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# 12.2 Divide and conquer search strategy
We have learned that search algorithms fall into two main categories.
- **Brute-force search**: It is implemented by traversing the data structure, with a time complexity of $O(n)$.
- **Adaptive search**: It utilizes a unique data organization form or prior information, and its time complexity can reach $O(\log n)$ or even $O(1)$.
In fact, **search algorithms with a time complexity of $O(\log n)$ are usually based on the divide-and-conquer strategy**, such as binary search and trees.
- Each step of binary search divides the problem (searching for a target element in an array) into a smaller problem (searching for the target element in half of the array), continuing until the array is empty or the target element is found.
- Trees represent the divide-and-conquer idea, where in data structures like binary search trees, AVL trees, and heaps, the time complexity of various operations is $O(\log n)$.
The divide-and-conquer strategy of binary search is as follows.
- **The problem can be divided**: Binary search recursively divides the original problem (searching in an array) into subproblems (searching in half of the array), achieved by comparing the middle element with the target element.
- **Subproblems are independent**: In binary search, each round handles one subproblem, unaffected by other subproblems.
- **The solutions of subproblems do not need to be merged**: Binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. When a subproblem is solved, the original problem is also solved.
Divide-and-conquer can enhance search efficiency because brute-force search can only eliminate one option per round, **whereas divide-and-conquer can eliminate half of the options**.
### 1. Implementing binary search based on divide-and-conquer
In previous chapters, binary search was implemented based on iteration. Now, we implement it based on divide-and-conquer (recursion).
!!! question
Given an ordered array `nums` of length $n$, where all elements are unique, please find the element `target`.
From a divide-and-conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$.
Starting from the original problem $f(0, n-1)$, perform the binary search through the following steps.
1. Calculate the midpoint $m$ of the search interval $[i, j]$, and use it to eliminate half of the search interval.
2. Recursively solve the subproblem reduced by half in size, which could be $f(i, m-1)$ or $f(m+1, j)$.
3. Repeat steps `1.` and `2.`, until `target` is found or the interval is empty and returns.
Figure 12-4 shows the divide-and-conquer process of binary search for element $6$ in an array.
{ class="animation-figure" }
Figure 12-4 The divide-and-conquer process of binary search
In the implementation code, we declare a recursive function `dfs()` to solve the problem $f(i, j)$:
=== "Python"
```python title="binary_search_recur.py"
def dfs(nums: list[int], target: int, i: int, j: int) -> int:
"""Binary search: problem f(i, j)"""
# If the interval is empty, indicating no target element, return -1
if i > j:
return -1
# Calculate midpoint index m
m = (i + j) // 2
if nums[m] < target:
# Recursive subproblem f(m+1, j)
return dfs(nums, target, m + 1, j)
elif nums[m] > target:
# Recursive subproblem f(i, m-1)
return dfs(nums, target, i, m - 1)
else:
# Found the target element, thus return its index
return m
def binary_search(nums: list[int], target: int) -> int:
"""Binary search"""
n = len(nums)
# Solve problem f(0, n-1)
return dfs(nums, target, 0, n - 1)
```
=== "C++"
```cpp title="binary_search_recur.cpp"
/* Binary search: problem f(i, j) */
int dfs(vector &nums, int target, int i, int j) {
// If the interval is empty, indicating no target element, return -1
if (i > j) {
return -1;
}
// Calculate midpoint index m
int m = i + (j - i) / 2;
if (nums[m] < target) {
// Recursive subproblem f(m+1, j)
return dfs(nums, target, m + 1, j);
} else if (nums[m] > target) {
// Recursive subproblem f(i, m-1)
return dfs(nums, target, i, m - 1);
} else {
// Found the target element, thus return its index
return m;
}
}
/* Binary search */
int binarySearch(vector &nums, int target) {
int n = nums.size();
// Solve problem f(0, n-1)
return dfs(nums, target, 0, n - 1);
}
```
=== "Java"
```java title="binary_search_recur.java"
/* Binary search: problem f(i, j) */
int dfs(int[] nums, int target, int i, int j) {
// If the interval is empty, indicating no target element, return -1
if (i > j) {
return -1;
}
// Calculate midpoint index m
int m = i + (j - i) / 2;
if (nums[m] < target) {
// Recursive subproblem f(m+1, j)
return dfs(nums, target, m + 1, j);
} else if (nums[m] > target) {
// Recursive subproblem f(i, m-1)
return dfs(nums, target, i, m - 1);
} else {
// Found the target element, thus return its index
return m;
}
}
/* Binary search */
int binarySearch(int[] nums, int target) {
int n = nums.length;
// Solve problem f(0, n-1)
return dfs(nums, target, 0, n - 1);
}
```
=== "C#"
```csharp title="binary_search_recur.cs"
[class]{binary_search_recur}-[func]{DFS}
[class]{binary_search_recur}-[func]{BinarySearch}
```
=== "Go"
```go title="binary_search_recur.go"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "Swift"
```swift title="binary_search_recur.swift"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "JS"
```javascript title="binary_search_recur.js"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "TS"
```typescript title="binary_search_recur.ts"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "Dart"
```dart title="binary_search_recur.dart"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "Rust"
```rust title="binary_search_recur.rs"
[class]{}-[func]{dfs}
[class]{}-[func]{binary_search}
```
=== "C"
```c title="binary_search_recur.c"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "Kotlin"
```kotlin title="binary_search_recur.kt"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```
=== "Ruby"
```ruby title="binary_search_recur.rb"
[class]{}-[func]{dfs}
[class]{}-[func]{binary_search}
```
=== "Zig"
```zig title="binary_search_recur.zig"
[class]{}-[func]{dfs}
[class]{}-[func]{binarySearch}
```