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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.

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 C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby Zig
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)

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);
    }

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);
    }

binary_search_recur.cs

    [class]{binary_search_recur}-[func]{DFS}

    [class]{binary_search_recur}-[func]{BinarySearch}

binary_search_recur.go

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.swift

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.js

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.ts

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.dart

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.rs

    [class]{}-[func]{dfs}

    [class]{}-[func]{binary_search}

binary_search_recur.c

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.kt

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

binary_search_recur.rb

    [class]{}-[func]{dfs}

    [class]{}-[func]{binary_search}

binary_search_recur.zig

    [class]{}-[func]{dfs}

    [class]{}-[func]{binarySearch}

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