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# 2.4 Space Complexity

Space complexity measures the growth trend of memory space occupied by an algorithm as the data size increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".

## 2.4.1 Algorithm-Related Space

The memory space used by an algorithm during execution mainly includes the following types.

- Input space: Used to store the input data of the algorithm.
- Temporary space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.
- Output space: Used to store the output data of the algorithm.

In general, the scope of space complexity statistics is "temporary space" plus "output space".

Temporary space can be further divided into three parts.

- Temporary data: Used to save various constants, variables, objects, etc., during the algorithm's execution.
- Stack frame space: Used to save the context data of called functions. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.
- Instruction space: Used to save compiled program instructions, which are usually ignored in actual statistics.

When analyzing the space complexity of a program, we usually consider three parts: temporary data, stack frame space, and output data, as shown in the following figure.

Figure 2-15 Algorithm-related space

The related code is as follows:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby

    class Node:
        """Class"""
        def __init__(self, x: int):
            self.val: int = x              # Node value
            self.next: Node | None = None  # Reference to the next node

    def function() -> int:
        """Function"""
        # Perform some operations...
        return 0

    def algorithm(n) -> int:  # Input data
        A = 0                 # Temporary data (constant, usually represented by uppercase letters)
        b = 0                 # Temporary data (variable)
        node = Node(0)        # Temporary data (object)
        c = function()        # Stack frame space (function call)
        return A + b + c      # Output data

    /* Structure */
    struct Node {
        int val;
        Node *next;
        Node(int x) : val(x), next(nullptr) {}
    };

    /* Function */
    int func() {
        // Perform some operations...
        return 0;
    }

    int algorithm(int n) {        // Input data
        const int a = 0;          // Temporary data (constant)
        int b = 0;                // Temporary data (variable)
        Node* node = new Node(0); // Temporary data (object)
        int c = func();           // Stack frame space (function call)
        return a + b + c;         // Output data
    }

    /* Class */
    class Node {
        int val;
        Node next;
        Node(int x) { val = x; }
    }

    /* Function */
    int function() {
        // Perform some operations...
        return 0;
    }

    int algorithm(int n) {        // Input data
        final int a = 0;          // Temporary data (constant)
        int b = 0;                // Temporary data (variable)
        Node node = new Node(0);  // Temporary data (object)
        int c = function();       // Stack frame space (function call)
        return a + b + c;         // Output data
    }

    /* Class */
    class Node(int x) {
        int val = x;
        Node next;
    }

    /* Function */
    int Function() {
        // Perform some operations...
        return 0;
    }

    int Algorithm(int n) {        // Input data
        const int a = 0;          // Temporary data (constant)
        int b = 0;                // Temporary data (variable)
        Node node = new(0);       // Temporary data (object)
        int c = Function();       // Stack frame space (function call)
        return a + b + c;         // Output data
    }

    /* Structure */
    type node struct {
        val  int
        next *node
    }

    /* Create node structure */
    func newNode(val int) *node {
        return &node{val: val}
    }

    /* Function */
    func function() int {
        // Perform some operations...
        return 0
    }

    func algorithm(n int) int { // Input data
        const a = 0             // Temporary data (constant)
        b := 0                  // Temporary data (variable)
        newNode(0)              // Temporary data (object)
        c := function()         // Stack frame space (function call)
        return a + b + c        // Output data
    }

    /* Class */
    class Node {
        var val: Int
        var next: Node?

        init(x: Int) {
            val = x
        }
    }

    /* Function */
    func function() -> Int {
        // Perform some operations...
        return 0
    }

    func algorithm(n: Int) -> Int { // Input data
        let a = 0             // Temporary data (constant)
        var b = 0             // Temporary data (variable)
        let node = Node(x: 0) // Temporary data (object)
        let c = function()    // Stack frame space (function call)
        return a + b + c      // Output data
    }

    /* Class */
    class Node {
        val;
        next;
        constructor(val) {
            this.val = val === undefined ? 0 : val; // Node value
            this.next = null;                       // Reference to the next node
        }
    }

    /* Function */
    function constFunc() {
        // Perform some operations
        return 0;
    }

    function algorithm(n) {       // Input data
        const a = 0;              // Temporary data (constant)
        let b = 0;                // Temporary data (variable)
        const node = new Node(0); // Temporary data (object)
        const c = constFunc();    // Stack frame space (function call)
        return a + b + c;         // Output data
    }

    /* Class */
    class Node {
        val: number;
        next: Node | null;
        constructor(val?: number) {
            this.val = val === undefined ? 0 : val; // Node value
            this.next = null;                       // Reference to the next node
        }
    }

    /* Function */
    function constFunc(): number {
        // Perform some operations
        return 0;
    }

    function algorithm(n: number): number { // Input data
        const a = 0;                        // Temporary data (constant)
        let b = 0;                          // Temporary data (variable)
        const node = new Node(0);           // Temporary data (object)
        const c = constFunc();              // Stack frame space (function call)
        return a + b + c;                   // Output data
    }

    /* Class */
    class Node {
      int val;
      Node next;
      Node(this.val, [this.next]);
    }

    /* Function */
    int function() {
      // Perform some operations...
      return 0;
    }

    int algorithm(int n) {  // Input data
      const int a = 0;      // Temporary data (constant)
      int b = 0;            // Temporary data (variable)
      Node node = Node(0);  // Temporary data (object)
      int c = function();   // Stack frame space (function call)
      return a + b + c;     // Output data
    }

    use std::rc::Rc;
    use std::cell::RefCell;

    /* Structure */
    struct Node {
        val: i32,
        next: Option<Rc<RefCell<Node>>>,
    }

    /* Create Node structure */
    impl Node {
        fn new(val: i32) -> Self {
            Self { val: val, next: None }
        }
    }

    /* Function */
    fn function() -> i32 {
        // Perform some operations...
        return 0;
    }

    fn algorithm(n: i32) -> i32 {       // Input data
        const a: i32 = 0;               // Temporary data (constant)
        let mut b = 0;                  // Temporary data (variable)
        let node = Node::new(0);        // Temporary data (object)
        let c = function();             // Stack frame space (function call)
        return a + b + c;               // Output data
    }

    /* Function */
    int func() {
        // Perform some operations...
        return 0;
    }

    int algorithm(int n) { // Input data
        const int a = 0;   // Temporary data (constant)
        int b = 0;         // Temporary data (variable)
        int c = func();    // Stack frame space (function call)
        return a + b + c;  // Output data
    }

    /* Class */
    class Node(var _val: Int) {
        var next: Node? = null
    }

    /* Function */
    fun function(): Int {
        // Perform some operations...
        return 0
    }

    fun algorithm(n: Int): Int { // Input data
        val a = 0                // Temporary data (constant)
        var b = 0                // Temporary data (variable)
        val node = Node(0)       // Temporary data (object)
        val c = function()       // Stack frame space (function call)
        return a + b + c         // Output data
    }

    ### Class ###
    class Node
        attr_accessor :val      # Node value
        attr_accessor :next     # Reference to the next node

        def initialize(x)
            @val = x
        end
    end

    ### Function ###
    def function
        # Perform some operations...
        0
    end

    ### Algorithm ###
    def algorithm(n)        # Input data
        a = 0               # Temporary data (constant)
        b = 0               # Temporary data (variable)
        node = Node.new(0)  # Temporary data (object)
        c = function        # Stack frame space (function call)
        a + b + c           # Output data
    end

## 2.4.2 Calculation Method

The calculation method for space complexity is roughly the same as for time complexity, except that what we measure changes from the "number of operations" to the "amount of space used".

Unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that sufficient memory space is reserved for all input data.

Observe the following code. Here, "worst case" in worst-case space complexity has two meanings.

1. Based on the worst input data: When \(n < 10\), the space complexity is \(O(1)\); but when \(n > 10\), the initialized array `nums` occupies \(O(n)\) space, so the worst-case space complexity is \(O(n)\).
2. Based on the peak memory during algorithm execution: For example, before executing the last line, the program occupies \(O(1)\) space; when initializing the array `nums`, the program occupies \(O(n)\) space, so the worst-case space complexity is \(O(n)\).

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby

    def algorithm(n: int):
        a = 0               # O(1)
        b = [0] * 10000     # O(1)
        if n > 10:
            nums = [0] * n  # O(n)

    void algorithm(int n) {
        int a = 0;               // O(1)
        vector<int> b(10000);    // O(1)
        if (n > 10)
            vector<int> nums(n); // O(n)
    }

    void algorithm(int n) {
        int a = 0;                   // O(1)
        int[] b = new int[10000];    // O(1)
        if (n > 10)
            int[] nums = new int[n]; // O(n)
    }

    void Algorithm(int n) {
        int a = 0;                   // O(1)
        int[] b = new int[10000];    // O(1)
        if (n > 10) {
            int[] nums = new int[n]; // O(n)
        }
    }

    func algorithm(n int) {
        a := 0                      // O(1)
        b := make([]int, 10000)     // O(1)
        var nums []int
        if n > 10 {
            nums := make([]int, n)  // O(n)
        }
        fmt.Println(a, b, nums)
    }

    func algorithm(n: Int) {
        let a = 0 // O(1)
        let b = Array(repeating: 0, count: 10000) // O(1)
        if n > 10 {
            let nums = Array(repeating: 0, count: n) // O(n)
        }
    }

    function algorithm(n) {
        const a = 0;                   // O(1)
        const b = new Array(10000);    // O(1)
        if (n > 10) {
            const nums = new Array(n); // O(n)
        }
    }

    function algorithm(n: number): void {
        const a = 0;                   // O(1)
        const b = new Array(10000);    // O(1)
        if (n > 10) {
            const nums = new Array(n); // O(n)
        }
    }

    void algorithm(int n) {
      int a = 0;                            // O(1)
      List<int> b = List.filled(10000, 0);  // O(1)
      if (n > 10) {
        List<int> nums = List.filled(n, 0); // O(n)
      }
    }

    fn algorithm(n: i32) {
        let a = 0;                              // O(1)
        let b = [0; 10000];                     // O(1)
        if n > 10 {
            let nums = vec![0; n as usize];     // O(n)
        }
    }

    void algorithm(int n) {
        int a = 0;               // O(1)
        int b[10000];            // O(1)
        if (n > 10)
            int nums[n] = {0};   // O(n)
    }

    fun algorithm(n: Int) {
        val a = 0                    // O(1)
        val b = IntArray(10000)      // O(1)
        if (n > 10) {
            val nums = IntArray(n)   // O(n)
        }
    }

    def algorithm(n)
        a = 0                           # O(1)
        b = Array.new(10000)            # O(1)
        nums = Array.new(n) if n > 10   # O(n)
    end

In recursive functions, it is necessary to count the stack frame space. Observe the following code:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby

    def function() -> int:
        # Perform some operations
        return 0

    def loop(n: int):
        """Loop has space complexity of O(1)"""
        for _ in range(n):
            function()

    def recur(n: int):
        """Recursion has space complexity of O(n)"""
        if n == 1:
            return
        return recur(n - 1)

    int func() {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    void loop(int n) {
        for (int i = 0; i < n; i++) {
            func();
        }
    }
    /* Recursion has space complexity of O(n) */
    void recur(int n) {
        if (n == 1) return;
        recur(n - 1);
    }

    int function() {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    void loop(int n) {
        for (int i = 0; i < n; i++) {
            function();
        }
    }
    /* Recursion has space complexity of O(n) */
    void recur(int n) {
        if (n == 1) return;
        recur(n - 1);
    }

    int Function() {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    void Loop(int n) {
        for (int i = 0; i < n; i++) {
            Function();
        }
    }
    /* Recursion has space complexity of O(n) */
    int Recur(int n) {
        if (n == 1) return 1;
        return Recur(n - 1);
    }

    func function() int {
        // Perform some operations
        return 0
    }

    /* Loop has space complexity of O(1) */
    func loop(n int) {
        for i := 0; i < n; i++ {
            function()
        }
    }

    /* Recursion has space complexity of O(n) */
    func recur(n int) {
        if n == 1 {
            return
        }
        recur(n - 1)
    }

    @discardableResult
    func function() -> Int {
        // Perform some operations
        return 0
    }

    /* Loop has space complexity of O(1) */
    func loop(n: Int) {
        for _ in 0 ..< n {
            function()
        }
    }

    /* Recursion has space complexity of O(n) */
    func recur(n: Int) {
        if n == 1 {
            return
        }
        recur(n: n - 1)
    }

    function constFunc() {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    function loop(n) {
        for (let i = 0; i < n; i++) {
            constFunc();
        }
    }
    /* Recursion has space complexity of O(n) */
    function recur(n) {
        if (n === 1) return;
        return recur(n - 1);
    }

    function constFunc(): number {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    function loop(n: number): void {
        for (let i = 0; i < n; i++) {
            constFunc();
        }
    }
    /* Recursion has space complexity of O(n) */
    function recur(n: number): void {
        if (n === 1) return;
        return recur(n - 1);
    }

    int function() {
      // Perform some operations
      return 0;
    }
    /* Loop has space complexity of O(1) */
    void loop(int n) {
      for (int i = 0; i < n; i++) {
        function();
      }
    }
    /* Recursion has space complexity of O(n) */
    void recur(int n) {
      if (n == 1) return;
      recur(n - 1);
    }

    fn function() -> i32 {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    fn loop(n: i32) {
        for i in 0..n {
            function();
        }
    }
    /* Recursion has space complexity of O(n) */
    fn recur(n: i32) {
        if n == 1 {
            return;
        }
        recur(n - 1);
    }

    int func() {
        // Perform some operations
        return 0;
    }
    /* Loop has space complexity of O(1) */
    void loop(int n) {
        for (int i = 0; i < n; i++) {
            func();
        }
    }
    /* Recursion has space complexity of O(n) */
    void recur(int n) {
        if (n == 1) return;
        recur(n - 1);
    }

    fun function(): Int {
        // Perform some operations
        return 0
    }
    /* Loop has space complexity of O(1) */
    fun loop(n: Int) {
        for (i in 0..<n) {
            function()
        }
    }
    /* Recursion has space complexity of O(n) */
    fun recur(n: Int) {
        if (n == 1) return
        return recur(n - 1)
    }

    def function
        # Perform some operations
        0
    end

    ### Loop has space complexity of O(1) ###
    def loop(n)
        (0...n).each { function }
    end

    ### Recursion has space complexity of O(n) ###
    def recur(n)
        return if n == 1
        recur(n - 1)
    end

The time complexity of both functions `loop()` and `recur()` is \(O(n)\), but their space complexities are different.

- The function `loop()` calls `function()` \(n\) times in a loop. In each iteration, `function()` returns and releases its stack frame space, so the space complexity remains \(O(1)\).
- The recursive function `recur()` has \(n\) unreturned `recur()` instances existing simultaneously during execution, thus occupying \(O(n)\) stack frame space.

## 2.4.3 Common Types

Let the input data size be \(n\). The following figure shows common types of space complexity (arranged from low to high).

\[ \begin{aligned} & O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline & \text{Constant} < \text{Logarithmic} < \text{Linear} < \text{Quadratic} < \text{Exponential} \end{aligned} \]

Figure 2-16 Common types of space complexity

### 1. Constant Order \(O(1)\)

Constant order is common for constants, variables, and objects whose number is independent of the input data size \(n\).

It should be noted that memory occupied by initializing variables or calling functions in a loop is released when entering the next iteration, so it does not accumulate space, and the space complexity remains \(O(1)\):

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def function() -> int:
        """Function"""
        # Perform some operations
        return 0

    def constant(n: int):
        """Constant order"""
        # Constants, variables, objects occupy O(1) space
        a = 0
        nums = [0] * 10000
        node = ListNode(0)
        # Variables in the loop occupy O(1) space
        for _ in range(n):
            c = 0
        # Functions in the loop occupy O(1) space
        for _ in range(n):
            function()

space_complexity.cpp

    /* Function */
    int func() {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    void constant(int n) {
        // Constants, variables, objects occupy O(1) space
        const int a = 0;
        int b = 0;
        vector<int> nums(10000);
        ListNode node(0);
        // Variables in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            int c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            func();
        }
    }

space_complexity.java

    /* Function */
    int function() {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    void constant(int n) {
        // Constants, variables, objects occupy O(1) space
        final int a = 0;
        int b = 0;
        int[] nums = new int[10000];
        ListNode node = new ListNode(0);
        // Variables in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            int c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            function();
        }
    }

space_complexity.cs

    /* Function */
    int Function() {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    void Constant(int n) {
        // Constants, variables, objects occupy O(1) space
        int a = 0;
        int b = 0;
        int[] nums = new int[10000];
        ListNode node = new(0);
        // Variables in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            int c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            Function();
        }
    }

space_complexity.go

    /* Function */
    func function() int {
        // Perform some operations...
        return 0
    }

    /* Constant order */
    func spaceConstant(n int) {
        // Constants, variables, objects occupy O(1) space
        const a = 0
        b := 0
        nums := make([]int, 10000)
        node := newNode(0)
        // Variables in the loop occupy O(1) space
        var c int
        for i := 0; i < n; i++ {
            c = 0
        }
        // Functions in the loop occupy O(1) space
        for i := 0; i < n; i++ {
            function()
        }
        b += 0
        c += 0
        nums[0] = 0
        node.val = 0
    }

space_complexity.swift

    /* Function */
    @discardableResult
    func function() -> Int {
        // Perform some operations
        return 0
    }

    /* Constant order */
    func constant(n: Int) {
        // Constants, variables, objects occupy O(1) space
        let a = 0
        var b = 0
        let nums = Array(repeating: 0, count: 10000)
        let node = ListNode(x: 0)
        // Variables in the loop occupy O(1) space
        for _ in 0 ..< n {
            let c = 0
        }
        // Functions in the loop occupy O(1) space
        for _ in 0 ..< n {
            function()
        }
    }

space_complexity.js

    /* Function */
    function constFunc() {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    function constant(n) {
        // Constants, variables, objects occupy O(1) space
        const a = 0;
        const b = 0;
        const nums = new Array(10000);
        const node = new ListNode(0);
        // Variables in the loop occupy O(1) space
        for (let i = 0; i < n; i++) {
            const c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (let i = 0; i < n; i++) {
            constFunc();
        }
    }

space_complexity.ts

    /* Function */
    function constFunc(): number {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    function constant(n: number): void {
        // Constants, variables, objects occupy O(1) space
        const a = 0;
        const b = 0;
        const nums = new Array(10000);
        const node = new ListNode(0);
        // Variables in the loop occupy O(1) space
        for (let i = 0; i < n; i++) {
            const c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (let i = 0; i < n; i++) {
            constFunc();
        }
    }

space_complexity.dart

    /* Function */
    int function() {
      // Perform some operations
      return 0;
    }

    /* Constant order */
    void constant(int n) {
      // Constants, variables, objects occupy O(1) space
      final int a = 0;
      int b = 0;
      List<int> nums = List.filled(10000, 0);
      ListNode node = ListNode(0);
      // Variables in the loop occupy O(1) space
      for (var i = 0; i < n; i++) {
        int c = 0;
      }
      // Functions in the loop occupy O(1) space
      for (var i = 0; i < n; i++) {
        function();
      }
    }

space_complexity.rs

    /* Function */
    fn function() -> i32 {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    #[allow(unused)]
    fn constant(n: i32) {
        // Constants, variables, objects occupy O(1) space
        const A: i32 = 0;
        let b = 0;
        let nums = vec![0; 10000];
        let node = ListNode::new(0);
        // Variables in the loop occupy O(1) space
        for i in 0..n {
            let c = 0;
        }
        // Functions in the loop occupy O(1) space
        for i in 0..n {
            function();
        }
    }

space_complexity.c

    /* Function */
    int func() {
        // Perform some operations
        return 0;
    }

    /* Constant order */
    void constant(int n) {
        // Constants, variables, objects occupy O(1) space
        const int a = 0;
        int b = 0;
        int nums[1000];
        ListNode *node = newListNode(0);
        free(node);
        // Variables in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            int c = 0;
        }
        // Functions in the loop occupy O(1) space
        for (int i = 0; i < n; i++) {
            func();
        }
    }

space_complexity.kt

    /* Function */
    fun function(): Int {
        // Perform some operations
        return 0
    }

    /* Constant order */
    fun constant(n: Int) {
        // Constants, variables, objects occupy O(1) space
        val a = 0
        var b = 0
        val nums = Array(10000) { 0 }
        val node = ListNode(0)
        // Variables in the loop occupy O(1) space
        for (i in 0..<n) {
            val c = 0
        }
        // Functions in the loop occupy O(1) space
        for (i in 0..<n) {
            function()
        }
    }

space_complexity.rb

    ### Function ###
    def function
      # Perform some operations
      0
    end

    ### Constant time ###
    def constant(n)
      # Constants, variables, objects occupy O(1) space
      a = 0
      nums = [0] * 10000
      node = ListNode.new

      # Variables in the loop occupy O(1) space
      (0...n).each { c = 0 }
      # Functions in the loop occupy O(1) space
      (0...n).each { function }
    end

### 2. Linear Order \(O(n)\)

Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \(n\):

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def linear(n: int):
        """Linear order"""
        # A list of length n occupies O(n) space
        nums = [0] * n
        # A hash table of length n occupies O(n) space
        hmap = dict[int, str]()
        for i in range(n):
            hmap[i] = str(i)

space_complexity.cpp

    /* Linear order */
    void linear(int n) {
        // Array of length n uses O(n) space
        vector<int> nums(n);
        // A list of length n occupies O(n) space
        vector<ListNode> nodes;
        for (int i = 0; i < n; i++) {
            nodes.push_back(ListNode(i));
        }
        // A hash table of length n occupies O(n) space
        unordered_map<int, string> map;
        for (int i = 0; i < n; i++) {
            map[i] = to_string(i);
        }
    }

space_complexity.java

    /* Linear order */
    void linear(int n) {
        // Array of length n uses O(n) space
        int[] nums = new int[n];
        // A list of length n occupies O(n) space
        List<ListNode> nodes = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            nodes.add(new ListNode(i));
        }
        // A hash table of length n occupies O(n) space
        Map<Integer, String> map = new HashMap<>();
        for (int i = 0; i < n; i++) {
            map.put(i, String.valueOf(i));
        }
    }

space_complexity.cs

    /* Linear order */
    void Linear(int n) {
        // Array of length n uses O(n) space
        int[] nums = new int[n];
        // A list of length n occupies O(n) space
        List<ListNode> nodes = [];
        for (int i = 0; i < n; i++) {
            nodes.Add(new ListNode(i));
        }
        // A hash table of length n occupies O(n) space
        Dictionary<int, string> map = [];
        for (int i = 0; i < n; i++) {
            map.Add(i, i.ToString());
        }
    }

space_complexity.go

    /* Linear order */
    func spaceLinear(n int) {
        // Array of length n uses O(n) space
        _ = make([]int, n)
        // A list of length n occupies O(n) space
        var nodes []*node
        for i := 0; i < n; i++ {
            nodes = append(nodes, newNode(i))
        }
        // A hash table of length n occupies O(n) space
        m := make(map[int]string, n)
        for i := 0; i < n; i++ {
            m[i] = strconv.Itoa(i)
        }
    }

space_complexity.swift

    /* Linear order */
    func linear(n: Int) {
        // Array of length n uses O(n) space
        let nums = Array(repeating: 0, count: n)
        // A list of length n occupies O(n) space
        let nodes = (0 ..< n).map { ListNode(x: $0) }
        // A hash table of length n occupies O(n) space
        let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, "\($0)") })
    }

space_complexity.js

    /* Linear order */
    function linear(n) {
        // Array of length n uses O(n) space
        const nums = new Array(n);
        // A list of length n occupies O(n) space
        const nodes = [];
        for (let i = 0; i < n; i++) {
            nodes.push(new ListNode(i));
        }
        // A hash table of length n occupies O(n) space
        const map = new Map();
        for (let i = 0; i < n; i++) {
            map.set(i, i.toString());
        }
    }

space_complexity.ts

    /* Linear order */
    function linear(n: number): void {
        // Array of length n uses O(n) space
        const nums = new Array(n);
        // A list of length n occupies O(n) space
        const nodes: ListNode[] = [];
        for (let i = 0; i < n; i++) {
            nodes.push(new ListNode(i));
        }
        // A hash table of length n occupies O(n) space
        const map = new Map();
        for (let i = 0; i < n; i++) {
            map.set(i, i.toString());
        }
    }

space_complexity.dart

    /* Linear order */
    void linear(int n) {
      // Array of length n uses O(n) space
      List<int> nums = List.filled(n, 0);
      // A list of length n occupies O(n) space
      List<ListNode> nodes = [];
      for (var i = 0; i < n; i++) {
        nodes.add(ListNode(i));
      }
      // A hash table of length n occupies O(n) space
      Map<int, String> map = HashMap();
      for (var i = 0; i < n; i++) {
        map.putIfAbsent(i, () => i.toString());
      }
    }

space_complexity.rs

    /* Linear order */
    #[allow(unused)]
    fn linear(n: i32) {
        // Array of length n uses O(n) space
        let mut nums = vec![0; n as usize];
        // A list of length n occupies O(n) space
        let mut nodes = Vec::new();
        for i in 0..n {
            nodes.push(ListNode::new(i))
        }
        // A hash table of length n occupies O(n) space
        let mut map = HashMap::new();
        for i in 0..n {
            map.insert(i, i.to_string());
        }
    }

space_complexity.c

    /* Hash table */
    typedef struct {
        int key;
        int val;
        UT_hash_handle hh; // Implemented using uthash.h
    } HashTable;

    /* Linear order */
    void linear(int n) {
        // Array of length n uses O(n) space
        int *nums = malloc(sizeof(int) * n);
        free(nums);

        // A list of length n occupies O(n) space
        ListNode **nodes = malloc(sizeof(ListNode *) * n);
        for (int i = 0; i < n; i++) {
            nodes[i] = newListNode(i);
        }
        // Memory release
        for (int i = 0; i < n; i++) {
            free(nodes[i]);
        }
        free(nodes);

        // A hash table of length n occupies O(n) space
        HashTable *h = NULL;
        for (int i = 0; i < n; i++) {
            HashTable *tmp = malloc(sizeof(HashTable));
            tmp->key = i;
            tmp->val = i;
            HASH_ADD_INT(h, key, tmp);
        }

        // Memory release
        HashTable *curr, *tmp;
        HASH_ITER(hh, h, curr, tmp) {
            HASH_DEL(h, curr);
            free(curr);
        }
    }

space_complexity.kt

    /* Linear order */
    fun linear(n: Int) {
        // Array of length n uses O(n) space
        val nums = Array(n) { 0 }
        // A list of length n occupies O(n) space
        val nodes = mutableListOf<ListNode>()
        for (i in 0..<n) {
            nodes.add(ListNode(i))
        }
        // A hash table of length n occupies O(n) space
        val map = mutableMapOf<Int, String>()
        for (i in 0..<n) {
            map[i] = i.toString()
        }
    }

space_complexity.rb

    ### Linear time ###
    def linear(n)
      # A list of length n occupies O(n) space
      nums = Array.new(n, 0)

      # A hash table of length n occupies O(n) space
      hmap = {}
      for i in 0...n
        hmap[i] = i.to_s
      end
    end

As shown in the following figure, the recursion depth of this function is \(n\), meaning that there are \(n\) unreturned `linear_recur()` functions existing simultaneously, using \(O(n)\) stack frame space:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def linear_recur(n: int):
        """Linear order (recursive implementation)"""
        print("Recursion n =", n)
        if n == 1:
            return
        linear_recur(n - 1)

space_complexity.cpp

    /* Linear order (recursive implementation) */
    void linearRecur(int n) {
        cout << "Recursion n = " << n << endl;
        if (n == 1)
            return;
        linearRecur(n - 1);
    }

space_complexity.java

    /* Linear order (recursive implementation) */
    void linearRecur(int n) {
        System.out.println("Recursion n = " + n);
        if (n == 1)
            return;
        linearRecur(n - 1);
    }

space_complexity.cs

    /* Linear order (recursive implementation) */
    void LinearRecur(int n) {
        Console.WriteLine("Recursion n = " + n);
        if (n == 1) return;
        LinearRecur(n - 1);
    }

space_complexity.go

    /* Linear order (recursive implementation) */
    func spaceLinearRecur(n int) {
        fmt.Println("Recursion n =", n)
        if n == 1 {
            return
        }
        spaceLinearRecur(n - 1)
    }

space_complexity.swift

    /* Linear order (recursive implementation) */
    func linearRecur(n: Int) {
        print("Recursion n = \(n)")
        if n == 1 {
            return
        }
        linearRecur(n: n - 1)
    }

space_complexity.js

    /* Linear order (recursive implementation) */
    function linearRecur(n) {
        console.log(`Recursion n = ${n}`);
        if (n === 1) return;
        linearRecur(n - 1);
    }

space_complexity.ts

    /* Linear order (recursive implementation) */
    function linearRecur(n: number): void {
        console.log(`Recursion n = ${n}`);
        if (n === 1) return;
        linearRecur(n - 1);
    }

space_complexity.dart

    /* Linear order (recursive implementation) */
    void linearRecur(int n) {
      print('Recursion n = $n');
      if (n == 1) return;
      linearRecur(n - 1);
    }

space_complexity.rs

    /* Linear order (recursive implementation) */
    fn linear_recur(n: i32) {
        println!("Recursion n = {}", n);
        if n == 1 {
            return;
        };
        linear_recur(n - 1);
    }

space_complexity.c

    /* Linear order (recursive implementation) */
    void linearRecur(int n) {
        printf("Recursion n = %d\r\n", n);
        if (n == 1)
            return;
        linearRecur(n - 1);
    }

space_complexity.kt

    /* Linear order (recursive implementation) */
    fun linearRecur(n: Int) {
        println("Recursion n = $n")
        if (n == 1)
            return
        linearRecur(n - 1)
    }

space_complexity.rb

    ### Linear space (recursive) ###
    def linear_recur(n)
      puts "Recursion n = #{n}"
      return if n == 1
      linear_recur(n - 1)
    end

Figure 2-17 Linear order space complexity generated by recursive function

### 3. Quadratic Order \(O(n^2)\)

Quadratic order is common in matrices and graphs, where the number of elements is quadratically related to \(n\):

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def quadratic(n: int):
        """Quadratic order"""
        # A 2D list occupies O(n^2) space
        num_matrix = [[0] * n for _ in range(n)]

space_complexity.cpp

    /* Exponential order */
    void quadratic(int n) {
        // 2D list uses O(n^2) space
        vector<vector<int>> numMatrix;
        for (int i = 0; i < n; i++) {
            vector<int> tmp;
            for (int j = 0; j < n; j++) {
                tmp.push_back(0);
            }
            numMatrix.push_back(tmp);
        }
    }

space_complexity.java

    /* Exponential order */
    void quadratic(int n) {
        // Matrix uses O(n^2) space
        int[][] numMatrix = new int[n][n];
        // 2D list uses O(n^2) space
        List<List<Integer>> numList = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            List<Integer> tmp = new ArrayList<>();
            for (int j = 0; j < n; j++) {
                tmp.add(0);
            }
            numList.add(tmp);
        }
    }

space_complexity.cs

    /* Exponential order */
    void Quadratic(int n) {
        // Matrix uses O(n^2) space
        int[,] numMatrix = new int[n, n];
        // 2D list uses O(n^2) space
        List<List<int>> numList = [];
        for (int i = 0; i < n; i++) {
            List<int> tmp = [];
            for (int j = 0; j < n; j++) {
                tmp.Add(0);
            }
            numList.Add(tmp);
        }
    }

space_complexity.go

    /* Exponential order */
    func spaceQuadratic(n int) {
        // Matrix uses O(n^2) space
        numMatrix := make([][]int, n)
        for i := 0; i < n; i++ {
            numMatrix[i] = make([]int, n)
        }
    }

space_complexity.swift

    /* Exponential order */
    func quadratic(n: Int) {
        // 2D list uses O(n^2) space
        let numList = Array(repeating: Array(repeating: 0, count: n), count: n)
    }

space_complexity.js

    /* Exponential order */
    function quadratic(n) {
        // Matrix uses O(n^2) space
        const numMatrix = Array(n)
            .fill(null)
            .map(() => Array(n).fill(null));
        // 2D list uses O(n^2) space
        const numList = [];
        for (let i = 0; i < n; i++) {
            const tmp = [];
            for (let j = 0; j < n; j++) {
                tmp.push(0);
            }
            numList.push(tmp);
        }
    }

space_complexity.ts

    /* Exponential order */
    function quadratic(n: number): void {
        // Matrix uses O(n^2) space
        const numMatrix = Array(n)
            .fill(null)
            .map(() => Array(n).fill(null));
        // 2D list uses O(n^2) space
        const numList = [];
        for (let i = 0; i < n; i++) {
            const tmp = [];
            for (let j = 0; j < n; j++) {
                tmp.push(0);
            }
            numList.push(tmp);
        }
    }

space_complexity.dart

    /* Exponential order */
    void quadratic(int n) {
      // Matrix uses O(n^2) space
      List<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));
      // 2D list uses O(n^2) space
      List<List<int>> numList = [];
      for (var i = 0; i < n; i++) {
        List<int> tmp = [];
        for (int j = 0; j < n; j++) {
          tmp.add(0);
        }
        numList.add(tmp);
      }
    }

space_complexity.rs

    /* Exponential order */
    #[allow(unused)]
    fn quadratic(n: i32) {
        // Matrix uses O(n^2) space
        let num_matrix = vec![vec![0; n as usize]; n as usize];
        // 2D list uses O(n^2) space
        let mut num_list = Vec::new();
        for i in 0..n {
            let mut tmp = Vec::new();
            for j in 0..n {
                tmp.push(0);
            }
            num_list.push(tmp);
        }
    }

space_complexity.c

    /* Exponential order */
    void quadratic(int n) {
        // 2D list uses O(n^2) space
        int **numMatrix = malloc(sizeof(int *) * n);
        for (int i = 0; i < n; i++) {
            int *tmp = malloc(sizeof(int) * n);
            for (int j = 0; j < n; j++) {
                tmp[j] = 0;
            }
            numMatrix[i] = tmp;
        }

        // Memory release
        for (int i = 0; i < n; i++) {
            free(numMatrix[i]);
        }
        free(numMatrix);
    }

space_complexity.kt

    /* Exponential order */
    fun quadratic(n: Int) {
        // Matrix uses O(n^2) space
        val numMatrix = arrayOfNulls<Array<Int>?>(n)
        // 2D list uses O(n^2) space
        val numList = mutableListOf<MutableList<Int>>()
        for (i in 0..<n) {
            val tmp = mutableListOf<Int>()
            for (j in 0..<n) {
                tmp.add(0)
            }
            numList.add(tmp)
        }
    }

space_complexity.rb

    ### Quadratic time ###
    def quadratic(n)
      # 2D list uses O(n^2) space
      Array.new(n) { Array.new(n, 0) }
    end

As shown in the following figure, the recursion depth of this function is \(n\), and an array is initialized in each recursive function with lengths of \(n\), \(n-1\), \(\dots\), \(2\), \(1\), with an average length of \(n / 2\), thus occupying \(O(n^2)\) space overall:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def quadratic_recur(n: int) -> int:
        """Quadratic order (recursive implementation)"""
        if n <= 0:
            return 0
        # Array nums length is n, n-1, ..., 2, 1
        nums = [0] * n
        return quadratic_recur(n - 1)

space_complexity.cpp

    /* Quadratic order (recursive implementation) */
    int quadraticRecur(int n) {
        if (n <= 0)
            return 0;
        vector<int> nums(n);
        cout << "In recursion n = " << n << ", nums length = " << nums.size() << endl;
        return quadraticRecur(n - 1);
    }

space_complexity.java

    /* Quadratic order (recursive implementation) */
    int quadraticRecur(int n) {
        if (n <= 0)
            return 0;
        // Array nums has length n, n-1, ..., 2, 1
        int[] nums = new int[n];
        System.out.println("In recursion n = " + n + ", nums length = " + nums.length);
        return quadraticRecur(n - 1);
    }

space_complexity.cs

    /* Quadratic order (recursive implementation) */
    int QuadraticRecur(int n) {
        if (n <= 0) return 0;
        int[] nums = new int[n];
        Console.WriteLine("Recursion n = " + n + ", nums length = " + nums.Length);
        return QuadraticRecur(n - 1);
    }

space_complexity.go

    /* Quadratic order (recursive implementation) */
    func spaceQuadraticRecur(n int) int {
        if n <= 0 {
            return 0
        }
        nums := make([]int, n)
        fmt.Printf("In recursion n = %d, nums length = %d \n", n, len(nums))
        return spaceQuadraticRecur(n - 1)
    }

space_complexity.swift

    /* Quadratic order (recursive implementation) */
    @discardableResult
    func quadraticRecur(n: Int) -> Int {
        if n <= 0 {
            return 0
        }
        // Array nums has length n, n-1, ..., 2, 1
        let nums = Array(repeating: 0, count: n)
        print("In recursion n = \(n), nums length = \(nums.count)")
        return quadraticRecur(n: n - 1)
    }

space_complexity.js

    /* Quadratic order (recursive implementation) */
    function quadraticRecur(n) {
        if (n <= 0) return 0;
        const nums = new Array(n);
        console.log(`In recursion n = ${n}, nums length = ${nums.length}`);
        return quadraticRecur(n - 1);
    }

space_complexity.ts

    /* Quadratic order (recursive implementation) */
    function quadraticRecur(n: number): number {
        if (n <= 0) return 0;
        const nums = new Array(n);
        console.log(`In recursion n = ${n}, nums length = ${nums.length}`);
        return quadraticRecur(n - 1);
    }

space_complexity.dart

    /* Quadratic order (recursive implementation) */
    int quadraticRecur(int n) {
      if (n <= 0) return 0;
      List<int> nums = List.filled(n, 0);
      print('In recursion n = $n, nums length = ${nums.length}');
      return quadraticRecur(n - 1);
    }

space_complexity.rs

    /* Quadratic order (recursive implementation) */
    fn quadratic_recur(n: i32) -> i32 {
        if n <= 0 {
            return 0;
        };
        // Array nums has length n, n-1, ..., 2, 1
        let nums = vec![0; n as usize];
        println!("In recursion n = {}, nums length = {}", n, nums.len());
        return quadratic_recur(n - 1);
    }

space_complexity.c

    /* Quadratic order (recursive implementation) */
    int quadraticRecur(int n) {
        if (n <= 0)
            return 0;
        int *nums = malloc(sizeof(int) * n);
        printf("In recursion n = %d, nums length = %d\r\n", n, n);
        int res = quadraticRecur(n - 1);
        free(nums);
        return res;
    }

space_complexity.kt

    /* Quadratic order (recursive implementation) */
    tailrec fun quadraticRecur(n: Int): Int {
        if (n <= 0)
            return 0
        // Array nums has length n, n-1, ..., 2, 1
        val nums = Array(n) { 0 }
        println("In recursion n = $n, nums length = ${nums.size}")
        return quadraticRecur(n - 1)
    }

space_complexity.rb

    ### Quadratic space (recursive) ###
    def quadratic_recur(n)
      return 0 unless n > 0

      # Array nums has length n, n-1, ..., 2, 1
      nums = Array.new(n, 0)
      quadratic_recur(n - 1)
    end

Figure 2-18 Quadratic order space complexity generated by recursive function

### 4. Exponential Order \(O(2^n)\)

Exponential order is common in binary trees. Observe the following figure: a "full binary tree" with \(n\) levels has \(2^n - 1\) nodes, occupying \(O(2^n)\) space:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
space_complexity.py

    def build_tree(n: int) -> TreeNode | None:
        """Exponential order (build full binary tree)"""
        if n == 0:
            return None
        root = TreeNode(0)
        root.left = build_tree(n - 1)
        root.right = build_tree(n - 1)
        return root

space_complexity.cpp

    /* Driver Code */
    TreeNode *buildTree(int n) {
        if (n == 0)
            return nullptr;
        TreeNode *root = new TreeNode(0);
        root->left = buildTree(n - 1);
        root->right = buildTree(n - 1);
        return root;
    }

space_complexity.java

    /* Driver Code */
    TreeNode buildTree(int n) {
        if (n == 0)
            return null;
        TreeNode root = new TreeNode(0);
        root.left = buildTree(n - 1);
        root.right = buildTree(n - 1);
        return root;
    }

space_complexity.cs

    /* Driver Code */
    TreeNode? BuildTree(int n) {
        if (n == 0) return null;
        TreeNode root = new(0) {
            left = BuildTree(n - 1),
            right = BuildTree(n - 1)
        };
        return root;
    }

space_complexity.go

    /* Driver Code */
    func buildTree(n int) *TreeNode {
        if n == 0 {
            return nil
        }
        root := NewTreeNode(0)
        root.Left = buildTree(n - 1)
        root.Right = buildTree(n - 1)
        return root
    }

space_complexity.swift

    /* Driver Code */
    func buildTree(n: Int) -> TreeNode? {
        if n == 0 {
            return nil
        }
        let root = TreeNode(x: 0)
        root.left = buildTree(n: n - 1)
        root.right = buildTree(n: n - 1)
        return root
    }

space_complexity.js

    /* Driver Code */
    function buildTree(n) {
        if (n === 0) return null;
        const root = new TreeNode(0);
        root.left = buildTree(n - 1);
        root.right = buildTree(n - 1);
        return root;
    }

space_complexity.ts

    /* Driver Code */
    function buildTree(n: number): TreeNode | null {
        if (n === 0) return null;
        const root = new TreeNode(0);
        root.left = buildTree(n - 1);
        root.right = buildTree(n - 1);
        return root;
    }

space_complexity.dart

    /* Driver Code */
    TreeNode? buildTree(int n) {
      if (n == 0) return null;
      TreeNode root = TreeNode(0);
      root.left = buildTree(n - 1);
      root.right = buildTree(n - 1);
      return root;
    }

space_complexity.rs

    /* Driver Code */
    fn build_tree(n: i32) -> Option<Rc<RefCell<TreeNode>>> {
        if n == 0 {
            return None;
        };
        let root = TreeNode::new(0);
        root.borrow_mut().left = build_tree(n - 1);
        root.borrow_mut().right = build_tree(n - 1);
        return Some(root);
    }

space_complexity.c

    /* Driver Code */
    TreeNode *buildTree(int n) {
        if (n == 0)
            return NULL;
        TreeNode *root = newTreeNode(0);
        root->left = buildTree(n - 1);
        root->right = buildTree(n - 1);
        return root;
    }

space_complexity.kt

    /* Driver Code */
    fun buildTree(n: Int): TreeNode? {
        if (n == 0)
            return null
        val root = TreeNode(0)
        root.left = buildTree(n - 1)
        root.right = buildTree(n - 1)
        return root
    }

space_complexity.rb

    ### Exponential space (build full binary tree) ###
    def build_tree(n)
      return if n == 0

      TreeNode.new.tap do |root|
        root.left = build_tree(n - 1)
        root.right = build_tree(n - 1)
      end
    end

Figure 2-19 Exponential order space complexity generated by full binary tree

### 5. Logarithmic Order \(O(\log n)\)

Logarithmic order is common in divide-and-conquer algorithms. For example, merge sort: given an input array of length \(n\), each recursion divides the array in half from the midpoint, forming a recursion tree of height \(\log n\), using \(O(\log n)\) stack frame space.

Another example is converting a number to a string. Given a positive integer \(n\), it has \(\lfloor \log_{10} n \rfloor + 1\) digits, i.e., the corresponding string length is \(\lfloor \log_{10} n \rfloor + 1\), so the space complexity is \(O(\log_{10} n + 1) = O(\log n)\).

## 2.4.4 Trading Time for Space

Ideally, we hope that both the time complexity and space complexity of an algorithm can reach optimal. However, in practice, optimizing both time complexity and space complexity simultaneously is usually very difficult.

Reducing time complexity usually comes at the cost of increasing space complexity, and vice versa. Sacrificing memory space to improve execution speed is called "trading space for time"; the reverse is called "trading time for space".

The choice of which approach depends on which aspect we value more. In most cases, time is more precious than space, so "trading space for time" is usually the more common strategy. Of course, when the data volume is very large, controlling space complexity is also very important.

##### Feel free to drop your insights, questions or suggestions