Computer Science
Grade 9
20 min
Space Complexity
Space Complexity
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1
Introduction & Learning Objectives
Learning Objectives
Define space complexity in the context of algorithms.
Identify the memory usage of primitive data types and simple data structures.
Differentiate between constant space complexity O(1) and linear space complexity O(n).
Analyze simple functions to determine their space complexity.
Explain why managing space complexity is important for writing efficient programs.
Distinguish between input space and auxiliary space.
Ever wonder why some apps are tiny and fast, while others take up all your phone's storage and slow it down? 📱 It's all about how they use memory!
In this lesson, you'll learn about 'Space Complexity,' which is a fancy way of measuring how much memory an algorithm needs to run. Understanding this helps you write smarter...
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Key Concepts & Vocabulary
TermDefinitionExample
Space ComplexityA measure of the total amount of memory space an algorithm or program uses, relative to the size of its input.An algorithm that just adds two numbers uses the same small amount of memory no matter how big the numbers are. An algorithm that copies a list needs more memory for a longer list.
Big O Notation (for Space)A simplified way to describe how the memory usage of an algorithm grows as the input size grows. It focuses on the big picture, not the exact number of bytes.We use O(1) to say memory usage is constant, and O(n) to say it grows in a straight line with the input size 'n'.
Constant Space - O(1)The algorithm uses a fixed amount of memory, regardless of the input size. The memory usage does not grow.A function that swaps two numbers....
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Core Syntax & Patterns
Rule of Primitives
Primitive types (integers, booleans, characters) use a constant amount of space, O(1).
When you see a variable that holds a single number or true/false value, its memory footprint is fixed and doesn't depend on the input size.
Rule of Data Structures
The space for data structures (like arrays, lists, or strings) depends on the number of elements they hold. A list of size 'n' takes O(n) space.
Use this rule when your algorithm creates a new data structure. If the new structure's size depends on the input size 'n', it contributes O(n) to the space complexity.
Rule of Dominance
Add complexities for separate memory allocations, and keep the largest (most dominant) term. O(n) + O(1) becomes O(n).
When analyzing an algorithm...
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Challenging
A function modifies a list of `n` numbers *in-place* by doubling each number (e.g., `[1, 2, 3]` becomes `[2, 4, 6]`). It does not create a new list. What is the function's auxiliary space complexity?
A.O(1)
B.O(n)
C.O(log n)
D.O(0)
Challenging
An algorithm's auxiliary space usage is precisely `4n + 200` bytes, where `n` is the input size. How is this represented using Big O notation?
A.O(4n + 200)
B.O(4n)
C.O(n)
D.O(1)
Challenging
A recursive function calls itself `n` times before reaching its base case. Each function call adds a new 'frame' to the program's call stack to store its local variables. What is the space complexity caused by the call stack in this scenario?
A.O(1), because local variables are constant space.
B.O(log n), which is typical for all recursive functions.
C.O(n), because `n` function calls are stacked in memory at the deepest point.
D.The call stack does not contribute to space complexity.
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