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[Algorithms II] Week 6-1 Reductions
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-reductions" title="1. Introduction to Reductions">1. Introduction to Reductions</a></li><li><a class="toc-href" href="#2-designing-algorithms" title="2. Designing Algorithms">2. Designing Algorithms</a><ul><li><a class="toc-href" href="#ex1-convex-hull-reduces-to-sorting" title="ex1. convex hull reduces to sorting">ex1. convex hull reduces to sorting</a></li><li><a class="toc-href" href="#ex2-undirected-shortest-path-nonneg-weights-reduces-to-directed-shortest-path" title="ex2. undirected shortest path (nonneg weights) reduces to directed shortest path">ex2. undirected shortest path (nonneg weights) reduces to directed shortest path</a></li></ul></li><li><a class="toc-href" href="#3-establishing-lower-bounds_1" title="3. Establishing Lower Bounds">3. Establishing Lower Bounds</a><ul><li><a class="toc-href" href="#ex-proof-of-lower-bound-for-convex-hull" title="ex. proof of lower bound for convex hull">ex. proof of lower bound for convex hull</a></li></ul></li><li><a class="toc-href" href="#4-classifying-problems_1" title="4. Classifying Problems">4. Classifying Problems</a><ul><li><a class="toc-href" href="#ex-integer-arithmetic-reductions-integer-multiplication" title="ex. integer arithmetic reductions: integer multiplication">ex. integer arithmetic reductions: integer multiplication</a></li><li><a class="toc-href" href="#ex-linear-algebra-reductions-matrix-multiplication" title="ex. linear-algebra reductions: matrix multiplication">ex. linear-algebra reductions: matrix multiplication</a></li><li><a class="toc-href" href="#summary" title="summary">summary</a></li></ul></li></ul></div>
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<p>Goal: classify problems according to computational requirements. <br/>
bad new: for huge number of pbs we don't know... </p>
<h1 id="1-introduction-to-reductions">1. Introduction to Reductions</h1>
<p>shifing gears: </p>
<ul>
<li>from individual problems to problem-solving models. </li>
<li>from linear/quard to polynomial/exponential pbs </li>
<li>from implementation details to conceptual framwork </li>
</ul>
<p>suppose we could (not) solve pb X efficiently <br/>
⇒ what else pbs could (not) we solve efficiently ? </p>
<blockquote>
<p>def. reduction<br/>
Pb X <strong>reduces to</strong> pb Y if you can use an algo that solves Y to solve X. </p>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image.png"/><br/>
for an instance of pb X → transform it into an instance of pb Y → translate the solution for Y to solution for X.<br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image001.png"/> </p>
<p>ex1. finding median can reduce to sorting... cost = NlogN+1<br/>
ex1. element distinctness can reduce to sorting... cost = NlogN + N </p>
<h1 id="2-designing-algorithms">2. Designing Algorithms</h1>
<p>algo design: by reduction to problems that we know how to solve (sorting/shortest path/flow/...) </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image002.png"/> </p>
<h3 id="ex1-convex-hull-reduces-to-sorting">ex1. convex hull reduces to sorting</h3>
<p>Gram scan algo... (discussed in algo-I course)<br/>
cost = NlogN + N<br/>
algo. Gram scan </p>
<blockquote>
<ul>
<li>pick a point with smallest y-coord </li>
<li>sort all points by polar angle wrt the picked point </li>
<li>consider points in this order, discard points that creates clockwise turn </li>
</ul>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image003.png"/> </p>
<h3 id="ex2-undirected-shortest-path-nonneg-weights-reduces-to-directed-shortest-path">ex2. undirected shortest path (nonneg weights) reduces to directed shortest path</h3>
<p>cost: ElogV + E<br/>
algo. replace each undir-edge by 2 dir-edge... </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image004.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image005.png"/> </p>
<h1 id="3-establishing-lower-bounds_1">3. Establishing Lower Bounds</h1>
<p>goal: prove that a pb requires (at least) a certain nb of steps. </p>
<p>ex. any compare-based sorting requires NlogN compares. log(N!) = NlogN </p>
<p>Bad news: very hard to estibalish lower bounds.<br/>
Good new: can spread the lower bound NlogN by reducing to sorting (if cost of reduction is small). </p>
<blockquote>
<p>def. linear-time reduction<br/>
pb X linear-time reduces to pb Y if X can be solved with: <br/>
1. linear nb of op for reduction<br/>
2. constant nb of calles to Y </p>
</blockquote>
<p>ex. almost all reductions we've seen so far... </p>
<h3 id="ex-proof-of-lower-bound-for-convex-hull">ex. proof of lower bound for convex hull</h3>
<p>prop. sorting linear-time reduces to convex hull <br/>
(注意这次是反向的! )<br/>
pf. <br/>
for an instance of sorting: x1 ... xn<br/>
⇒ convert to convex hull instance: <em>(x1, x1^2), ... , (xn, xn^2)</em><br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image006.png"/> </p>
<p>⇒ implication: all (ccw-based) convex hull algo cannot be easier than NlgN ! (otherwise sorting would be easier..) </p>
<p>lesson: Establishing lower bounds through reduction is an important tool in guiding algorithm design efforts.<br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image007.png"/> </p>
<h1 id="4-classifying-problems_1">4. Classifying Problems</h1>
<p>prove that pb X and pb Y have the same complexity: </p>
<ul>
<li>show X linear-time reduces to Y </li>
<li>show Y linear-time reduces to X </li>
<li>conclude that X Y have the same complexity (even if we don't know what it is) </li>
</ul>
<p>ex. sorting and convex hull... <br/>
一个囧囧的脑洞: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image008.png"/> </p>
<h3 id="ex-integer-arithmetic-reductions-integer-multiplication">ex. integer arithmetic reductions: integer multiplication</h3>
<p>integer multiplication: of two N-bit integers. <br/>
Its complexity (unknown) is denoted as M(N)<br/>
brute force: N^2 ops → so M(N) = Omega(N2)<br/>
many other integer ops can reduce to integer multiplication: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image009.png"/><br/>
what is M(N)?<br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image010.png"/> </p>
<h3 id="ex-linear-algebra-reductions-matrix-multiplication">ex. linear-algebra reductions: matrix multiplication</h3>
<p>compute product of 2 N*N matrices. <br/>
Its complexity (unknown) is denoted as MM(N)<br/>
brute force: N^3<br/>
operations that can reduce to matrix-multiplication:<br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image011.png"/><br/>
what is MM(N)?<br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_1_reductions/pasted_image012.png"/> </p>
<h3 id="summary">summary</h3>
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-reductions" title="1. Introduction to Reductions">1. Introduction to Reductions</a></li><li><a class="toc-href" href="#2-designing-algorithms" title="2. Designing Algorithms">2. Designing Algorithms</a><ul><li><a class="toc-href" href="#ex1-convex-hull-reduces-to-sorting" title="ex1. convex hull reduces to sorting">ex1. convex hull reduces to sorting</a></li><li><a class="toc-href" href="#ex2-undirected-shortest-path-nonneg-weights-reduces-to-directed-shortest-path" title="ex2. undirected shortest path (nonneg weights) reduces to directed shortest path">ex2. undirected shortest path (nonneg weights) reduces to directed shortest path</a></li></ul></li><li><a class="toc-href" href="#3-establishing-lower-bounds_1" title="3. Establishing Lower Bounds">3. Establishing Lower Bounds</a><ul><li><a class="toc-href" href="#ex-proof-of-lower-bound-for-convex-hull" title="ex. proof of lower bound for convex hull">ex. proof of lower bound for convex hull</a></li></ul></li><li><a class="toc-href" href="#4-classifying-problems_1" title="4. Classifying Problems">4. Classifying Problems</a><ul><li><a class="toc-href" href="#ex-integer-arithmetic-reductions-integer-multiplication" title="ex. integer arithmetic reductions: integer multiplication">ex. integer arithmetic reductions: integer multiplication</a></li><li><a class="toc-href" href="#ex-linear-algebra-reductions-matrix-multiplication" title="ex. linear-algebra reductions: matrix multiplication">ex. linear-algebra reductions: matrix multiplication</a></li><li><a class="toc-href" href="#summary" title="summary">summary</a></li></ul></li></ul></div>
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