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[Algorithms II] Week 3-1 Maximum Flow
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-maxflow" title="1. Introduction to Maxflow">1. Introduction to Maxflow</a><ul><li><a class="toc-href" href="#min-cut-pb" title="Min-cut pb">Min-cut pb</a></li><li><a class="toc-href" href="#max-flow-pb" title="Max-flow pb">Max-flow pb</a></li></ul></li><li><a class="toc-href" href="#2-ford-fulkerson-algorithm_1" title="2. Ford-Fulkerson Algorithm">2. Ford-Fulkerson Algorithm</a></li><li><a class="toc-href" href="#3-maxflow-mincut-theorem" title="3. Maxflow-Mincut Theorem">3. Maxflow-Mincut Theorem</a></li><li><a class="toc-href" href="#4-running-time-analysis" title="4. Running Time Analysis">4. Running Time Analysis</a><ul><li><a class="toc-href" href="#integer-capacity-graphs" title="integer capacity graphs">integer capacity graphs</a></li><li><a class="toc-href" href="#bad-case-for-ff" title="Bad case for FF">Bad case for FF</a></li></ul></li><li><a class="toc-href" href="#5-java-implementation_1" title="5. Java Implementation">5. Java Implementation</a><ul><li><a class="toc-href" href="#representation-of-flow-graph" title="representation of flow graph">representation of flow graph</a></li><li><a class="toc-href" href="#residual-graph-gr" title="Residual graph Gr">Residual graph Gr</a></li><li><a class="toc-href" href="#apis" title="APIs">APIs</a></li></ul></li><li><a class="toc-href" href="#6-maxflow-applications_1" title="6. Maxflow Applications">6. Maxflow Applications</a><ul><li><a class="toc-href" href="#ex1-bipartite-matching-pb" title="ex1. bipartite matching pb">ex1. bipartite matching pb</a></li><li><a class="toc-href" href="#ex2-baseball-elimination" title="ex2. baseball elimination">ex2. baseball elimination</a></li></ul></li></ul></div>
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</div>
<h1 id="1-introduction-to-maxflow">1. Introduction to Maxflow</h1>
<h3 id="min-cut-pb">Min-cut pb</h3>
<ul>
<li>input: edge-weighted digraph G, each edge <code>e</code> has weight(<em>"capacity"</em>) <code>c[e]</code>>=0, a <em>source vertex</em> <code>s</code>, a <em>target vertex</em> <code>t</code>. </li>
<li><strong>def</strong>. an <strong>st-cut </strong><code>(A,B)</code> is a partition of vertices into 2 disjoint sets A and B, with <code>s</code> in set <code>A</code> and <code>t</code> in set <code>B</code>. </li>
<li><strong>def</strong>. the <strong>capacity</strong> of a cut <code>(A,B)</code> is sum of capacities of edges going <em>from A to B</em> <em>(not considering B to A)</em>. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image.png"/> <br/>
⇒ min-cut pb: find the cut <strong>(A,B)</strong> with min-capacity. </p>
<h3 id="max-flow-pb">Max-flow pb</h3>
<ul>
<li>same input: graph G, source s, target t </li>
<li><strong>def.</strong> an <strong>st-flow</strong> is an assignment of values to edges <code>f: e→f[e]</code> such that: <ul>
<li>capacity constraint: <code>0<=f[e]<=c[e]</code> for any e; </li>
<li>local equilibrium: for any vertex v (other than s or t), <em>inflow=outflow</em>; </li>
</ul>
</li>
<li><strong>def.</strong> the <strong>value</strong> of a flow <code>f</code> is the inflow at <code>t</code>. (assume no ingoing edge to s or outgoing edge to t) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image001.png"/> <br/>
⇒ max-flow pb: find <code>f</code> with max value. </p>
<p>remark: max-flow and min-cut are dual problems. </p>
<h1 id="2-ford-fulkerson-algorithm_1">2. Ford-Fulkerson Algorithm</h1>
<p><strong>def.</strong> given a flow <code>f</code> for a graph, an <strong>"augment path"</strong> is an <strong>undirected</strong> path form <code>s</code> to <code>t</code>, if there exist <code>df>0</code> (<em>"bottleneck capacity"</em>) such that: </p>
<ul>
<li>for forward edges e: can augment flow by <code>df</code> (not full: <code>f[e]+df<=c[e]</code>) </li>
<li>for backward edges: can decrease flow by <code>df</code> (not empty: <code>f[e]-df>=0</code>) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image002.png"/> </p>
<ul>
<li>def. <strong>residual capacity</strong> <ul>
<li>for forward edge e, <em>residual-cap = c[e]-f[e]</em> </li>
<li>for backward edge e, <em>residual-cap = f[e]</em> </li>
</ul>
</li>
</ul>
<p>⇒ an aug-path is a path where <em>each edge has residual capacity >0</em>. </p>
<p><strong>blocking edges</strong>: <em>full forward edge</em> or <em>empty backward edge.</em> </p>
<p>→ idea: increase flow along augment paths. <br/>
<strong>[algo]</strong> </p>
<blockquote>
<ul>
<li>start: 0 flow: <code>f[e]=0</code> for all e. </li>
<li>find an augment path (and the corresponding <code>df</code>) in graph, and change the flows along the path by <code>+/-df</code>. </li>
<li>loop until no augment path exists. (ie. all path s→t are blocked either by a <em>full forward edge</em> or an <em>empty backward edge, </em>ie. by an edge with 0 residual capacity) </li>
</ul>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image003.png"/> </p>
<p>FF is a gernel algorithm: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image004.png"/> </p>
<h1 id="3-maxflow-mincut-theorem">3. Maxflow-Mincut Theorem</h1>
<p><strong>def</strong>. for a cut (A,B), the <strong>net flow</strong> across the cut (<em>netflow(A,B)</em>) is the sum of flows from A to B minus flows from B to A. </p>
<blockquote>
<p><strong>[flow-value Lemma]</strong> <br/>
For any flow <code>f</code> and any cut <code>(A,B)</code>⇒ <em>netflow(A,B) = value(f).</em> <br/>
<em>pf.</em> <br/>
induction on the size of set B. <br/>
<em> base case, when B={t}, by def we have </em>netflow(A,B) = value(f)<em> <br/>
</em> when moving any vertex v from A to B: <br/>
* netflow(A, B) augment by <em>flow(A→v)+flow(B→v)=inflow(v)</em>, <br/>
* netflow(A, B) decrease by <em>flow(v→A)+flow(v→B)=outflow(v)</em>, <br/>
* by equilibrium of flow, <em>netflow(A',B')=netflow(A,B)=value(f)</em> </p>
</blockquote>
<p>ex. (A: gray vertices, B: white vertices) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image005.png"/> </p>
<p>[cor] <em>outflow(s)=inflow(t)=value(f)</em> </p>
<blockquote>
<p><strong>[weak duality]</strong> <br/>
For <em>any</em> flow <code>f</code> and <em>any</em> cut <code>(A,B)</code>, ⇒ <em>value(f) <= capacity(A,B).</em> </p>
<p><strong>[Augmenting path Th]</strong> <br/>
A flow <code>f</code> is maxflow <em>iff</em> there is no augment path. </p>
<p><strong>[maxflow-mincut Th]</strong> <br/>
<strong>value(maxflow) = capacity(mincut).</strong> </p>
</blockquote>
<p><em>pf.</em> <br/>
for any flow <code>f</code>, prove the equivalence of the 3 following statements: <br/>
i. there exists a cut st: <em>capacity(cut) = value(f).</em> <br/>
ii. <code>f</code> is a maxflow. <br/>
iii. there is no augmenting path wrt <code>f</code>. </p>
<ul>
<li>[i⇒ii] </li>
</ul>
<p>suppose cut(A,B) st: <em>capacity(A,B)=value(f)</em> <br/>
⇒ by weak duality, for any other flow f', <em>vlaue(f')<=capacity(A,B)=value(f)</em> </p>
<ul>
<li>[ii⇒iii] (eqv to prove ~iii⇒~ii) </li>
</ul>
<p>suppose there is an aug-path from s to t, of bottleneck capacity=df, <br/>
⇒ by improving f with df, we get a f' > f </p>
<ul>
<li>[iii⇒i] </li>
</ul>
<p>suppose there is no aug-path, ie, <em>all path from s to t are blocked by some full-forward edge or empty backward edge.</em> <br/>
⇒ let A:=vertices connected with s by a path with no blocking edges, and B := the rest <br/>
(<em>so</em> <em>once we get a maxflow, we can <strong><em>compute the mincut</em></strong> in this way</em>) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image006.png"/> <br/>
→ for all edges across A and B, all forward edges are full, all backward edges are empty <br/>
⇒ capacity(A,B) = netflow(A,B) = value(f) by flow-value lemma <br/>
CQFD... 过瘾... </p>
<h1 id="4-running-time-analysis">4. Running Time Analysis</h1>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image007.png"/> </p>
<ul>
<li>getting a mincut form maxflow? → easy (as discussed in the pf above) </li>
<li>computing an aug-path? → <strong>BFS</strong> </li>
<li>does FF algo always terminate? how many augmentations? → ... </li>
</ul>
<h3 id="integer-capacity-graphs">integer capacity graphs</h3>
<p><em>special case of FF algo</em>: edge capacities are <em>integers</em> between 1 and U. </p>
<p>invariant: flow is always integer all along FF algo. </p>
<p>[prop] nb of augmentations <= value of maxflow. <br/>
pf. each augmentation will add flow by >=1. </p>
<p>[integrality Th] There exist an integer-valued maxflow. </p>
<h3 id="bad-case-for-ff">Bad case for FF</h3>
<p>nb of augmentation == value of maxflow <br/>
(each time, the path through the middle edge is chosen as aug-path) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image008.png"/> <br/>
<strong>can be easily avoided</strong>⇒ by using shortest(nb of edges)/fastest(biggest df) path <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image010.png"/> </p>
<p>Performance of FF depends on the algo for choosing aug-path: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image009.png"/> </p>
<h1 id="5-java-implementation_1">5. Java Implementation</h1>
<h3 id="representation-of-flow-graph">representation of flow graph</h3>
<ul>
<li>flow edge: </li>
</ul>
<p>each e= v→w, have flow f[e] and capacity c[e]. </p>
<ul>
<li>flow graph: </li>
</ul>
<p>put e in both v and w's adj-list. </p>
<ul>
<li>flow augmentation (by delta) <ul>
<li>for forward edge e, f[e] += delta </li>
<li>for backward edge e, f[e] -= delta </li>
</ul>
</li>
</ul>
<h3 id="residual-graph-gr">Residual graph Gr</h3>
<p><strong>def</strong>. For a flow <code>f</code> and a graph <code>G</code>, the <strong>residual graph</strong> <code>Gr</code> is obtained by: </p>
<blockquote>
<p>for each edge <code>e=v→w</code>, (with <code>c[e]</code> and <code>f[e]</code>) in <code>G</code>, put in <code>Gr</code>: <br/>
<em> <code>e1=v→w</code>, with weight=<code>c[e]-f[e]</code> <br/>
</em> <code>e2=w→v</code>, with weight=<code>f[e]</code> (即两个方向上的weight都为residual capacity) </p>
</blockquote>
<p>(rmq: <code>Gr</code> is just a weighted digraph, not a flow graph) </p>
<p>[prop] <strong>Augment path in </strong><code>G</code><strong> is equivalent to a path in </strong><code>Gr</code> (<code>df</code> of aug-path in <code>G</code> = min edge weight in <code>Gr</code>)<strong>.</strong> <br/>
(但是实现的时候其实不用显式构造Gr, 只需BFS的时候修改一下即可) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image011.png"/> </p>
<h3 id="apis">APIs</h3>
<p>这里的API设计的非常合理... 导致每一部分的代码量都不大... NB </p>
<ul>
<li>
<p>flow-edge: <br/>
rmq. both calculate residual-cap and augmentation need to specify a <em>direction</em>, so we need a index v as parameter for these 2 functions. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class FlowEdge{ </span></span>
<span class="code-line"><span class="err"> private final int v, w; </span></span>
<span class="code-line"><span class="err"> private final double capacity; </span></span>
<span class="code-line"><span class="err"> private double flow=0.0; </span></span>
<span class="code-line"><span class="err"> FlowEdge(int v, int w, double cap); </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"> int other(int v); </span></span>
<span class="code-line"><span class="err"> double capacity(); </span></span>
<span class="code-line"><span class="err"> double flow(); </span></span>
<span class="code-line"><span class="err"> double residualCapTo(int v);// residual capacity </span></span>
<span class="code-line"><span class="err"> void addFlowTo(int v, double delta);// augment residual flow </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
</li>
<li>
<p>flow graph: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class FlowNetwork{ </span></span>
<span class="code-line"><span class="err"> private Bag<FlowEdge>[] adj;//use adj-list representation for flow graph </span></span>
<span class="code-line"><span class="err"> FlowNetwork(int V); </span></span>
<span class="code-line"><span class="err"> void addEdge(FlowEdge e); </span></span>
<span class="code-line"><span class="err"> Iterable<FlowEdge> adj(int v);// both incoming and outgoing edges </span></span>
<span class="code-line"><span class="err"> ... </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
</li>
<li>
<p>FF algo: </p>
<ul>
<li>use a function <code>hasAugPath()</code> to test termination </li>
<li>use a function <code>bottleNeck()</code> to get delta </li>
<li>if a augpath is found, use two arrays <code>reached[]</code> and <code>edgeTo[]</code> to get the augpath (find the path <em>backwards</em>). </li>
</ul>
</li>
</ul>
<p>code: </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">FordFulkerson</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="k">boolean</span><span class="err">[]</span><span class="w"> </span><span class="n">reached</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="n">reached</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="w"> </span><span class="n">indicates</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">path</span><span class="w"> </span><span class="n">s</span><span class="c1">-->v exists in Gr, used in DFS </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="n">FlowEdge</span><span class="err">[]</span><span class="w"> </span><span class="n">edgeTo</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">edgeTo</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="k">last</span><span class="w"> </span><span class="n">edge</span><span class="w"> </span><span class="k">on</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="k">path</span><span class="w"> </span><span class="n">s</span><span class="c1">-->v </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="k">double</span><span class="w"> </span><span class="k">value</span><span class="o">=</span><span class="mf">0.0</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="k">value</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">flow</span><span class="w"> </span></span>
<span class="code-line"><span class="k">public</span><span class="w"> </span><span class="n">FordFulkerson</span><span class="p">(</span><span class="n">FlowNetwork</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="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">t</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">while</span><span class="p">(</span><span class="n">this</span><span class="p">.</span><span class="n">hasAugPath</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">s</span><span class="p">,</span><span class="n">t</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">double</span><span class="w"> </span><span class="n">delta</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">bottleNeck</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="o">=</span><span class="n">t</span><span class="p">;</span><span class="w"> </span><span class="n">v</span><span class="o">!=</span><span class="n">s</span><span class="p">;</span><span class="w"> </span><span class="n">v</span><span class="o">=</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="n">other</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">edgeTo</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="p">.</span><span class="n">addFlowTo</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="n">delta</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="k">value</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">delta</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="k">each</span><span class="w"> </span><span class="nc">time</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">flow</span><span class="w"> </span><span class="k">value</span><span class="w"> </span><span class="n">augments</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">delta</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>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="k">double</span><span class="w"> </span><span class="n">bottleNeck</span><span class="p">()</span><span class="err">{</span><span class="o">//</span><span class="n">bottleneck</span><span class="o">-</span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">min</span><span class="w"> </span><span class="n">residual</span><span class="w"> </span><span class="n">flow</span><span class="w"> </span><span class="k">on</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">aut</span><span class="o">-</span><span class="k">path</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">double</span><span class="w"> </span><span class="n">bottleneck</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">9999999</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">assert</span><span class="p">(</span><span class="n">reached</span><span class="o">[</span><span class="n">t</span><span class="o">]</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">aug</span><span class="o">-</span><span class="k">path</span><span class="w"> </span><span class="n">should</span><span class="w"> </span><span class="n">exsit</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="o">=</span><span class="n">t</span><span class="p">;</span><span class="w"> </span><span class="n">v</span><span class="o">!=</span><span class="n">s</span><span class="p">;</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">edgeTo</span><span class="o">[</span><span class="n">v</span><span class="o">]</span><span class="p">.</span><span class="n">other</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">bottleneck</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="nf">min</span><span class="p">(</span><span class="n">bottleneck</span><span class="p">,</span><span class="w"> </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>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">bottleneck</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="n">private</span><span class="w"> </span><span class="k">boolean</span><span class="w"> </span><span class="n">hasAugPath</span><span class="p">(</span><span class="n">FlowNetwork</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="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">t</span><span class="p">)</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">perform</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">BFS</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Queue</span><span class="o"><</span><span class="k">Integer</span><span class="o">></span><span class="w"> </span><span class="n">q</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">LinkedList</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="n">this</span><span class="p">.</span><span class="n">reached</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="k">boolean</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="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">FlowEdge</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">q</span><span class="p">.</span><span class="k">add</span><span class="p">(</span><span class="n">s</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">q</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">q</span><span class="p">.</span><span class="n">deque</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">FlowEdge</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="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">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="n">other</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">if</span><span class="p">(</span><span class="err">!</span><span class="n">reached</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="n">residualCapTo</span><span class="p">(</span><span class="n">w</span><span class="p">)</span><span class="o">></span><span class="mi">0</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="w"> </span><span class="n">modified</span><span class="w"> </span><span class="nl">BFS</span><span class="p">:</span><span class="w"> </span><span class="n">valid</span><span class="w"> </span><span class="n">edges</span><span class="w"> </span><span class="k">are</span><span class="w"> </span><span class="n">those</span><span class="w"> </span><span class="k">with</span><span class="w"> </span><span class="n">residualCap</span><span class="o">></span><span class="mi">0</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="n">reached</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="k">true</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">w</span><span class="o">==</span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">true</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">reached</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">BFS</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">q</span><span class="p">.</span><span class="n">enqueue</span><span class="p">(</span><span class="n">w</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="n">BFS</span><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="n">loop</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">false</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="err">}</span><span class="o">//</span><span class="k">class</span><span class="w"> </span><span class="n">FF</span><span class="w"></span></span>
</pre></div>
<h1 id="6-maxflow-applications_1">6. Maxflow Applications</h1>
<p>关键是建模很巧妙... <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image012.png"/> </p>
<h3 id="ex1-bipartite-matching-pb">ex1. bipartite matching pb</h3>
<p>二分图的最大匹配问题. (有点像marriage stable问题...但是不一样 因为没有preference order) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image013.png"/> <br/>
⇒ is there a way to match all students to a job? <br/>
ie. <em>given a bipartite graph, find a perfect matching.</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image014.png"/> <br/>
<strong>modeling</strong> </p>
<blockquote>
<ul>
<li>add source <code>s</code> and target <code>t</code> </li>
<li>all edges from <code>s</code> to students: capacity=1 </li>
<li>all edges from companies to <code>t</code>: capacity=1 </li>
<li>all edges from student to company: capacity=INF </li>
</ul>
</blockquote>
<p>⇒ find maxflow in the graph <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image015.png"/> </p>
<p><strong>when no perfect matching: mincut can explain why</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image016.png"/> <br/>
in the above case, student 2,4,5 can only be matched to 7,10 <br/>
⇒ mincut can help us find such cases! </p>
<p><em>recall: how to get mincut from maxflow</em> </p>
<blockquote>
<p>mincut = (A,B), where: <br/>
A:=vertices connected with s by a path with non blocking edges, <br/>
B := the rest <br/>
(<em>blocking edges: full forward edge or empty backward edge on path</em>) </p>
</blockquote>
<p>ex. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image017.png"/> </p>
<ul>
<li>let S=students on s side of mincut (in above case, <em>S={2,4,5}</em>) </li>
<li>let T=companies on s side of mincut (in above case, T<em>={7,10}</em>) </li>
<li>|S|>|T|, that's why no perfect matching! </li>
</ul>
<h3 id="ex2-baseball-elimination">ex2. baseball elimination</h3>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image018.png"/> <br/>
(前三列是目前成绩, 后面四列是接下来赛程矩阵) <br/>
Montreal is mathematically eliminated → easy to see <br/>
→ Philly is mathematically eliminated also ! </p>
<ul>
<li>another case: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image019.png"/> <br/>
Detroit is mathematically eliminated ! <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image020.png"/> <br/>
whether team-4 still has a chance to win? <br/>
<strong>modelling</strong> </p>
<ul>
<li>remaining games flow from s to t. </li>
<li>use team-pairs ans teams as vertices </li>
<li><em>carefully chosen capacities</em>(see below) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_1/pasted_image021.png"/> <br/>
⇒ team 4 could win iff all flow from s are full (ie. all match points can be repartitioned over other teams <em>without depassing team 4's maximum wins</em>). <br/>
总之很巧妙.... </p>
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-maxflow" title="1. Introduction to Maxflow">1. Introduction to Maxflow</a><ul><li><a class="toc-href" href="#min-cut-pb" title="Min-cut pb">Min-cut pb</a></li><li><a class="toc-href" href="#max-flow-pb" title="Max-flow pb">Max-flow pb</a></li></ul></li><li><a class="toc-href" href="#2-ford-fulkerson-algorithm_1" title="2. Ford-Fulkerson Algorithm">2. Ford-Fulkerson Algorithm</a></li><li><a class="toc-href" href="#3-maxflow-mincut-theorem" title="3. Maxflow-Mincut Theorem">3. Maxflow-Mincut Theorem</a></li><li><a class="toc-href" href="#4-running-time-analysis" title="4. Running Time Analysis">4. Running Time Analysis</a><ul><li><a class="toc-href" href="#integer-capacity-graphs" title="integer capacity graphs">integer capacity graphs</a></li><li><a class="toc-href" href="#bad-case-for-ff" title="Bad case for FF">Bad case for FF</a></li></ul></li><li><a class="toc-href" href="#5-java-implementation_1" title="5. Java Implementation">5. Java Implementation</a><ul><li><a class="toc-href" href="#representation-of-flow-graph" title="representation of flow graph">representation of flow graph</a></li><li><a class="toc-href" href="#residual-graph-gr" title="Residual graph Gr">Residual graph Gr</a></li><li><a class="toc-href" href="#apis" title="APIs">APIs</a></li></ul></li><li><a class="toc-href" href="#6-maxflow-applications_1" title="6. Maxflow Applications">6. Maxflow Applications</a><ul><li><a class="toc-href" href="#ex1-bipartite-matching-pb" title="ex1. bipartite matching pb">ex1. bipartite matching pb</a></li><li><a class="toc-href" href="#ex2-baseball-elimination" title="ex2. baseball elimination">ex2. baseball elimination</a></li></ul></li></ul></div>
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