Decision-Tree Model in Sorting Algorithms
In the context of sorting algorithms, a decision tree is a full binary tree where each internal node represents a comparison between elements, and each leaf node represents a unique permutation of the input elements. The decision tree models the sequence of comparisons made by the sorting algorithm for a given input size. This tree can be used to analyze the efficiency and behavior of the algorithm.
Let’s take the example of a simple sorting algorithm, like Bubble Sort, and see how a decision tree might look for a list of 3 elements [a, b, c].
Java
Here’s a Java example that mimics the decision tree for Bubble Sort on an array of 3 elements.
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| public class DecisionTreeBubbleSort {
static void bubbleSort(int[] arr) {
int n = arr.length;
for (int i = 0; i < n - 1; i++) {
for (int j = 0; j < n - i - 1; j++) {
if (arr[j] > arr[j + 1]) {
// swap
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
}
}
}
}
public static void main(String[] args) {
int[] arr = {3, 1, 2};
bubbleSort(arr);
for (int i : arr) {
System.out.print(i + " ");
}
}
}
|
C++
The C++ example:
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| #include <iostream>
using namespace std;
void bubbleSort(int arr[], int n) {
for (int i = 0; i < n - 1; i++) {
for (int j = 0; j < n - i - 1; j++) {
if (arr[j] > arr[j + 1]) {
// swap
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
}
}
}
}
int main() {
int arr[] = {3, 1, 2};
int n = sizeof(arr) / sizeof(arr[0]);
bubbleSort(arr, n);
for (int i = 0; i < n; i++) {
cout << arr[i] << " ";
}
return 0;
}
|
Python
Python example:
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| def bubble_sort(arr):
n = len(arr)
for i in range(n - 1):
for j in range(0, n - i - 1):
if arr[j] > arr[j + 1]:
# swap
arr[j], arr[j + 1] = arr[j + 1], arr[j]
arr = [3, 1, 2]
bubble_sort(arr)
print(arr)
|
In each of these implementations, the nested loop structure is what decides the order of element comparisons, effectively making the decisions a full binary tree would. Each comparison if (arr[j] > arr[j + 1])
could be seen as an internal node in the decision tree, directing the sorting algorithm along different paths depending on the outcome of the comparison.