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[Algorithms II] Week 2-2 Shortest Paths
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-shortest-paths-apis" title="1. Shortest Paths APIs">1. Shortest Paths APIs</a><ul><li><a class="toc-href" href="#shortest-path-variants" title="shortest path variants">shortest path variants</a></li><li><a class="toc-href" href="#apis" title="APIs">APIs</a></li></ul></li><li><a class="toc-href" href="#2-shortest-path-properties_1" title="2. Shortest path properties">2. Shortest path properties</a><ul><li><a class="toc-href" href="#edge-relaxation" title="Edge relaxation">Edge relaxation</a></li><li><a class="toc-href" href="#optimality-conditions" title="optimality conditions">optimality conditions</a></li><li><a class="toc-href" href="#generic-algo" title="Generic algo">Generic algo</a></li></ul></li><li><a algorithm'="" class="toc-href" href="#3-dijkstras-algorithm_1" s="" title="3. Dijkstra">3. Dijkstra's Algorithm</a><ul><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#4-edge-weighted-dags_1" title="4. Edge-Weighted DAGs">4. Edge-Weighted DAGs</a><ul><li><a class="toc-href" href="#correctness" title="correctness">correctness</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#application" title="application">application</a></li></ul></li><li><a class="toc-href" href="#5-negative-weights_1" title="5. Negative Weights">5. Negative Weights</a><ul><li><a class="toc-href" href="#bellman-ford" title="Bellman-Ford">Bellman-Ford</a></li><li><a class="toc-href" href="#improvement" title="improvement">improvement</a></li><li><a class="toc-href" href="#finding-negative-cycle" title="Finding negative cycle">Finding negative cycle</a></li><li><a class="toc-href" href="#negative-cycle-application" title="negative cycle application">negative cycle application</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="Summery">Summery</a></li></ul></div>
</div>
</div>
<h1 id="1-shortest-paths-apis">1. Shortest Paths APIs</h1>
<p>context: <em>directe, weighted graphs</em>. </p>
<h3 id="shortest-path-variants">shortest path variants</h3>
<p>in terms of vertices: </p>
<ul>
<li>source-sink: form one vertex to another </li>
<li><strong>single source</strong>: from one vertex to all others (considered in this lecture) </li>
<li>all pairs </li>
</ul>
<p>constraints on edge weights: </p>
<ul>
<li>nonnegative weights </li>
<li>arbitary weights </li>
<li>eculidean </li>
</ul>
<p>cycles: </p>
<ul>
<li>no directed cycles </li>
<li>no negative cycles </li>
</ul>
<h3 id="apis">APIs</h3>
<p>for weighted directed edge: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class DirectedEdge{ </span></span>
<span class="code-line"><span class="err"> DirectedEdge(int v, int w, double weight); </span></span>
<span class="code-line"><span class="err"> int from(); </span></span>
<span class="code-line"><span class="err"> int to(); </span></span>
<span class="code-line"><span class="err"> double weight(); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>for edge-weighted digraph: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class EdgeWeightedDigraph{ </span></span>
<span class="code-line"><span class="err"> private final Bag<DirectedEdge>[] adj; </span></span>
<span class="code-line"><span class="err"> EdgeWeightedDigraph(int V); </span></span>
<span class="code-line"><span class="err"> void addEdge(DirectedEdge e); </span></span>
<span class="code-line"><span class="err"> Iterable<DirectedEdge> adj(int v); </span></span>
<span class="code-line"><span class="err"> int V(); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>for single source shortest-path: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class SP{ </span></span>
<span class="code-line"><span class="err"> SP(EdgeWeightedDigraph G, int s);//s is the source node </span></span>
<span class="code-line"><span class="err"> double distTo(int v);//dist from s to v </span></span>
<span class="code-line"><span class="err"> Iterable<DirectedEdge> pathTo(int v);// shortest path from s to v </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h1 id="2-shortest-path-properties_1">2. Shortest path properties</h1>
<p>goal: single-source shortest path <br/>
<strong>prop</strong>. a <strong>shortest-path-tree</strong> (SPT) exists. <br/>
这个结论以前没见过... 不太evident吧... 没有太想明白 <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image.png"/> <br/>
⇒ consequence: can represent this SPT by 2 vertex-indexted arrays: </p>
<ul>
<li><code>double distTo[v]</code> = shortest path length s → v </li>
<li><code>DirectedEdge edgeTo[v]</code> = last edge to v in shortest path, <code>edgeTo[s] = null</code> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image001.png"/> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">distTo</span><span class="err">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">V</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="n">DirectedEdge</span><span class="w"> </span><span class="n">edgeTo</span><span class="err">[]</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">DirectedEdge</span><span class="o">[</span><span class="n">V</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="k">public</span><span class="w"> </span><span class="k">double</span><span class="w"> </span><span class="n">distTo</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">v</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">return</span><span class="w"> </span><span class="n">this</span><span class="p">.</span><span class="n">distTo</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="k">public</span><span class="w"> </span><span class="n">Iterable</span><span class="o"><</span><span class="n">DirectedEdge</span><span class="o">></span><span class="w"> </span><span class="n">pathTo</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">v</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">Stack</span><span class="o"><</span><span class="n">DirectedEdge</span><span class="o">></span><span class="w"> </span><span class="k">path</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="n">DirectedEdge</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="n">DirectedEdge</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">this</span><span class="p">.</span><span class="n">edgeTo</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="n">e</span><span class="o">!=</span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">edgeTo</span><span class="o">[</span><span class="n">e.from</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">path</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">e</span><span class="p">);</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="k">path</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="edge-relaxation">Edge relaxation</h3>
<p>dynamic prog: </p>
<ul>
<li>distTo[v] = length of <strong>known</strong> shortest path from s to v </li>
<li>distTo[w] = length of <strong>known</strong> shortest path from s to w </li>
<li>edgeTo[w] = last edge in the <strong>known</strong> shortest path form s to w </li>
</ul>
<p>(consider edges one by one) <br/>
<strong>def</strong>. edge e = v-->w <strong>relaxes</strong> if e.weight+distTo[v] < distTo[w]. → update distTo[w] and edgeTo[w]. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image003.png"/> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">relax</span><span class="p">(</span><span class="n">DirectedEdge</span><span class="w"> </span><span class="n">e</span><span class="p">)</span><span class="err">{</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">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</span><span class="p">.</span><span class="k">from</span><span class="p">(),</span><span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</span><span class="p">.</span><span class="k">to</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">distTo</span><span class="o">[</span><span class="n">w</span><span class="o">]></span><span class="n">distTo</span><span class="o">[</span><span class="n">v</span><span class="o">]+</span><span class="n">e</span><span class="p">.</span><span class="n">weight</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">distTo</span><span class="o">[</span><span class="n">w</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">distTo</span><span class="o">[</span><span class="n">v</span><span class="o">]+</span><span class="n">e</span><span class="p">.</span><span class="n">weight</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">edgeTo</span><span class="o">[</span><span class="n">w</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</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="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="optimality-conditions">optimality conditions</h3>
<p><strong>prop</strong>. optimality conditions <br/>
distTo[] is the solution <em>iff</em>: </p>
<blockquote>
<ul>
<li>distTo[s] = 0 </li>
<li>distTo[v] is the weight of some path from s to v </li>
<li>for any edge e = v → w, distTo[w] <= distTo[v]+e.weight() </li>
</ul>
</blockquote>
<h3 id="generic-algo">Generic algo</h3>
<p>just relax all edges... <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image004.png"/> <br/>
<strong>prop</strong>. the above generic algo gives the SPT from s. </p>
<p>implementations of the algo: </p>
<ul>
<li>Dijkstra (nonnegative edge) </li>
<li>Topological sort (!) (no directed cycles) </li>
<li>Bellman-Ford (no negative directed cycles) </li>
</ul>
<h1 id="3-dijkstras-algorithm_1">3. Dijkstra's Algorithm</h1>
<p><em>(non-negative edges)</em> </p>
<blockquote>
<ul>
<li>consider vertices in increasing distance from s.* </li>
<li>add vertex to the SPT, relax all edges from that vertex. </li>
<li>each time: take the closest vertex to s that is not in the SPT (ie. whose dist is not determined yet), add the vertex, and relax all its outgoing edges. </li>
</ul>
</blockquote>
<p><strong>prop</strong>. Dijkstra works. <br/>
pf. <br/>
<em> each edge e = v → w is considered exactly once (when vertex v is added to SPT). distTo[w] <= distTo[v]+e.weight() after relaxing e <br/>
</em> ineq holds until algo terminates <br/>
* distTo[w] decrease monotonely, <br/>
* distTo[v] will not change because each time we choose distTo[v] smallest, and all edges non-negative. <br/>
* CQFD by optimality condition. </p>
<h3 id="implementation">implementation</h3>
<p>Use an <code>IndexMinPQ</code> to store vertices. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="k">class</span><span class="w"> </span><span class="n">DijkstraSP</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">dist</span><span class="err">[]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">DirectedEdge</span><span class="w"> </span><span class="n">edgeTo</span><span class="err">[]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">IndexMinPQ</span><span class="o"><</span><span class="k">Integer</span><span class="p">,</span><span class="k">Double</span><span class="o">></span><span class="w"> </span><span class="n">pq</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">DijkstraSP</span><span class="p">(</span><span class="n">EdgeWeightedDigraph</span><span class="w"> </span><span class="n">G</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">s</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">this</span><span class="p">.</span><span class="n">dist</span><span class="err">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">G.V()</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">this</span><span class="p">.</span><span class="n">edgeTo</span><span class="err">[]</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">DirectedEdge</span><span class="o">[</span><span class="n">G.V()</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">this</span><span class="p">.</span><span class="n">pq</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">IndexMinPQ</span><span class="o"><</span><span class="k">Integer</span><span class="p">,</span><span class="k">Double</span><span class="o">></span><span class="p">(</span><span class="n">G</span><span class="p">.</span><span class="n">V</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">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">v</span><span class="o"><</span><span class="n">G</span><span class="p">.</span><span class="n">V</span><span class="p">();</span><span class="w"> </span><span class="n">v</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">dist</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">999999</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">dist</span><span class="o">[</span><span class="n">s</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </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="n">pq</span><span class="p">.</span><span class="k">insert</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="mf">0.0</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">while</span><span class="p">(</span><span class="err">!</span><span class="n">pq</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">())</span><span class="err">{</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">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="n">delMin</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="n">DirectedEdge</span><span class="w"> </span><span class="nl">e</span><span class="p">:</span><span class="n">G</span><span class="p">.</span><span class="n">adj</span><span class="p">(</span><span class="n">v</span><span class="p">))</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">relax</span><span class="p">(</span><span class="n">e</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">decrease</span><span class="w"> </span><span class="k">key</span><span class="w"> </span><span class="ow">or</span><span class="w"> </span><span class="k">insert</span><span class="w"> </span><span class="n">vertices</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="n">pq</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="k">constructor</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="n">private</span><span class="w"> </span><span class="n">relax</span><span class="p">(</span><span class="n">DirectedEdge</span><span class="w"> </span><span class="n">e</span><span class="p">)</span><span class="err">{</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">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</span><span class="p">.</span><span class="k">from</span><span class="p">(),</span><span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</span><span class="p">.</span><span class="k">to</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">dist</span><span class="o">[</span><span class="n">w</span><span class="o">]<</span><span class="n">dist</span><span class="o">[</span><span class="n">v</span><span class="o">]+</span><span class="n">e</span><span class="p">.</span><span class="n">weight</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">dist</span><span class="o">[</span><span class="n">w</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dist</span><span class="o">[</span><span class="n">v</span><span class="o">]+</span><span class="n">e</span><span class="p">.</span><span class="n">weight</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">edgeTo</span><span class="o">[</span><span class="n">w</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">e</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">pq</span><span class="p">.</span><span class="k">contains</span><span class="p">(</span><span class="n">w</span><span class="p">))</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="n">decreaseKey</span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">dist</span><span class="o">[</span><span class="n">w</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">else</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="k">insert</span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">dist</span><span class="o">[</span><span class="n">w</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>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="o">//</span><span class="n">relax</span><span class="p">()</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="o">//</span><span class="k">class</span><span class="w"></span></span>
</pre></div>
<p><strong>Dijkstra is in fact a Prim algorithm ! </strong><em>Both are algos that compute a spanning tree.</em> </p>
<ul>
<li>in Prim: each time takes the vertex <em>closest to the tree </em>(and is for undirected graph). </li>
<li>in Dijkstra: each time takes the vertex <em>closest to source s </em>(and is for directed graph). </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image005.png"/> <br/>
DFS BFS are also computing spanning tree! <br/>
...NB!! </p>
<h3 id="complexity">complexity</h3>
<ul>
<li>V insertions to PQ (each vertex is added to the SPT) </li>
<li>V delMin from PQ </li>
<li>E decrease key </li>
</ul>
<p>→ depends on the PQ implementation. <br/>
in our implementation, <strong>O(ElogV)</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image006.png"/> </p>
<h1 id="4-edge-weighted-dags_1">4. Edge-Weighted DAGs</h1>
<p>in a DAG, it's easier to find the shortest path ? <br/>
⇒ yes! <br/>
simple algo: </p>
<blockquote>
<ul>
<li>consider vertices in topological order </li>
<li>relax outgoing edges form this vertex. </li>
</ul>
</blockquote>
<h3 id="correctness">correctness</h3>
<p><strong>prop</strong>. </p>
<blockquote>
<p>topo-sort algo computes SPT in any DAG (<em>even with negative weights</em>) in time <strong>O(E+V) </strong><em>(linear time!)</em>. </p>
</blockquote>
<p><em>pf.</em> <br/>
<em> each edge is relaxed exactely once. distTo[w] <= distTo[v]+e.weight() after relaxing e <br/>
</em> ineq holds until the algo terminates because: <br/>
* distTo[w] decrease monotonely, <br/>
* distTo[v] will not change because of topo-order, no edge pointing to v after v is relaxed. <br/>
* CQFD </p>
<h3 id="implementation_1">implementation</h3>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class AsyclicSP{ </span></span>
<span class="code-line"><span class="err"> AsyclicSP(EdgeWeightedDigraph G, int s){ </span></span>
<span class="code-line"><span class="err"> // init dist[] and edgeTo[]... </span></span>
<span class="code-line"><span class="err"> Topological topo = new Topological(G); </span></span>
<span class="code-line"><span class="err"> for(int v: topo.order()) </span></span>
<span class="code-line"><span class="err"> for(DirectedEdge e:G.adj(v)) </span></span>
<span class="code-line"><span class="err"> relax(e); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h3 id="application">application</h3>
<ul>
<li><em>seam-carving</em> </li>
</ul>
<p>Resizing images <em>non-uniformly</em> without distortion. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image008.png"/> <br/>
好神奇! <br/>
Grid DAG of pixels, edges are pointed to 3 downward neighbors. <br/>
横向缩小: 删掉一个"seam" <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image009.png"/> </p>
<ul>
<li>
<p>longest path in DAG → <em>just negate all edge weights</em>, as this algo is OK for negative edges. </p>
</li>
<li>
<p>parallel job scheduling <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image010.png"/> <br/>
感觉有点运筹的意思, 关键是建立一个DAG, 比较有技巧性: </p>
<ul>
<li>add source and sink </li>
<li>split each job to 2 vertices (begin and end) </li>
</ul>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image011.png"/> <br/>
⇒ use longest path to schedule jobs </p>
<h1 id="5-negative-weights_1">5. Negative Weights</h1>
<p>只是给所有边增加weight并<strong>不能</strong>解决负权边问题. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image013.png"/> <br/>
⇒ need a different algo. </p>
<p>prop. SPT exists <em>iff</em> there is no <em>negative cycles.</em> </p>
<h3 id="bellman-ford">Bellman-Ford</h3>
<p>一句话总结Bellman算法: 每次relax所有的边, 进行V次. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image014.png"/> <br/>
(<em>因为relax k次以后, s到v的最短路径的长度小于等于k.</em>) </p>
<p><strong>prop</strong>. Bellman-Ford works, complexity is <strong>O(EV)</strong>. <br/>
pf. <br/>
after i passes, found shortest path contain at most i edges. </p>
<p>code: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="n">BellmanFordSP</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">BellmanFordSP</span><span class="p">(</span><span class="n">EdgeWeightedDigraph</span> <span class="k">G</span><span class="p">,</span> <span class="nb">int</span> <span class="n">s</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="o">//</span> <span class="n">init</span> <span class="n">dist</span><span class="p">[]</span> <span class="k">and</span> <span class="n">edgeTo</span><span class="p">[]...</span> </span>
<span class="code-line"> <span class="k">for</span><span class="p">(</span><span class="nb">int</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="k">G</span><span class="p">.</span><span class="n">V</span><span class="p">();</span><span class="n">i</span><span class="o">++</span><span class="p">)</span> </span>
<span class="code-line"> <span class="k">for</span><span class="p">(</span><span class="n">DirectedEdge</span> <span class="n">e</span><span class="p">:</span><span class="k">G</span><span class="p">.</span><span class="n">edges</span><span class="p">())</span> </span>
<span class="code-line"> <span class="n">relax</span><span class="p">(</span><span class="n">e</span><span class="p">);</span> </span>
<span class="code-line"> <span class="err">}</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h3 id="improvement">improvement</h3>
<p><em>if dist[v] is not changed during one pasee → no need to relax incoming edges any more.</em> <br/>
→ much faster in practice. </p>
<p>FIFO implementation: maintain a queue of vertices whoses dist is changed. </p>
<h3 id="finding-negative-cycle">Finding negative cycle</h3>
<p>add two method to SP: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">boolean hasNegCycle(); </span></span>
<span class="code-line"><span class="err">Iterable<DirectedEdge> negCycle();</span></span>
</pre></div>
<p><img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image016.png"/> </p>
<p><strong>prop</strong>. </p>
<blockquote>
<p>if a vertex u is updated in phase V, there exists a negative cycle, and can track edgeTo[u] to find the cycle. </p>
</blockquote>
<p>原因是最短路径的长度小于V, 如果大于V则说明存在negative cycle. </p>
<h3 id="negative-cycle-application">negative cycle application</h3>
<p><em>arbitrage detection</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image018.png"/> <br/>
建模:
graph of currencies, edge weight = exchange weight (complete graph). <br/>
⇒ <em>find a cycle whose product of edge is >1.</em> <br/>
⇒ <strong>take logs</strong> to make a shortest-path pb. <br/>
⇒ take minus log, then try to find a negative cycle. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image020.png"/> <br/>
NB!! </p>
<h1 id="summery_1">Summery</h1>
<p><img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image015.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week2_2/pasted_image021.png"/> </p>
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<div id="toc"><ul><li><a class="toc-href" href="#1-shortest-paths-apis" title="1. Shortest Paths APIs">1. Shortest Paths APIs</a><ul><li><a class="toc-href" href="#shortest-path-variants" title="shortest path variants">shortest path variants</a></li><li><a class="toc-href" href="#apis" title="APIs">APIs</a></li></ul></li><li><a class="toc-href" href="#2-shortest-path-properties_1" title="2. Shortest path properties">2. Shortest path properties</a><ul><li><a class="toc-href" href="#edge-relaxation" title="Edge relaxation">Edge relaxation</a></li><li><a class="toc-href" href="#optimality-conditions" title="optimality conditions">optimality conditions</a></li><li><a class="toc-href" href="#generic-algo" title="Generic algo">Generic algo</a></li></ul></li><li><a algorithm'="" class="toc-href" href="#3-dijkstras-algorithm_1" s="" title="3. Dijkstra">3. Dijkstra's Algorithm</a><ul><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#4-edge-weighted-dags_1" title="4. Edge-Weighted DAGs">4. Edge-Weighted DAGs</a><ul><li><a class="toc-href" href="#correctness" title="correctness">correctness</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#application" title="application">application</a></li></ul></li><li><a class="toc-href" href="#5-negative-weights_1" title="5. Negative Weights">5. Negative Weights</a><ul><li><a class="toc-href" href="#bellman-ford" title="Bellman-Ford">Bellman-Ford</a></li><li><a class="toc-href" href="#improvement" title="improvement">improvement</a></li><li><a class="toc-href" href="#finding-negative-cycle" title="Finding negative cycle">Finding negative cycle</a></li><li><a class="toc-href" href="#negative-cycle-application" title="negative cycle application">negative cycle application</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="Summery">Summery</a></li></ul></div>
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