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84 lines
5.4 KiB
84 lines
5.4 KiB
# Binary search
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<u>Binary search</u> is an efficient search algorithm based on the divide-and-conquer strategy. It utilizes the orderliness of data, reducing the search range by half each round until the target element is found or the search interval is empty.
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!!! question
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Given an array `nums` of length $n$, with elements arranged in ascending order and non-repeating. Please find and return the index of element `target` in this array. If the array does not contain the element, return $-1$. An example is shown in the figure below.
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As shown in the figure below, we first initialize pointers $i = 0$ and $j = n - 1$, pointing to the first and last elements of the array, representing the search interval $[0, n - 1]$. Please note that square brackets indicate a closed interval, which includes the boundary values themselves.
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Next, perform the following two steps in a loop.
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1. Calculate the midpoint index $m = \lfloor {(i + j) / 2} \rfloor$, where $\lfloor \: \rfloor$ denotes the floor operation.
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2. Compare the size of `nums[m]` and `target`, divided into the following three scenarios.
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1. If `nums[m] < target`, it indicates that `target` is in the interval $[m + 1, j]$, thus set $i = m + 1$.
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2. If `nums[m] > target`, it indicates that `target` is in the interval $[i, m - 1]$, thus set $j = m - 1$.
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3. If `nums[m] = target`, it indicates that `target` is found, thus return index $m$.
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If the array does not contain the target element, the search interval will eventually reduce to empty. In this case, return $-1$.
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=== "<1>"
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=== "<2>"
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=== "<3>"
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=== "<4>"
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=== "<5>"
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=== "<6>"
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=== "<7>"
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It's worth noting that since $i$ and $j$ are both of type `int`, **$i + j$ might exceed the range of `int` type**. To avoid large number overflow, we usually use the formula $m = \lfloor {i + (j - i) / 2} \rfloor$ to calculate the midpoint.
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The code is as follows:
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```src
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[file]{binary_search}-[class]{}-[func]{binary_search}
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```
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**Time complexity is $O(\log n)$** : In the binary loop, the interval reduces by half each round, hence the number of iterations is $\log_2 n$.
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**Space complexity is $O(1)$** : Pointers $i$ and $j$ use constant size space.
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## Interval representation methods
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Besides the aforementioned closed interval, a common interval representation is the "left-closed right-open" interval, defined as $[0, n)$, where the left boundary includes itself, and the right boundary does not include itself. In this representation, the interval $[i, j)$ is empty when $i = j$.
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We can implement a binary search algorithm with the same functionality based on this representation:
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```src
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[file]{binary_search}-[class]{}-[func]{binary_search_lcro}
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```
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As shown in the figure below, in the two types of interval representations, the initialization of the binary search algorithm, the loop condition, and the narrowing interval operation are different.
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Since both boundaries in the "closed interval" representation are defined as closed, the operations to narrow the interval through pointers $i$ and $j$ are also symmetrical. This makes it less prone to errors, **therefore, it is generally recommended to use the "closed interval" approach**.
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## Advantages and limitations
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Binary search performs well in both time and space aspects.
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- Binary search is time-efficient. With large data volumes, the logarithmic time complexity has a significant advantage. For instance, when the data size $n = 2^{20}$, linear search requires $2^{20} = 1048576$ iterations, while binary search only requires $\log_2 2^{20} = 20$ iterations.
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- Binary search does not require extra space. Compared to search algorithms that rely on additional space (like hash search), binary search is more space-efficient.
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However, binary search is not suitable for all situations, mainly for the following reasons.
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- Binary search is only applicable to ordered data. If the input data is unordered, it is not worth sorting it just to use binary search, as sorting algorithms typically have a time complexity of $O(n \log n)$, which is higher than both linear and binary search. For scenarios with frequent element insertion to maintain array order, inserting elements into specific positions has a time complexity of $O(n)$, which is also quite costly.
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- Binary search is only applicable to arrays. Binary search requires non-continuous (jumping) element access, which is inefficient in linked lists, thus not suitable for use in linked lists or data structures based on linked lists.
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- With small data volumes, linear search performs better. In linear search, each round only requires 1 decision operation; whereas in binary search, it involves 1 addition, 1 division, 1 to 3 decision operations, 1 addition (subtraction), totaling 4 to 6 operations; therefore, when data volume $n$ is small, linear search can be faster than binary search.
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