I. Introduction
Before diving deeper into data structures and algorithms, it's essential to understand the general methods of algorithm analysis. Algorithm analysis primarily involves examining the time and space complexity of an algorithm, but in some cases, we are more interested in the actual runtime performance of the algorithm. Additionally, algorithm visualization is a practical skill that helps us better understand how an algorithm executes step by step. This technique becomes particularly useful when dealing with complex or iterative algorithms.
In this article, we will first explore how to quantify the actual running performance of an algorithm through experimental design. Then, we'll introduce the method for analyzing time complexity. We will also present a multiplier experiment that can predict an algorithm’s performance efficiently. Finally, we will walk through some common interview questions from top internet companies to reinforce our understanding and apply what we've learned.
II. General Methods of Algorithm Analysis
1. Quantifying the Actual Running Performance of an Algorithm
Before discussing the time-space complexity of an algorithm, let's first look at how to measure its real-world performance. The most common metric used is the actual running time. Usually, this time depends on the size of the problem being solved. For example, sorting 1 million elements takes longer than sorting 100,000 elements. Therefore, when observing the running time of an algorithm, we must consider the scale of the problem and how the actual runtime grows as the input size increases.
Let’s take an example from the book "Algorithms (4th Edition)" (Douban). The code below counts the number of triplets in an array that sum to zero:
public class ThreeSum {
public static int count(int[] a) {
int N = a.length;
int cnt = 0;
for (int i = 0; i < N; i++) {
for (int j = i + 1; j < N; j++) {
for (int k = j + 1; k < N; k++) {
if (a[i] + a[j] + a[k] == 0) {
cnt++;
}
}
}
}
return cnt;
}
public static void main(String[] args) {
int[] a = StdIn.readAllInts();
StdOut.println(count(a));
}
}
The classes `StdIn` and `StdOut` are provided in the project repository: https://github.com/absfree/Algo. The function of this code is to count the number of triplets in the array that sum to zero. The algorithm is straightforward—iterating through all possible triplets and checking their sum.
To measure the runtime, we can record the start and end times of the process. The difference gives the actual running time. Here is the code used to measure the runtime:
public static void main(String[] args) {
int[] a = In.readInts(args[0]);
long startTime = System.currentTimeMillis();
int count = count(a);
long endTime = System.currentTimeMillis();
double time = (endTime - startTime) / 1000.0;
StdOut.println("The result is: " + count + ", and takes " + time + " seconds.");
}
We tested the algorithm with inputs of sizes 1000, 2000, and 4000, and the results were:
- 1000 integers: 70 triplets, 1.017 seconds
- 2000 integers: 528 triplets, 7.894 seconds
- 4000 integers: 4039 triplets, 64.348 seconds
From these results, we observe that when the problem size doubles, the running time increases by approximately eight times. This suggests that the running time of the ThreeSum algorithm follows a cubic relationship with the input size, T(N) = k * N^3.
2. Time Complexity Analysis of the Algorithm
(1) Basic Concepts
Time complexity refers to the growth rate of an algorithm's runtime as the input size increases. Common notations include Big O, Big Ω, and Big Θ.
Big O notation represents the upper bound of the runtime, indicating the worst-case scenario. It defines that f(n) is O(g(n)) if there exist constants c and N₀ such that |f(n)| ≤ c * g(n) for all n > N₀.
Big Ω notation provides the lower bound, representing the best-case scenario. It states that f(n) is Ω(g(n)) if there exist constants c and N₀ such that |f(n)| ≥ c * g(n) for all n > N₀.
Big Θ notation represents the tight bound, meaning f(n) is Θ(g(n)) if there exist constants c₠and c₂ such that c₠* g(n) ≤ |f(n)| ≤ c₂ * g(n) for all n > N₀.
In practice, Big O notation is most commonly used to describe the upper bound of the algorithm's performance.
(2) Time Complexity Analysis Method
To analyze the time complexity of the ThreeSum algorithm, we focus on the most time-consuming operation—in this case, the `if` statement inside the nested loops. The number of times this statement is executed is N*(N-1)*(N-2)/6, which simplifies to O(N³). This confirms our earlier hypothesis about the cubic growth of the algorithm's runtime.
The two steps to determine time complexity are:
1. Identify the key operation that dominates the runtime.
2. Calculate the number of times this operation is executed.
For instance, in the ThreeSum algorithm, the `if` statement is the key operation, and its execution count determines the time complexity.
3. Expected Runtime of the Algorithm
The expected runtime reflects the typical performance of an algorithm under normal conditions. While the worst-case time complexity might be high, the average case is often more relevant in practice. For example, quicksort has a worst-case time complexity of O(n²), but its average case is O(n log n), making it faster in many real-world scenarios.
4. Magnification Experiment
A magnification experiment is a powerful tool to predict an algorithm’s performance. By doubling the input size and measuring the runtime, we can estimate the growth order of the algorithm. For example, if the runtime increases by a factor of 8 when the input size doubles, the growth order is likely O(n³).
This approach is based on the magnification theorem, which states that if T(N) ~ a * N^b * log N, then T(2N)/T(N) ~ 2^b.
5. Amortized Analysis
Amortized analysis is used to evaluate the average cost of operations over a sequence. For example, in a dynamic array, pushing an element may occasionally require resizing the array, leading to a higher cost. However, this cost is spread across multiple operations, resulting in an average cost of O(1) per push.
III. Practice Problems and Interview Questions
Now that we’ve covered the basics of algorithm analysis, let’s apply our knowledge to solve some common interview questions.
[Tencent] What is the time complexity of the following algorithm?
```java
int foo(int n) {
if (n <= 1) {
return 1;
}
return n * foo(n - 1);
}
```
This is a recursive function that calculates the factorial of n. The number of recursive calls is n, so the time complexity is O(n).
[Jingdong] What is the time complexity of the following function?
```java
void recursive(int n, int m, int o) {
if (n <= 0) {
printf("%d, %d", m, o);
} else {
recursive(n - 1, m + 1, o);
recursive(n - 1, m, o + 1);
}
}
```
Each call branches into two recursive calls, leading to a total of 2^n operations. Thus, the time complexity is O(2^n).
[Jingdong] What is the time complexity of the following program?
```java
x = m;
y = 1;
while (x - y > e) {
x = (x + y) / 2;
y = m / x;
}
print(x);
```
The loop reduces the difference between x and y exponentially, resulting in a logarithmic number of iterations. Hence, the time complexity is O(log m).
[Sogou] Given the recurrence relation T(n) = 2T(n/2) + n, with T(1) = 1, what is the time complexity?
Using the master theorem, this recurrence falls into Case 2, where the time complexity is O(n log n).
By solving these problems, we reinforce our understanding of algorithm analysis and improve our ability to assess performance in real-world scenarios.
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