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<p><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.</p>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, with elements arranged in ascending order and non-repeating. Please find and return the index of element <code>target</code> in this array. If the array does not contain the element, return <span class="arithmatex">\(-1\)</span>. An example is shown below.</p>
<p>Given an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, with elements arranged in ascending order and non-repeating. Please find and return the index of element <code>target</code> in this array. If the array does not contain the element, return <span class="arithmatex">\(-1\)</span>. An example is shown in Figure 10-1.</p>
</div>
<p><a class="glightbox" href="../binary_search.assets/binary_search_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search example data" class="animation-figure" src="../binary_search.assets/binary_search_example.png" /></a></p>
<p align="center"> Figure 10-1 &nbsp; Binary search example data </p>
<p>As shown in the Figure 10-2 , we first initialize pointers <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = n - 1\)</span>, pointing to the first and last elements of the array, representing the search interval <span class="arithmatex">\([0, n - 1]\)</span>. Please note that square brackets indicate a closed interval, which includes the boundary values themselves.</p>
<p>As shown in Figure 10-2, we first initialize pointers <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = n - 1\)</span>, pointing to the first and last elements of the array, representing the search interval <span class="arithmatex">\([0, n - 1]\)</span>. Please note that square brackets indicate a closed interval, which includes the boundary values themselves.</p>
<p>Next, perform the following two steps in a loop.</p>
<ol>
<li>Calculate the midpoint index <span class="arithmatex">\(m = \lfloor {(i + j) / 2} \rfloor\)</span>, where <span class="arithmatex">\(\lfloor \: \rfloor\)</span> denotes the floor operation.</li>
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_lcro%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%EF%BC%88%E5%B7%A6%E9%97%AD%E5%8F%B3%E5%BC%80%E5%8C%BA%E9%97%B4%EF%BC%89%22%22%22%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%B7%A6%E9%97%AD%E5%8F%B3%E5%BC%80%E5%8C%BA%E9%97%B4%20%5B0,%20n%29%20%EF%BC%8C%E5%8D%B3%20i,%20j%20%E5%88%86%E5%88%AB%E6%8C%87%E5%90%91%E6%95%B0%E7%BB%84%E9%A6%96%E5%85%83%E7%B4%A0%E3%80%81%E5%B0%BE%E5%85%83%E7%B4%A0%2B1%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%0A%20%20%20%20%23%20%E5%BE%AA%E7%8E%AF%EF%BC%8C%E5%BD%93%E6%90%9C%E7%B4%A2%E5%8C%BA%E9%97%B4%E4%B8%BA%E7%A9%BA%E6%97%B6%E8%B7%B3%E5%87%BA%EF%BC%88%E5%BD%93%20i%20%3D%20j%20%E6%97%B6%E4%B8%BA%E7%A9%BA%EF%BC%89%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%AD%E7%82%B9%E7%B4%A2%E5%BC%95%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20%E6%AD%A4%E6%83%85%E5%86%B5%E8%AF%B4%E6%98%8E%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bm%2B1,%20j%29%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20%20%23%20%E6%AD%A4%E6%83%85%E5%86%B5%E8%AF%B4%E6%98%8E%20target%20%E5%9C%A8%E5%8C%BA%E9%97%B4%20%5Bi,%20m%29%20%E4%B8%AD%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20%E6%89%BE%E5%88%B0%E7%9B%AE%E6%A0%87%E5%85%83%E7%B4%A0%EF%BC%8C%E8%BF%94%E5%9B%9E%E5%85%B6%E7%B4%A2%E5%BC%95%0A%20%20%20%20return%20-1%20%20%23%20%E6%9C%AA%E6%89%BE%E5%88%B0%E7%9B%AE%E6%A0%87%E5%85%83%E7%B4%A0%EF%BC%8C%E8%BF%94%E5%9B%9E%20-1%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20%E4%BA%8C%E5%88%86%E6%9F%A5%E6%89%BE%EF%BC%88%E5%B7%A6%E9%97%AD%E5%8F%B3%E5%BC%80%E5%8C%BA%E9%97%B4%EF%BC%89%0A%20%20%20%20index%20%3D%20binary_search_lcro%28nums,%20target%29%0A%20%20%20%20print%28%22%E7%9B%AE%E6%A0%87%E5%85%83%E7%B4%A0%206%20%E7%9A%84%E7%B4%A2%E5%BC%95%20%3D%20%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>As shown in the Figure 10-3 , in the two types of interval representations, the initialization of the binary search algorithm, the loop condition, and the narrowing interval operation are different.</p>
<p>As shown in Figure 10-3, in the two types of interval representations, the initialization of the binary search algorithm, the loop condition, and the narrowing interval operation are different.</p>
<p>Since both boundaries in the "closed interval" representation are defined as closed, the operations to narrow the interval through pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are also symmetrical. This makes it less prone to errors, <strong>therefore, it is generally recommended to use the "closed interval" approach</strong>.</p>
<p><a class="glightbox" href="../binary_search.assets/binary_search_ranges.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Two types of interval definitions" class="animation-figure" src="../binary_search.assets/binary_search_ranges.png" /></a></p>
<p align="center"> Figure 10-3 &nbsp; Two types of interval definitions </p>