https://yahel1216.github.io/categories/algorithms/Recent content in Algorithms on Yahel UffenheimerHugoen-USTue, 31 Mar 2026 00:00:00 +0000https://yahel1216.github.io/posts/clarksons-lp-algorithm/Tue, 31 Mar 2026 00:00:00 +0000https://yahel1216.github.io/posts/clarksons-lp-algorithm/<h1 id="clarksons-algorithm-for-linear-programming">Clarkson&rsquo;s Algorithm for Linear Programming</h1> <div class="alert alert-info my-6 overflow-hidden rounded-lg transition-all duration-200 ease-out hover:-translate-y-0.5 hover:shadow-md" style="background-color: color-mix(in srgb, var(--color-) 10%, transparent); border-left-color: var(--color-); --hover-bg: color-mix(in srgb, var(--color-) 15%, transparent);" onmouseover="this.style.backgroundColor = this.style.getPropertyValue('--hover-bg')" onmouseout="this.style.backgroundColor = 'color-mix(in srgb, var(--color-) 10%, transparent)'" role="alert" aria-labelledby="alert-0-title"> <div class=" flex items-center justify-between px-6 py-6" > <div class="flex items-center gap-3"> <h4 id="alert-0-title" class="m-0 font-semibold text-foreground/90"> Prerequisites </h4> </div> </div> <div id="alert-0-content" class="alert-content px-6 pb-6"> <div class="prose prose-sm text-foreground/90 max-w-none"> <ul> <li><strong>Linear Programming:</strong> Feasibility, optimality, and the standard form of an LP.</li> <li><strong>Linear Algebra:</strong> Rank, linear independence, matrix inverses, and spanning sets.</li> <li><strong>Probability:</strong> Expectation and Markov&rsquo;s inequality.</li> </ul> </div> </div> </div><script> function toggleAlert(alertId) { const content = document.getElementById(alertId + '-content'); const chevron = document.getElementById(alertId + '-chevron'); const header = content.previousElementSibling; if (content.classList.contains('hidden')) { content.classList.remove('hidden'); chevron.style.transform = 'rotate(0deg)'; header.setAttribute('aria-expanded', 'true'); } else { content.classList.add('hidden'); chevron.style.transform = 'rotate(-90deg)'; header.setAttribute('aria-expanded', 'false'); } } document.addEventListener('DOMContentLoaded', function() { const collapsedAlerts = document.querySelectorAll('.alert-content.hidden'); collapsedAlerts.forEach(function(content) { const alertId = content.id.replace('-content', ''); const chevron = document.getElementById(alertId + '-chevron'); if (chevron) { chevron.style.transform = 'rotate(-90deg)'; } }); }); </script> <hr> <p>Linear programs arise throughout combinatorial optimization, machine learning, and operations research. In many practical settings the number of constraints $n$ is enormous compared to the ambient dimension $d$ — think of $n = 10^6$ constraints in $d = 50$ dimensions. Standard algorithms like the simplex method take time proportional to $n$ per pivot, and interior-point methods scale as $O(n^{3.5})$ in the worst case. When $n \gg d$, most constraints are redundant: the optimal solution is determined by only $d$ of them.</p>