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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.

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.

The figure below shows the divide-and-conquer process of binary search for element \(6\) in an array.

The divide-and-conquer process of binary search

In the implementation code, we declare a recursive function dfs() to solve the problem \(f(i, j)\):

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