only works on numbers, not text.

Solution

1. Step 1: Understand linear search method

   Linear search goes through each item in the list one by one to find the target.

2. Step 2: Compare with other search methods

   Binary search splits the list and requires sorting, but linear search does not.

3. Final Answer:

   It checks each item one by one until it finds the target. -> Option A

4. Quick Check:

   Linear search = check items one by one [OK]

Hint: Linear search checks items one by one [OK]

Common Mistakes:

• Thinking linear search needs sorted list
• Confusing linear search with binary search
• Believing linear search only works on numbers

2. Which of the following is the correct syntax for a linear search loop in Python to find target in arr?

easy

A. for i in arr:
if i == target
return True

B. for i in range(len(arr)):
if arr[i] = target:
return True

C. while i < len(arr):
if arr[i] == target
return True

D. for i in arr:
if i == target:
return True

Solution

1. Step 1: Check correct loop syntax

   for i in arr: if i == target: return True uses a for loop to iterate over each element in arr correctly.

2. Step 2: Verify condition and syntax

   for i in arr: if i == target: return True uses '==' for comparison and proper indentation, which is correct.

3. Final Answer:

   for i in arr: if i == target: return True -> Option D

4. Quick Check:

   Correct for loop and comparison syntax [OK]

Hint: Use '==' for comparison and proper indentation [OK]

Common Mistakes:

• Using single '=' instead of '==' for comparison
• Missing colon ':' after if statement
• Incorrect indentation causing syntax errors

3. What will be the output of the following Python code?

def binary_search(arr, target):
    low, high = 0, len(arr) - 1
    while low <= high:
        mid = (low + high) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            low = mid + 1
        else:
            high = mid - 1
    return -1

arr = [2, 4, 6, 8, 10]
print(binary_search(arr, 6))

medium

A. -1

B. 1

C. 2

D. 3

Solution

1. Step 1: Understand binary search on sorted list

   The list is sorted: [2, 4, 6, 8, 10]. Target is 6.

2. Step 2: Trace the binary search steps

   Initial low=0, high=4, mid=2. arr[2]=6 matches target, so return 2.

3. Final Answer:

   2 -> Option C

4. Quick Check:

   Index of 6 in list = 2 [OK]

Hint: Binary search returns index of target if found [OK]

Common Mistakes:

• Not using zero-based index
• Confusing mid calculation
• Assuming binary search works on unsorted lists

4. The following code is intended to perform a binary search but has an error. What is the error?

def binary_search(arr, target):
    low, high = 0, len(arr)
    while low <= high:
        mid = (low + high) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            low = mid + 1
        else:
            high = mid - 1
    return -1

medium

A. The mid calculation should use float division.

B. The high index should be len(arr) - 1, not len(arr).

C. The loop condition should be low < high, not low <= high.

D. The function should return True instead of index.

Solution

1. Step 1: Check initialization of high index

   High is set to len(arr), which is out of valid index range (0 to len(arr)-1).

2. Step 2: Understand index range in Python lists

   List indices go from 0 to len(arr)-1, so high must be len(arr)-1 to avoid index error.

3. Final Answer:

   The high index should be len(arr) - 1, not len(arr). -> Option B

4. Quick Check:

   High index = len(arr) - 1 [OK]

Hint: High index must be last valid index (len-1) [OK]

Common Mistakes:

• Setting high to len(arr) causes index out of range
• Using float division for mid index
• Wrong loop condition causing infinite loop

5. You have a sorted list of 1024 numbers. You want to find if the number 500 is in the list. Which search method is faster and why?

hard

A. Binary search, because it splits the list and reduces search steps quickly.

B. Binary search, but only if the list is unsorted.

C. Linear search, because it works only on sorted lists.

D. Linear search, because it checks each item one by one.

Solution

1. Step 1: Identify list size and sorting

   The list has 1024 numbers and is sorted, which suits binary search.

2. Step 2: Compare search methods speed

   Binary search halves the search space each step, so it finds the target in about 10 steps (log2(1024) = 10), much faster than linear search.

3. Final Answer:

   Binary search, because it splits the list and reduces search steps quickly. -> Option A

4. Quick Check:

   Binary search faster on sorted large lists [OK]

Hint: Use binary search on sorted big lists for speed [OK]

Common Mistakes:

• Choosing linear search for large sorted lists
• Thinking binary search works on unsorted lists
• Ignoring the list size impact on search speed