### 2. Linear Order $O(n)$ {data-toc-label="Linear Order"}
### 2. Linear Order $O(n)$ {data-toc-label="2. Linear Order"}
Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to $n$:
@ -1589,7 +1589,7 @@ As shown below, this function's recursive depth is $n$, meaning there are $n$ in
<palign="center"> Figure 2-17 Recursive Function Generating Linear Order Space Complexity </p>
### 3. Quadratic Order $O(n^2)$ {data-toc-label="Quadratic Order"}
### 3. Quadratic Order $O(n^2)$ {data-toc-label="3. Quadratic Order"}
Quadratic order is common in matrices and graphs, where the number of elements is quadratic to $n$:
@ -2025,7 +2025,7 @@ As shown below, the recursive depth of this function is $n$, and in each recursi
<palign="center"> Figure 2-18 Recursive Function Generating Quadratic Order Space Complexity </p>
### 4. Exponential Order $O(2^n)$ {data-toc-label="Exponential Order"}
### 4. Exponential Order $O(2^n)$ {data-toc-label="4. Exponential Order"}
Exponential order is common in binary trees. Observe the below image, a "full binary tree" with $n$ levels has $2^n - 1$ nodes, occupying $O(2^n)$ space:
@ -2225,7 +2225,7 @@ Exponential order is common in binary trees. Observe the below image, a "full bi
<palign="center"> Figure 2-19 Full Binary Tree Generating Exponential Order Space Complexity </p>
### 5. Logarithmic Order $O(\log n)$ {data-toc-label="Logarithmic Order"}
### 5. Logarithmic Order $O(\log n)$ {data-toc-label="5. Logarithmic Order"}
Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length $n$ is recursively divided in half each round, forming a recursion tree of height $\log n$, using $O(\log n)$ stack frame space.
<palign="center"> Figure 2-9 Common Types of Time Complexity </p>
### 1. Constant Order $O(1)$ {data-toc-label="Constant Order"}
### 1. Constant Order $O(1)$ {data-toc-label="1. Constant Order"}
Constant order means the number of operations is independent of the input data size $n$. In the following function, although the number of operations `size` might be large, the time complexity remains $O(1)$ as it's unrelated to $n$:
@ -1167,7 +1167,7 @@ Constant order means the number of operations is independent of the input data s
### 2. Linear Order $O(n)$ {data-toc-label="Linear Order"}
### 2. Linear Order $O(n)$ {data-toc-label="2. Linear Order"}
Linear order indicates the number of operations grows linearly with the input data size $n$. Linear order commonly appears in single-loop structures:
@ -1538,7 +1538,7 @@ Operations like array traversal and linked list traversal have a time complexity
It's important to note that **the input data size $n$ should be determined based on the type of input data**. For example, in the first example, $n$ represents the input data size, while in the second example, the length of the array $n$ is the data size.
### 3. Quadratic Order $O(n^2)$ {data-toc-label="Quadratic Order"}
### 3. Quadratic Order $O(n^2)$ {data-toc-label="3. Quadratic Order"}
Quadratic order means the number of operations grows quadratically with the input data size $n$. Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of $O(n)$, resulting in an overall complexity of $O(n^2)$:
@ -2073,7 +2073,7 @@ For instance, in bubble sort, the outer loop runs $n - 1$ times, and the inner l
### 4. Exponential Order $O(2^n)$ {data-toc-label="Exponential Order"}
### 4. Exponential Order $O(2^n)$ {data-toc-label="4. Exponential Order"}
Biological "cell division" is a classic example of exponential order growth: starting with one cell, it becomes two after one division, four after two divisions, and so on, resulting in $2^n$ cells after $n$ divisions.
@ -2491,7 +2491,7 @@ In practice, exponential order often appears in recursive functions. For example
Exponential order growth is extremely rapid and is commonly seen in exhaustive search methods (brute force, backtracking, etc.). For large-scale problems, exponential order is unacceptable, often requiring dynamic programming or greedy algorithms as solutions.
### 5. Logarithmic Order $O(\log n)$ {data-toc-label="Logarithmic Order"}
### 5. Logarithmic Order $O(\log n)$ {data-toc-label="5. Logarithmic Order"}
In contrast to exponential order, logarithmic order reflects situations where "the size is halved each round." Given an input data size $n$, since the size is halved each round, the number of iterations is $\log_2 n$, the inverse function of $2^n$.
@ -2861,7 +2861,7 @@ Logarithmic order is typical in algorithms based on the divide-and-conquer strat
This means the base $m$ can be changed without affecting the complexity. Therefore, we often omit the base $m$ and simply denote logarithmic order as $O(\log n)$.
### 6. Linear-Logarithmic Order $O(n \log n)$ {data-toc-label="Linear-Logarithmic Order"}
Linear-logarithmic order often appears in nested loops, with the complexities of the two loops being $O(\log n)$ and $O(n)$ respectively. The related code is as follows:
@ -3076,7 +3076,7 @@ The image below demonstrates how linear-logarithmic order is generated. Each lev
Mainstream sorting algorithms typically have a time complexity of $O(n \log n)$, such as quicksort, mergesort, and heapsort.
### 7. Factorial Order $O(n!)$ {data-toc-label="Factorial Order"}
### 7. Factorial Order $O(n!)$ {data-toc-label="7. Factorial Order"}
Factorial order corresponds to the mathematical problem of "full permutation." Given $n$ distinct elements, the total number of possible permutations is:
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