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[Algorithms I] Week 1-2 Analysis of Algorithms
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction" title="1. Introduction">1. Introduction</a></li><li><a class="toc-href" href="#2-observations" title="2. Observations">2. Observations</a></li><li><a class="toc-href" href="#3mathematical-models" title="3.Mathematical Models">3.Mathematical Models</a></li><li><a class="toc-href" href="#4-order-of-growth-classification" title="4. Order of Growth Classification">4. Order of Growth Classification</a></li><li><a class="toc-href" href="#5-theory-of-algorithms" title="5. Theory of Algorithms">5. Theory of Algorithms</a></li><li><a class="toc-href" href="#6-memory" title="6. Memory">6. Memory</a></li></ul></div>
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<h1 id="1-introduction">1. Introduction</h1>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image.png"/></p>
<h1 id="2-observations">2. Observations</h1>
<p>ex. <strong>3-SUM pb</strong><br/>
<em>given N distinct numbers, how many triples sum up to 0? (pb related to computatioal geogtry)</em></p>
<ul>
<li>brute force method:</li>
</ul>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">i</span><span class="o"><</span><span class="n">N</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span><span class="n">j</span><span class="o"><</span><span class="n">N</span><span class="p">;</span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">k</span><span class="o">=</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span><span class="n">k</span><span class="o"><</span><span class="n">N</span><span class="p">;</span><span class="n">k</span><span class="o">++</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="err">{</span><span class="k">if</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">]+</span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">]+</span><span class="n">a</span><span class="o">[</span><span class="n">k</span><span class="o">]==</span><span class="mi">0</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="nf">count</span><span class="o">++</span><span class="p">;</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"></span></span>
</pre></div>
<ul>
<li>mesuring running time:</li>
</ul>
<p>stdlib.jar里面提供了一个<code>Stopwatch</code>类用于记录运行时间. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image001.png"/></p>
<ul>
<li>log-log plot</li>
</ul>
<p>T(N) = running time for input of size N<br/>
log(N)-log(T(N)) plot:<br/>
often get a straight line — power law <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image002.png"/></p>
<ul>
<li><strong>doubling ratio</strong>:</li>
</ul>
<p>(for checking the power law relationship, checking the power order)<br/>
each time double the size of input, then take log of the time ratio of 2 runs: log( T(2N)/T(N) ) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image003.png"/> </p>
<h1 id="3mathematical-models">3.Mathematical Models</h1>
<ul>
<li>total running time: sum of cost*frequency of operations </li>
<li>
<p>cost of some basic operations:</p>
<ul>
<li>array allocation: c*N (because all array entries have to be set to 0/false/null)</li>
<li>string concatenation: c*N (proportional to the length of string !)</li>
</ul>
</li>
<li>
<p>simplification</p>
</li>
</ul>
<p>crude analysis<br/>
ignore lower terms <strong>tilde notation</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image005.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image004.png"/></p>
<ul>
<li>estimating discrete sum by relaxation</li>
</ul>
<p>Replace the sum with an integral, and use calculus — 很机智... <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image006.png"/></p>
<h1 id="4-order-of-growth-classification">4. Order of Growth Classification</h1>
<p>(discard the leading coefficient when considering the growth order)</p>
<ul>
<li>only a small set of growth functions: </li>
</ul>
<p><code>1, logN, N, NlogN, N^2, N^3, 2^N</code> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image007.png"/> </p>
<ul>
<li>exemples:</li>
</ul>
<p>binary search ⇒ logN<br/>
divide and conquer ⇒ NlogN<br/>
exhaustive search ⇒ 2^N <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image008.png"/></p>
<p>practical performance: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image009.png"/> </p>
<ul>
<li>
<p>ex. <strong>binary search</strong></p>
<p>public int binearch(int arr[], int key){//arr[] already sorted
int lo=0,hi=arr.length;
while(i<j){
int m = (lo+hi)/2;
if(arr[m]==key) return m;
else if(arr[m]<key) lo=m+1;
else hi=m-1;
}
return -1;
}</p>
</li>
</ul>
<p>(→ Bug in Java's Arrays.binarySearch() discovered in 2006......) <br/>
→ invariant: if key in arr, arr[lo]<=key<=arr[hi] <br/>
<strong>proposition. </strong>binary search uses at most logN+1 compares to search a sorted array of size N.
<strong>pf. </strong>
denote <em>T(N)</em> := nb of compares for array with size <=N<br/>
→ T(1)=1<br/>
→ recurrence relation: T(N)<=T(N/2)+1<br/>
⇒ T(N)=logN </p>
<ul>
<li><strong>a faster 3-SUM</strong></li>
</ul>
<p>→ first sort the array <em>(~NlogN)</em><br/>
→ for any pair a[i] and a[j], do binary search for -(a[i]+a[j]) <em>~(N2LogN)</em><br/>
⇒ reduce from N3 to N2logN ! (for 8k numbers, running time goes from 51s to 0.96s) </p>
<h1 id="5-theory-of-algorithms">5. Theory of Algorithms</h1>
<ul>
<li>
<p>types of analysis</p>
<p>-best case
-worst case
-average case(random input, "expected cost")</p>
</li>
<li>
<p>notations</p>
</li>
</ul>
<p><strong>big Theta/big O/big Omega</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image010.png"/> <br/>
- big O: <em>upper bound → * once a specific algo is found, find an upper bound<br/>
- big Omega: </em>lower bound <em>→ proove that no algo can do better<br/>
- big Theta: symptotic growth (same order, optimal algo) → lower and upper bound </em>match* <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image011.png"/> <br/>
⇒ in this course: use tilde notation: contain leading constants for highest order term</p>
<h1 id="6-memory">6. Memory</h1>
<p>KB: 2^10 bytes<br/>
MB: 2^20 bytes (1 million)
GB: 2^30 bytes (1 billion)
64-bit machines: <em>8 byte pointers</em> </p>
<ul>
<li>typical memory usage:</li>
</ul>
<p>for primary types: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image012.png"/><br/>
for arrays (with <em>array overhead=24bytes</em>) : <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image013.png"/> <br/>
<em>Obj overhead: 16 bytes</em> (obj的大小=16+obj内部filed的大小)<br/>
<em>references</em>: 8 bytes (ex. inner class has a ref to encolsing class)<br/>
<em>padding</em>: each obj uses a multiply of 8 bytes (obj大小=8 bytes的整数倍) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image014.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_2/pasted_image015.png"/> </p>
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