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46 lines
3.0 KiB
46 lines
3.0 KiB
7 months ago
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# Divide and conquer search strategy
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We have learned that search algorithms fall into two main categories.
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- **Brute-force search**: It is implemented by traversing the data structure, with a time complexity of $O(n)$.
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- **Adaptive search**: It utilizes a unique data organization form or prior information, and its time complexity can reach $O(\log n)$ or even $O(1)$.
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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.
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- 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.
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- 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 $O(\log n)$.
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The divide-and-conquer strategy of binary search is as follows.
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- **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.
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- **Subproblems are independent**: In binary search, each round handles one subproblem, unaffected by other subproblems.
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- **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.
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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**.
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### Implementing binary search based on divide-and-conquer
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In previous chapters, binary search was implemented based on iteration. Now, we implement it based on divide-and-conquer (recursion).
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!!! question
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Given an ordered array `nums` of length $n$, where all elements are unique, please find the element `target`.
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From a divide-and-conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$.
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Starting from the original problem $f(0, n-1)$, perform the binary search through the following steps.
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1. Calculate the midpoint $m$ of the search interval $[i, j]$, and use it to eliminate half of the search interval.
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2. Recursively solve the subproblem reduced by half in size, which could be $f(i, m-1)$ or $f(m+1, j)$.
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3. Repeat steps `1.` and `2.`, until `target` is found or the interval is empty and returns.
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The diagram below shows the divide-and-conquer process of binary search for element $6$ in an array.
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In the implementation code, we declare a recursive function `dfs()` to solve the problem $f(i, j)$:
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```src
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[file]{binary_search_recur}-[class]{}-[func]{binary_search}
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```
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