Solved #2592- Fractal Search

dsa-solutions/Searching-Algorithms/Fractal-Search.md

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+---
+id: Fractal-Search-Algorithm
+title: Fractal Search Algorithm
+sidebar_label: Fractal Search Algorithm
+tags:
+  - Advanced
+  - Search Algorithms
+  - Fractals
+  - CPP
+  - Python
+  - Java
+  - JavaScript
+  - DSA
+description: "This is a detailed explanation and implementation of the Fractal Search Algorithm."
+
+---
+
+## What is the Fractal Search Algorithm?
+
+The Fractal Search Algorithm (FSA) is an advanced search algorithm inspired by the fractal nature of various processes and patterns in nature. It leverages the self-similarity and recursive properties of fractals to efficiently search through complex spaces.
+
+## Algorithm for Fractal Search
+
+1. **Initialization**: Define the search space and initialize the fractal pattern.
+2. **Recursive Division**: Recursively divide the search space into smaller subspaces following the fractal pattern.
+3. **Search Subspaces**: Evaluate the subspaces to find the target element.
+4. **Merge Results**: Combine the results from the subspaces to determine the final position of the target element.
+
+## How does Fractal Search work?
+
+- FSA divides the search space into self-similar subspaces recursively.
+- It searches each subspace individually, combining results to identify the target element.
+- This approach reduces the search space significantly, improving efficiency.
+
+
+## Problem Description
+
+Given a complex search space, implement the Fractal Search Algorithm to find the target element. If the element is not present, the algorithm should indicate that as well.
+
+## Examples
+
+**Example 1:**
+Input:
+search_space = [1, 3, 5, 7, 9]
+target = 5
+Output: 2
+
+**Example 2:**
+Input:
+search_space = [2, 4, 6, 8, 10]
+target = 7
+Output: -1
+
+## Your Task
+
+Complete the function `fractal_search()` which takes a list `search_space` and an integer `target` as input parameters and returns the index of the target element. If the element is not present, return -1.
+
+Expected Time Complexity: $O(\log n)$
+Expected Auxiliary Space: $O(n)$
+
+## Constraints
+
+- $1 <= n <= 10^6$
+- $1 <= search_space[i] <= 10^9$
+- $1 <= target <= 10^9$
+
+## Implementation
+
+```python
+import numpy as np
+
+def fractal_search(search_space, target):
+    def recursive_search(subspace, depth):
+        if len(subspace) == 0:
+            return -1
+
+        mid_index = len(subspace) // 2
+        mid_value = subspace[mid_index]
+
+        if mid_value == target:
+            return mid_index
+
+        if target < mid_value:
+            return recursive_search(subspace[:mid_index], depth + 1)
+        else:
+            result = recursive_search(subspace[mid_index + 1:], depth + 1)
+            return mid_index + 1 + result if result != -1 else -1
+
+    return recursive_search(search_space, 0)
+
+# Example usage:
+search_space = [1, 3, 5, 7, 9]
+target = 5
+print(fractal_search(search_space, target))  # Output: 2
+```
+
+```cpp
+#include <iostream>
+#include <vector>
+
+int fractal_search(const std::vector<int>& search_space, int target) {
+    int recursive_search(const std::vector<int>& subspace, int depth) {
+        if (subspace.empty()) {
+            return -1;
+        }
+
+        int mid_index = subspace.size() / 2;
+        int mid_value = subspace[mid_index];
+
+        if (mid_value == target) {
+            return mid_index;
+        }
+
+        if (target < mid_value) {
+            return recursive_search({subspace.begin(), subspace.begin() + mid_index}, depth + 1);
+        } else {
+            int result = recursive_search({subspace.begin() + mid_index + 1, subspace.end()}, depth + 1);
+            return result != -1 ? mid_index + 1 + result : -1;
+        }
+    }
+
+    return recursive_search(search_space, 0);
+}
+
+// Example usage:
+int main() {
+    std::vector<int> search_space = {1, 3, 5, 7, 9};
+    int target = 5;
+    std::cout << fractal_search(search_space, target) << std::endl;  // Output: 2
+    return 0;
+}
+```
+
+```java
+import java.util.List;
+
+public class FractalSearch {
+    public static int fractalSearch(List<Integer> search_space, int target) {
+        int recursiveSearch(List<Integer> subspace, int depth) {
+            if (subspace.isEmpty()) {
+                return -1;
+            }
+
+            int midIndex = subspace.size() / 2;
+            int midValue = subspace.get(midIndex);
+
+            if (midValue == target) {
+                return midIndex;
+            }
+
+            if (target < midValue) {
+                return recursiveSearch(subspace.subList(0, midIndex), depth + 1);
+            } else {
+                int result = recursiveSearch(subspace.subList(midIndex + 1, subspace.size()), depth + 1);
+                return result != -1 ? midIndex + 1 + result : -1;
+            }
+        }
+
+        return recursiveSearch(search_space, 0);
+    }
+
+    public static void main(String[] args) {
+        List<Integer> search_space = List.of(1, 3, 5, 7, 9);
+        int target = 5;
+        System.out.println(fractalSearch(search_space, target));  // Output: 2
+    }
+}
+```
+
+```javascript
+function fractalSearch(search_space, target) {
+    function recursiveSearch(subspace, depth) {
+        if (subspace.length === 0) {
+            return -1;
+        }
+
+        const midIndex = Math.floor(subspace.length / 2);
+        const midValue = subspace[midIndex];
+
+        if (midValue === target) {
+            return midIndex;
+        }
+
+        if (target < midValue) {
+            return recursiveSearch(subspace.slice(0, midIndex), depth + 1);
+        } else {
+            const result = recursiveSearch(subspace.slice(midIndex + 1), depth + 1);
+            return result !== -1 ? midIndex + 1 + result : -1;
+        }
+    }
+
+    return recursiveSearch(search_space, 0);
+}
+
+// Example usage:
+const search_space = [1, 3, 5, 7, 9];
+const target = 5;
+console.log(fractalSearch(search_space, target));  // Output: 2
+```
+
+# Complexity Analysis
+### Time Complexity: $O(\log n)$, where $n$ is the number of elements in the search space. The recursive division reduces the search space exponentially.
+### Space Complexity: $O(n)$, due to the additional space required for recursive function calls and subspaces.
+# Advantages and Disadvantages
+## Advantages:
+
+Efficient for large and complex search spaces due to the recursive division.
+Exploits the self-similarity of fractals, making it suitable for certain types of data structures.
+## Disadvantages:
+
+More complex to implement compared to traditional search algorithms.
+Performance may vary depending on the nature of the search space and fractal pattern used.
+### References
+Wikipedia: Fractal
+Research Paper: Fractal Search Algorithm