https://unnamed.website/tags/data-structures/Recent content in Data-Structures on Unnamed WebsiteHugoen-usAnthony WangAnthony WangSat, 15 Mar 2025 21:30:06 -0400https://unnamed.website/posts/formally-verifying-fenwick-trees/Sat, 15 Mar 2025 21:30:06 -0400Anthony Wanghttps://unnamed.website/posts/formally-verifying-fenwick-trees/<style> .chardiv { float: left; width: 100px; } .charimg { height: 60px; } blockquote { min-height: 60px; } </style> <div class="chardiv"> <img src="https://unnamed.website/img/char/kublai.png" class="charimg"> </div> <blockquote><strong>Kublai</strong>: Hey, it&rsquo;s you again! That formal verification thing you mentioned last time sucks!</blockquote> <p>Huh? You mean our <a href="https://unnamed.website/posts/i-can-prove-it-can-sort/">proof of the ICan&rsquo;tBelieveItCanSort algorithm</a>?</p> <style> .chardiv { float: left; width: 100px; } .charimg { height: 60px; } blockquote { min-height: 60px; } </style> <div class="chardiv"> <img src="https://unnamed.website/img/char/kublai.png" class="charimg"> </div> <blockquote><strong>Kublai</strong>: Yeah! ICan&rsquo;tBelieveItCanSort? More like ICanBelieveItObviouslyCanSort! After watching that visualization a few times, it intuitively makes so much sense. It&rsquo;s trivial.</blockquote> <p>Trivial? I&rsquo;m banning that word. If you consider something to be trivial, you probably haven&rsquo;t pondered it deeply enough.</p>https://unnamed.website/posts/fenwick-trees-awesome/Tue, 14 Jan 2025 20:53:04 -0500Anthony Wanghttps://unnamed.website/posts/fenwick-trees-awesome/<link rel="stylesheet" href="https://unnamed.website/katex/katex.min.css" crossorigin="anonymous"> <script defer src="https://unnamed.website/katex/katex.min.js" crossorigin="anonymous"></script> <script defer src="https://unnamed.website/katex/contrib/auto-render.min.js" crossorigin="anonymous" onload="renderMathInElement(document.body, {delimiters: [{left: '$', right: '$', display: false}, {left: '\\(', right: '\\)', display: false}, {left: '\\[', right: '\\]', display: true}, {left: '\\begin{equation}', right: '\\end{equation}', display: true}, {left: '\\begin{equation*}', right: '\\end{equation*}', display: true}, {left: '\\begin{align}', right: '\\end{align}', display: true}, {left: '\\begin{align*}', right: '\\end{align*}', display: true}]});"></script> <p><em>This post will only use one-based indexing since that&rsquo;s what Fenwick trees traditionally use, although they can also be modified to use zero-based indexing.</em></p> <p>So imagine you have an array $A$ of size $N$, and you&rsquo;d like to support two operations. The first one, called $Update(i, v)$, is trivial: Given an index $i$, add $v$ to $A_i$. Easy peasy!</p>