hello-algo/en/docs/chapter_divide_and_conquer/binary_search_recur.md

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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<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(vector<int> &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}
```