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[Algorithms II] Week 5-1 Regular Expressions
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-regular-expressions" title="1. Regular Expressions">1. Regular Expressions</a><ul><li><a class="toc-href" href="#regular-expression" title="regular expression">regular expression</a></li></ul></li><li><a class="toc-href" href="#2-res-and-nfas_1" title="2. REs and NFAs">2. REs and NFAs</a><ul><li><a class="toc-href" href="#first-attempt-of-pattern-matching" title="first attempt of pattern matching">first attempt of pattern matching</a></li><li><a class="toc-href" href="#nfa" title="NFA">NFA</a></li></ul></li><li><a class="toc-href" href="#3-nfa-simulation_1" title="3. NFA Simulation">3. NFA Simulation</a><ul><li><a class="toc-href" href="#algorithm" title="algorithm">algorithm</a></li><li><a class="toc-href" href="#concrete-example" title="concrete example">concrete example</a></li><li><a class="toc-href" href="#reachability" title="reachability">reachability</a></li><li><a class="toc-href" href="#java-implementation" title="Java implementation">Java implementation</a></li><li><a class="toc-href" href="#analysis" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#4-nfa-construction_1" title="4. NFA Construction">4. NFA Construction</a><ul><li><a class="toc-href" href="#buiding-a-nfa-from-a-re-parsing" title="buiding a NFA from a re (parsing)">buiding a NFA from a re (parsing)</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#5-regular-expression-applications_1" title="5. Regular Expression Applications">5. Regular Expression Applications</a><ul><li><a class="toc-href" href="#grep" title="grep">grep</a></li><li><a class="toc-href" href="#regexp-in-other-languages" title="regexp in other languages">regexp in other languages</a></li><li><a class="toc-href" href="#harvesting-information" title="Harvesting information">Harvesting information</a></li><li><a class="toc-href" href="#not-so-regular-expressions" title="Not-so-regular expressions">Not-so-regular expressions</a></li><li><a class="toc-href" href="#summary" title="Summary">Summary</a></li></ul></li></ul></div>
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<h1 id="1-regular-expressions">1. Regular Expressions</h1>
<p>pb: pattern matching. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image.png"/> </p>
<h3 id="regular-expression">regular expression</h3>
<p>Is a notation to specify a set of strings. <br/>
basic operations: </p>
<ul>
<li>concatenation </li>
<li>or </li>
<li>closure: "0 or more appearances of chars" </li>
<li>parentheses </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image001.png"/><br/>
additional operations (<em>added for convinence</em>):<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image002.png"/><br/>
ex. <code>[A-C]+</code> is equivalent to <code>(A|B|C)(A|B|C)*</code>. </p>
<p>吐槽名句: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image003.png"/> </p>
<h1 id="2-res-and-nfas_1">2. REs and NFAs</h1>
<p><strong>duality</strong> between RE and DFA: </p>
<ul>
<li>RE: to decribe a set of strings. </li>
<li>DFA: machine to ecognize whether a string is in a given set. </li>
</ul>
<p><strong>[Kleene's therom]</strong> </p>
<blockquote>
<p>For any DFA, there exists a RE that describes the same set of strings;<br/>
For any RE, there exists a DFA that recognizes the same set of strings. </p>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image004.png"/> </p>
<h3 id="first-attempt-of-pattern-matching">first attempt of pattern matching</h3>
<p>(Ken Tompson) same as KMP — no backup.<br/>
basic plan: </p>
<ul>
<li>construct the DFA </li>
<li>simulate the DFA with text </li>
</ul>
<p>bad news: DFA may have exponential nb of states. <br/>
⇒ change to <strong>NFA</strong> (nondeterministic finite automaton). </p>
<h3 id="nfa">NFA</h3>
<blockquote></blockquote>
<ul>
<li>put RE into parentheses </li>
<li>every <em>char</em> as a state (start=0, success=M) — 这里和之前的DFA很不一样: 之前是每个transition(edge)关联一个char, 这里是每个状态(node)关联一个char. </li>
<li><strong>epsilon-transition</strong> (red links below): change of machine state <em>without scanning text</em> </li>
<li><strong>match-transition</strong> (black links below): change state, but also have to scan next char in text, <em>match transition is added after each alphabetic char</em> </li>
<li>success (accept) if <em>any sequence</em> of transitions (after scanning all text) end at state-M. </li>
</ul>
<p>亦可理解为, DFA是每一条边对应一个可能的(字母表内的)char, 而NFA只有match-transition对应于pattern里的(alphabetic) char, 其他epsilon transition的边对应空字符串(也就是epsilon string). <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image005.png"/><br/>
example:<br/>
is "AAAABD" a match ?<br/>
→ yes. (和上一节substring的插图进行一下比对, 还是有很大不同)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image006.png"/> </p>
<p>pb: non-determinism<br/>
How to determine whether a string is a match of a NFA (ie. <em>how to select the right sequence</em> of transition) ? <br/>
⇒ sysematically consider <em>all</em> possible transition sequences. </p>
<h1 id="3-nfa-simulation_1">3. NFA Simulation</h1>
<ul>
<li>state names: 0 to M. (M+1 states in total, M=length of RE string). </li>
<li>match-transitions: store in array <code>re[]</code> (the match transitions are naturally in order of the array). </li>
<li>epsilon -transitions: store in a digraph <code>G</code> </li>
</ul>
<p>idea: </p>
<blockquote>
<p>maintain a <em>set</em> of all state that NFA could be in after reading first i chars in text. </p>
</blockquote>
<p>at each iteration: check all reachable state wrt the transitions, then update reachable states. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image007.png"/> </p>
<h3 id="algorithm">algorithm</h3>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image008.png"/><br/>
(for the NFA above, 注意为了方便已经加了 必要的括号) </p>
<p><strong>[Algo]</strong> </p>
<blockquote>
<ul>
<li>initial: <code>rs</code>(reachable state)=reachable state from state 0 (left parenthese) using epsilon trantisions </li>
<li>
<p>consume a char <em>in text</em>: </p>
</li>
<li>
<p><code>nrs</code> (<em>new-reachable-states</em>) = empty set </p>
</li>
<li>from all reachable state of this character: add next state using the match-transition to <code>nrs</code> </li>
<li>add all reachable states (using epsilon transition) form the <code>nrs</code> set to <code>nrs</code> </li>
<li>set <code>rs = nrs</code>, and consume the next char in text </li>
<li>accept if at the end the state M is in <code>rs</code> </li>
</ul>
</blockquote>
<h3 id="concrete-example">concrete example</h3>
<ul>
<li>init: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image010.png"/> </p>
<ul>
<li>when <em>matching A</em> from text: state 2 or 6 </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image011.png"/><br/>
using match transition of A, we can get to state 3 or 7<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image012.png"/><br/>
if we add epsilon transitions: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image013.png"/><br/>
so reachable states after reading 1st A are: 2, 3, 4, 7<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image014.png"/> </p>
<ul>
<li>matching 2nd A from text: state 2 </li>
</ul>
<p>using match transition we can only get to state 3. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image015.png"/><br/>
using epsilon transitions from state 3: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image016.png"/><br/>
(<em>the only state after matching A is state 2 3 4</em>) </p>
<ul>
<li>etc... </li>
</ul>
<p>或者直接看这张图: </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image017.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image018.png"/> </p>
<h3 id="reachability">reachability</h3>
<p>All reachable vertices from <em>a set of</em> source vertices → just DFS. <br/>
⇒ directly use the API from the digraph section: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image019.png"/><br/>
running time linear to E+V </p>
<h3 id="java-implementation">Java implementation</h3>
<p>API: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class NFA{ </span></span>
<span class="code-line"><span class="err"> private int M; </span></span>
<span class="code-line"><span class="err"> private char[] re; </span></span>
<span class="code-line"><span class="err"> private Digraph G;// digraph of the epsilon-transitions </span></span>
<span class="code-line"><span class="err"> public NFA(String regexp){ </span></span>
<span class="code-line"><span class="err"> M = regexp.length(); </span></span>
<span class="code-line"><span class="err"> re = regexp.toCharArray(); </span></span>
<span class="code-line"><span class="err"> G = buildEpsilonTransitionGraph();// helper function to build the graph G </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> public boolean matches(String text);// does text match the regexp? </span></span>
<span class="code-line"><span class="err"> private Digraph buildEpsilonTransitionGraph();// private helper function </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>The function <code>buildEpsilonTransitionGraph()</code>will be attacked in next session, for now we focus on the NFA simulation code — that is, the <code>mathes()</code> method. </p>
<p>For simplicity let's assume we have a function <code>reachableVertices(Digraph G, Bag<Integer> sourceSet)</code> and <code>reachableVertices(Digraph G, int source)</code> that gives the reachable states from (a set of) source vertices, including the sources. Or we can directly use the <code>DirectedDFS</code> api as listed above. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="k">boolean</span><span class="w"> </span><span class="n">matches</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="nc">text</span><span class="p">)</span><span class="err">{</span><span class="w"> </span><span class="o">//</span><span class="n">does</span><span class="w"> </span><span class="nc">text</span><span class="w"> </span><span class="k">match</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">regexp</span><span class="vm">?</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Bag</span><span class="o"><</span><span class="k">Integer</span><span class="o">></span><span class="w"> </span><span class="n">rechableStates</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">reachableVertices</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">init</span><span class="w"> </span><span class="n">reachable</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">char</span><span class="w"> </span><span class="nl">c</span><span class="p">:</span><span class="w"> </span><span class="nc">text</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Bag</span><span class="o"><</span><span class="k">Integer</span><span class="o">></span><span class="w"> </span><span class="n">newRechableStatesBymatch</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Bag</span><span class="o"><</span><span class="k">Integer</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="nl">i</span><span class="p">:</span><span class="n">rechableStates</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="n">c</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">'.'</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">newRechableStatesBymatch</span><span class="p">.</span><span class="k">add</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="k">match</span><span class="w"> </span><span class="n">transition</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">rechableStates</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">reachableVertices</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">newRechableStates</span><span class="p">);</span><span class="o">//</span><span class="n">epsilon</span><span class="w"> </span><span class="n">transition</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">reachableStates</span><span class="p">.</span><span class="k">contains</span><span class="p">(</span><span class="n">M</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>
<p>(代码虽然短但是这个过程我理解了好久.. 另外上面的代码有点伪). </p>
<h3 id="analysis">Analysis</h3>
<p><strong>prop</strong>. the matches() method takes <code>O(MN)</code> time in worst case. </p>
<p>pf. N chars in text, each char can go through <= M states (DFS), and in the digraph, no node has >3 degree ⇒ <em>number of edges <= 3M</em>, so the time for each dfs is O(M), in total we have O(MN). </p>
<h1 id="4-nfa-construction_1">4. NFA Construction</h1>
<p>→ construct the epsilon transition digraph. </p>
<h3 id="buiding-a-nfa-from-a-re-parsing">buiding a NFA from a re (parsing)</h3>
<ul>
<li>states in a NFA: one state per char, plus an accept state (state M) </li>
<li>alphabet state: chars in alphabet (<code>A, B, C, D</code>) → (implicitly) put a match transition to next state </li>
<li>metacharacters: <code>( ) . * |</code> , 5 metacharacters in total </li>
</ul>
<p>⇒ to deal with the metacharacters: </p>
<ul>
<li><strong>paretheses </strong><code>( )</code> </li>
</ul>
<p>simply put a epsilon-transition to the next state<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image020.png"/> </p>
<ul>
<li><strong>closure </strong><code>*</code> </li>
</ul>
<p>星号前面只可能是字母(包括<code>.</code>)或者右括号<code>)</code>, 所以分两种情况讨论一下, 需要向前看一位, 这里就比较subtle<br/>
for each <code>*</code> state, add 3 transitions as below: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image021.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image022.png"/> </p>
<ul>
<li><strong>or</strong> <code>|</code> </li>
</ul>
<p>or符号肯定在一个括号里面<br/>
add 2 epsilon transitions wrt parethese: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image023.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image024.png"/> </p>
<p>以上就是NFA建立G的时候要处理的三种情形, 这三种情形都要知道一个左括号(<code>lp</code>)的位置 ⇒ use a stack ! </p>
<h3 id="implementation">implementation</h3>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image025.png"/> </p>
<ul>
<li>for alphabetic chars: do one-char <em>lookahead</em> → if next is <code>*</code>, add transitions. </li>
<li>for left parenthese <code>(</code>: add transition to next state, and push to stack </li>
<li>for or <code>|</code>: add transition to next state, and push to stack </li>
<li>for right parenthese <code>)</code>: pop the stack to deal with or and lp; and also do lookahead. </li>
</ul>
<p>code is not trival... look carefully: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="n">Digraph</span><span class="w"> </span><span class="n">buildEpsilonTransitionGraph</span><span class="p">()</span><span class="err">{</span><span class="o">//</span><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">helper</span><span class="w"> </span><span class="k">function</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Digraph</span><span class="w"> </span><span class="n">G</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Digraph</span><span class="p">(</span><span class="n">M</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Stack</span><span class="o"><</span><span class="k">Integer</span><span class="o">></span><span class="w"> </span><span class="n">stk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Stack</span><span class="o"><</span><span class="k">Integer</span><span class="o">></span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">lp</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">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">M</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">'|'</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">'('</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">stk</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">i</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">'('</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">')'</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">'*'</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">')'</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="w"> </span><span class="n">need</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="n">until</span><span class="w"> </span><span class="k">get</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">lp</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stk</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">j</span><span class="o">]==</span><span class="s1">'|'</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">lp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stk</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="ow">or</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">lp</span><span class="p">,</span><span class="w"> </span><span class="ow">or</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="k">add</span><span class="w"> </span><span class="n">edge</span><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="err">`</span><span class="ow">or</span><span class="err">`</span><span class="w"> </span><span class="k">case</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="ow">or</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="n">lp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</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>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">do</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="nl">lookahead</span><span class="p">:</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i+1</span><span class="o">]==</span><span class="s1">'*'</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">re</span><span class="o">[</span><span class="n">i</span><span class="o">]==</span><span class="s1">')'</span><span class="p">)</span><span class="err">{</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="k">case</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nl">closure</span><span class="p">:</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">rp</span><span class="w"> </span><span class="k">before</span><span class="w"> </span><span class="err">`</span><span class="o">*</span><span class="err">`</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">lp</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">lp</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">else</span><span class="err">{</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="k">case</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nl">closure</span><span class="p">:</span><span class="w"> </span><span class="n">an</span><span class="w"> </span><span class="n">alphabetic</span><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="k">before</span><span class="w"> </span><span class="err">`</span><span class="o">*</span><span class="err">`</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">addEdge</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">i</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>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="o">//</span><span class="w"> </span><span class="k">go</span><span class="w"> </span><span class="n">through</span><span class="w"> </span><span class="k">each</span><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">re</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">G</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="analysis_1">Analysis</h3>
<p><strong>prop</strong>. building an NFA takes linear time and space in M. <br/>
pf. for each char, the nb of operations is const. </p>
<p>真不愧是most ingenius algorithm we met in this course...... </p>
<h1 id="5-regular-expression-applications_1">5. Regular Expression Applications</h1>
<h3 id="grep">grep</h3>
<p>"<strong>G</strong>eneralized <strong>R</strong>egular <strong>E</strong>xpression <strong>P</strong>rint"<br/>
print out all lines (from stdin) having a substring of an RE.<br/>
⇒ equal to adding a <code>.*</code> to the beginning and end of the RE to make a match. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class GREP{ </span></span>
<span class="code-line"><span class="err"> public void main(String[] args){ </span></span>
<span class="code-line"><span class="err"> String re = ".*"+args[0]+".*"; </span></span>
<span class="code-line"><span class="err"> NFA nfa = new NFA(re); </span></span>
<span class="code-line"><span class="err"> while(StdIn.hasNextLine){ </span></span>
<span class="code-line"><span class="err"> String line = StdIn.readLine(); </span></span>
<span class="code-line"><span class="err"> if(nfa.matches(line)) StdOut.println(line); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>the grep has NM worst case running time — same as brute force substring search — amazing... </p>
<p>grep application: crossword puzzles<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image026.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image027.png"/> </p>
<h3 id="regexp-in-other-languages">regexp in other languages</h3>
<ul>
<li>unix: grep, awk </li>
<li>script: python, <em>perl</em> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image028.png"/> </p>
<ul>
<li>java: <code>String.matches(regexp)</code> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image029.png"/> </p>
<h3 id="harvesting-information">Harvesting information</h3>
<p>goal: print <em>all substrings</em> of input that match an RE. <br/>
use <code>Pattern</code> and <code>Matcher</code> class in <code>java.util.regexp</code>. <br/>
first compile the regexp, then build the matcher<br/>
→ so that we can iterate through all matches of the input using <code>find()</code> and <code>group()</code> of the matcher <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image030.png"/> </p>
<p><em>Caveat</em>: performance NOT guaranteed ! <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image031.png"/><br/>
→ exponential time growth! <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image032.png"/> </p>
<h3 id="not-so-regular-expressions">Not-so-regular expressions</h3>
<p>"not rugular" means Kleene's Th doesn't hold <br/>
→ efficient performance not tractable......<br/>
<strong>back-reference</strong><br/>
<code>\1</code> matches <em>subexpressions</em> that was matched earliser<br/>
limitations of regular languages: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image034.png"/> </p>
<h3 id="summary">Summary</h3>
<p>the substring and regexp are examples of compilers ! (from string to a NFA/DFA/bytecode)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image033.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_1/pasted_image035.png"/> </p>
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<div id="toc"><ul><li><a class="toc-href" href="#1-regular-expressions" title="1. Regular Expressions">1. Regular Expressions</a><ul><li><a class="toc-href" href="#regular-expression" title="regular expression">regular expression</a></li></ul></li><li><a class="toc-href" href="#2-res-and-nfas_1" title="2. REs and NFAs">2. REs and NFAs</a><ul><li><a class="toc-href" href="#first-attempt-of-pattern-matching" title="first attempt of pattern matching">first attempt of pattern matching</a></li><li><a class="toc-href" href="#nfa" title="NFA">NFA</a></li></ul></li><li><a class="toc-href" href="#3-nfa-simulation_1" title="3. NFA Simulation">3. NFA Simulation</a><ul><li><a class="toc-href" href="#algorithm" title="algorithm">algorithm</a></li><li><a class="toc-href" href="#concrete-example" title="concrete example">concrete example</a></li><li><a class="toc-href" href="#reachability" title="reachability">reachability</a></li><li><a class="toc-href" href="#java-implementation" title="Java implementation">Java implementation</a></li><li><a class="toc-href" href="#analysis" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#4-nfa-construction_1" title="4. NFA Construction">4. NFA Construction</a><ul><li><a class="toc-href" href="#buiding-a-nfa-from-a-re-parsing" title="buiding a NFA from a re (parsing)">buiding a NFA from a re (parsing)</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#5-regular-expression-applications_1" title="5. Regular Expression Applications">5. Regular Expression Applications</a><ul><li><a class="toc-href" href="#grep" title="grep">grep</a></li><li><a class="toc-href" href="#regexp-in-other-languages" title="regexp in other languages">regexp in other languages</a></li><li><a class="toc-href" href="#harvesting-information" title="Harvesting information">Harvesting information</a></li><li><a class="toc-href" href="#not-so-regular-expressions" title="Not-so-regular expressions">Not-so-regular expressions</a></li><li><a class="toc-href" href="#summary" title="Summary">Summary</a></li></ul></li></ul></div>
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