Network Flow Solver

.gitignore

bower_components
*.swp
*.swo

.tern-project

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README.md

Click in the open space to add a node, drag from one node to another to add an edge.
Alt-drag a node to move the graph layout. Click a node or an edge to select it.

When a node is selected: Delete or Backspace removes the node.
When an edge is selected: f fixes it in place, and Delete or Backspace removes the edge.

The inspiration for the digraph logic originally came from rkirsling's modal logic playground.

This project is an attempt to make the solutions to network flow problems accessible the general public. This project uses the Ford-Fulkerson algorithm to calculate the max flow. Special thanks to Dimitris Achilloptas for the introduction to such a great algorithm!

TODO:

• Multi-Edge Selection?
• Linear gravitation points?

bower.json

{
  "name": "nf",
  "version": "0.0.0",
  "homepage": "https://github.com/estk/nf",
  "authors": [
    "Evan Simmons <esims89@gmail.com>"
  ],
  "description": "Network Flow",
  "license": "MIT",
  "private": true,
  "ignore": [
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    "node_modules",
    "bower_components",
    "test",
    "tests"
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  "dependencies": {
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    "es6-shim": "~0.10.1"
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}

d3.min.js

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1!=n.getDate()}],["%B",function(n){return n.getMonth()}],["%Y",Ae]]),$s={range:function(n,t,e){return Go.range(Math.ceil(n/e)*e,+t,e).map(Bo)},floor:At,ceil:At};Zs.year=ic.year,ic.scale=function(){return Xo(Go.scale.linear(),Zs,Vs)};var Xs=Zs.map(function(n){return[n[0].utc,n[1]]}),Bs=Os.multi([[".%L",function(n){return n.getUTCMilliseconds()}],[":%S",function(n){return n.getUTCSeconds()}],["%I:%M",function(n){return n.getUTCMinutes()}],["%I %p",function(n){return n.getUTCHours()}],["%a %d",function(n){return n.getUTCDay()&&1!=n.getUTCDate()}],["%b %d",function(n){return 1!=n.getUTCDate()}],["%B",function(n){return n.getUTCMonth()}],["%Y",Ae]]);Xs.year=ic.year.utc,ic.scale.utc=function(){return Xo(Go.scale.linear(),Xs,Bs)},Go.text=Ct(function(n){return n.responseText}),Go.json=function(n,t){return Nt(n,"application/json",Jo,t)},Go.html=function(n,t){return Nt(n,"text/html",Wo,t)},Go.xml=Ct(function(n){return n.responseXML}),"function"==typeof define&&define.amd?define(Go):"object"==typeof module&&module.exports?module.exports=Go:this.d3=Go}();

editor.css

/* === Shared === */
a {
  color: steelblue;
}
#app-container {
  text-align: center;
  text-rendering: geometricPrecision;
}
textarea {
  text-align: center;
  display:block;
  margin: auto;
}
textarea.dialog {
  height: 2em;
}
textarea.transcribe {
  text-align: left;
  display: block;
  margin: auto;
  width: 80%;
  height: 1em;
}
.transbtncontainer, .solvebtncontainer {
  display:table;
  margin:auto;
}
.maxflow {
  text-align: center;
  display:block;
  margin:auto;
}

text {
  font: 13px sans-serif;
  pointer-events: none;
}

/* SVG Elements */
svg {
  background-color: #FFF;
  cursor: default;
  border: solid;
  border-width: 1px;
  -webkit-user-select: none;
  -moz-user-select: none;
  -moz-user-select: -moz-none;
  -ms-user-select: none;
  -o-user-select: none;
  user-select: none;
}

.node {
  text-anchor: middle;
}
.node circle {
  stroke-width: 1.5px;
}

.link text {
  text-anchor: middle;
  dominant-baseline: text-after-edge;
}
path.link {
  fill: none;
  stroke: #666;
  stroke-width: 2px;
}

/* === Editor === */
svg.editor:not(.active):not(.ctrl) {
  cursor: crosshair;
}

.editor .node circle {
  cursor: pointer;
}
.editor .node.selected circle {
  stroke-width: 2.5px;
}

.editor .link {
  cursor: pointer;
}
.editor .link .halo {
  stroke-width: 15px !important;
  stroke: transparent !important;
}
.editor path.selected.link {
  stroke-dasharray: 10,2;
}

/* Dragline */
.editor path.hidden.link {
  stroke-width: 0;
}
.editor path.dragline.link {
  pointer-events: none;
}

/* === Show Steps === */
.link.sending path {
  stroke-width: 3px;
  stroke: steelblue;
}
.link.resending path {
  stroke-width: 3px;
  stroke: brown;
}
.link.sending path{
  stroke-width: 3px;
  stroke: steelblue;
}
.link.sending {
  fill: steelblue;
}
.link.resending {
  fill: brown;
}

editor.js

  window.homePath = window.location.pathname;
  window.onpopstate = function(e) {
    if (e.target.location.pathname === homePath) {
      e.preventDefault();
      makeEditor();
    } else if (e.target.location.pathname === "/flowview") {
      e.preventDefault();
      flowView();
    }
  };

  d3.select('body')
    .append('a')
      .attr('href', 'more.html')
    .append('h3')
      .attr('style', 'text-align:center;')
      .text('What is this?');


function makeEditor(){
  "use strict";
  var margin = {top: 20, right: 10, bottom: 20, left: 10};
  var width = 960 - margin.left - margin.right,
      height = 500 - margin.top - margin.bottom,
      colors = d3.scale.category20(),
      percentile = width/10;


  d3.select('#app-container').remove();
  var body = d3.select('body')
      .append('div')
    .attr('id', 'app-container');


  // === UI ===

  // TextBox
  var textBox = body.append('textarea')
      .attr('class', 'dialog')
      .onKey('return', textboxHandler);


  // MaxFlow
  var maxFlow = body.append('div')
      .attr('class', 'maxflow')
      .text("Max Flow: ")
    .append("span");

  var solveBtnContainer = body.append('div')
      .attr('class', 'solvebtncontainer');

  var solveButton = solveBtnContainer.append('button')
      .attr('class', 'solve')
      .attr("name", "solve")
      .attr("type", "button")
      .text("Solve")
      .on('click', solveFlow);
          
  function solveFlow (){
    var res = fordFulkerson(networkGraph.nodes(), networkGraph.links());
    updateGraph(res[0]);
  }

  function updateGraph(res) {
    var flow = res.maxflow,
        tmpLinks = res.edges,
        tmpNodes = res.vertices;
    networkGraph = new Graph([], []);
    maxFlow.text("");
    restart();

    networkGraph = new Graph(tmpNodes, tmpLinks);
    maxFlow.text(flow);
    restart();
  }

  var resetButton = solveBtnContainer.append('button')
      .attr('class', 'reset')
      .attr("name", "reset")
      .attr("type", "button")
      .text("Reset")
      .on('click', resetFlow);

  function resetFlow() {
    networkGraph.resetFlow();
    maxFlow.text("");
    restart();
  }

  var showStepsButton = solveBtnContainer.append('button')
      .attr('class', 'show-steps')
      .attr("name", "showSteps")
      .attr("type", "button")
      .text("Show Steps")
      .on('click', makeFlowView);

  function makeFlowView() {
    var url = '/flowview';
    window.history.pushState(null,null, url);
    // perform ford fulkerson
    var res = fordFulkerson(networkGraph.nodes(), networkGraph.links()),
        finalFlow = res[0],
        log = res[1];

    window.log = log;
    window.networkGraph = networkGraph;

    flowView();
  }

  function textboxHandler() {
    var contents = textBox[0][0].value;
    function setName(n, v) {
      n.name = v;
      restart();
    }
    function setCapacity(e, v) {
      e.capacity = v;
      restart();
    }

    // === Dispatch ===

    d3.event.preventDefault();
    // Empty
    if (contents === "") {
      alert("No name!");
      return;

    // Nothing selected
    } else if (!selected_node && !selected_link) {
      alert("Please select a link or node to name.");
    }

    // Changing Node Name
    if (selected_node) {
      if (contents.length > 2) {
        alert("Need Length <= 2");
      }
      else {
        setName(selected_node, contents);
      }

    // Changing Edge Capacity
    } else if (selected_link) {
      resetFlow();
      var newCap = parseInt(contents, null);
      if (isNaN(newCap)) {
        alert("Please enter a number");
      }
      else {
        setCapacity(selected_link, newCap);
      }
    }

    // Clear textbox
    textBox[0][0].value = "";
  }

  // === Graph ===

  // Svg
  var svg = body
    .append('svg')
    .attr('class', 'editor')
    .attr('width', width)
    .attr('height', height);

  // set up initial nodes and links
  //  - Source has id === 0, Sink id === 1.
  //  - nodes are known by 'id', not by index in array.
  //  - links are always source < target; edge directions are set by 'left' and 'right'.
  var tmpNodes = [
      {id: 0, name: 's', fixed: true, x: (percentile), y: height/2},
      {id: 1, name: 't', fixed: true, x: (width-percentile), y: height/2},
      {id: 2, name: 'a', },
      {id: 3, name: 'b', },
      {id: 4, name: 'c', },
      {id: 5, name: 'd', }
    ],
    tmpLinks = [
      {source: tmpNodes[0], target: tmpNodes[2], capacity: 10 },
      {source: tmpNodes[0], target: tmpNodes[3], capacity: 10 },
      {source: tmpNodes[2], target: tmpNodes[3], capacity: 2 },
      {source: tmpNodes[2], target: tmpNodes[4], capacity: 4 },
      {source: tmpNodes[2], target: tmpNodes[5], capacity: 8 },
      {source: tmpNodes[3], target: tmpNodes[5], capacity: 9 },
      {source: tmpNodes[4], target: tmpNodes[1], capacity: 10 },
      {source: tmpNodes[5], target: tmpNodes[4], capacity: 6 },
      {source: tmpNodes[5], target: tmpNodes[1], capacity: 10 },
    ];

  var networkGraph;
  if (window.networkGraph) {
    networkGraph = Graph.fromJSON( window.networkGraph.toJSON() );
  } else {
    networkGraph = new Graph(tmpNodes, tmpLinks);
  }

  // init D3 force layout
  var force = d3.layout.force()
      .nodes(networkGraph.nodes)
      .links(networkGraph.links)
      .size([width, height])
      .linkDistance(100)
      .gravity(0.1)
      .charge(-1000)
      .linkStrength(0.05)
      .on('tick', tick);

  var drag = force.drag();

  // define arrow markers for graph links
  var defs = svg.append('svg:defs');

  defs.append('svg:marker')
      .attr('id', 'end-arrow')
      .attr('viewBox', '0 -5 10 10')
      .attr('refX', 6)
      .attr('markerWidth', 10)
      .attr('markerHeight', 10)
      .attr('markerUnits', "userSpaceOnUse")
      .attr('orient', 'auto')
    .append('svg:path')
      .attr('d', 'M0,-5L10,0L0,5');

  // line displayed when dragging new nodes
  var drag_line = svg.append('svg:path')
    .attr('class', 'link dragline hidden')
    .attr('d', 'M0,0L0,0');

  // mouse event vars
  var selected_node = null,
      selected_link = null,
      mousedown_link = null,
      mousedown_node = null,
      mouseup_node = null;
        

  function resetMouseVars() {
    mousedown_node = null;
    mouseup_node = null;
    mousedown_link = null;
  }

  // === Business Time ===

  // handles to link and node element groups
  var path = svg.append('svg:g').selectAll('g'),
      circle = svg.append('svg:g').selectAll('g');

  // update force layout (called automatically each iteration)
  function tick() {
    // draw directed edges with proper padding from node centers
    path.selectAll('path').attr('d', function(d) {
      var deltaX = d.target.x - d.source.x,
          deltaY = d.target.y - d.source.y,
          dist = Math.sqrt(deltaX * deltaX + deltaY * deltaY),
          normX = deltaX / dist,
          normY = deltaY / dist,
          sourcePadding = 12,
          targetPadding = 17,
          sourceX = d.source.x + (sourcePadding * normX),
          sourceY = d.source.y + (sourcePadding * normY),
          targetX = d.target.x - (targetPadding * normX),
          targetY = d.target.y - (targetPadding * normY);
      return 'M' + sourceX + ',' + sourceY + 'L' + targetX + ',' + targetY;
    });

    circle.attr('transform', function(d) {
      return 'translate(' + d.x + ',' + d.y + ')';
    });
  }

  // update graph (called when needed)
  function restart() {
    // Re-seed force layout with new nodes (/links).
    force.nodes(networkGraph.nodes())
        .links(networkGraph.links());

    // path (link) group
    path = path.data(networkGraph.links(), function(d) {return d.id});

    // add new links
    var g = path.enter().append('g')
      .attr('class', 'link')
      .on('mousedown', function(d) {
        if(d3.event.altKey) {return}

        // select link
        mousedown_link = d;
        if(mousedown_link === selected_link) {selected_link = null}
        else {selected_link = mousedown_link;}
        selected_node = null;

        textBox.attr('placeholder', "New Max capacity");
        restart();
      });

    g.append('svg:path')
      .attr('class', 'link')
      .attr('id', function(d) {return d.id})
      .classed('selected', function(d) { return d === selected_link; })
      .style('marker-end', 'url(#end-arrow)');

    g.append("svg:path")
        .attr('class', 'halo');

    g.append('text')
      .append('textPath')
        .attr('xlink:href', function (d) {return "#" + d.id})
        .attr('startOffset', '50%')
      .append('tspan')
        .attr('dy', -5);

    // remove old links
    path.exit().remove();

    // update existing links
    path.select('path').classed('selected', function(d) { return d === selected_link; })
      .style('marker-end', 'url(#end-arrow)');

    path.select('text')
      .select('tspan')
        .text(function(d) {
          if (d.flow) {return d.flow + " / " + d.capacity;}
          else {return 0 + " / " + d.capacity;}
        });
        

    // circle (node) group
    circle = circle.data(networkGraph.nodes(), function(d) {return d.id});

    circle
      .classed('selected', function(d){ return d === selected_node; });

    circle.selectAll('text')
        .attr('y', 4)
        .attr('class', 'id')
        .text(function(d) { return d.name; });

    // update existing nodes (selected visual states)
    circle.selectAll('circle')
      .style('fill', function(d) {
        return (d === selected_node) ? d3.rgb(colors(d.id)).brighter().toString() : colors(d.id);
      });

    // add new nodes
    var gC = circle.enter().append('svg:g');

    // Add the circle where it should be.
    gC.attr('class','node')
        .classed('selected', function(d){ return d === selected_node })
        .attr('transform', function(d) {
          if (!d.x || !d.y) {return ""}
          return 'translate(' + d.x + ',' + d.y + ')';
        });

    gC.append('svg:circle')
      .attr('r', 12)
      .style('fill', function(d) {
        return (d === selected_node) ? d3.rgb(colors(d.id)).brighter().toString() : colors(d.id);
      })
      .style('stroke', function(d) { return d3.rgb(colors(d.id)).darker().toString(); });

    // show node names
    gC.append('svg:text')
        .attr('y', 4)
        .attr('class', 'name')
        .text(function(d) { return d.name; });

    gC.on('mouseover', function(d) {
        if(!mousedown_node || d === mousedown_node) {return}
        // e.target target node
        d3.select(this).select('circle').transition().attr('transform', 'scale(1.1)');
      })
      .on('mouseout', function(d) {
        if(!mousedown_node || d === mousedown_node) {return}
        // unenlarge target node
        d3.select(this).select('circle').transition().attr('transform', '');
      })
      .on('mousedown', function(d) {
        if(d3.event.altKey) {return}

        // select node
        mousedown_node = d;
        selected_node = mousedown_node;
        selected_link = null;

        textBox.attr('placeholder', "New Node Name");

        // reposition drag line
        drag_line
          .style('marker-end', 'url(#end-arrow)')
          .classed('hidden', false)
          .attr('d', 'M' + mousedown_node.x + ',' + mousedown_node.y + 'L' + mousedown_node.x + ',' + mousedown_node.y);

        restart();
      })
      .on('mouseup', function(d) {

        if(!mousedown_node) {return}

        // needed by FF
        drag_line
          .classed('hidden', true)
          .style('marker-end', '');

        // check for drag-to-self
        mouseup_node = d;
        if(mouseup_node === mousedown_node) { resetMouseVars(); return; }

        // unenlarge target node
        d3.select(this).select('circle').transition().attr('transform', '');

        // add link to graph (update if exists)
        // NB: links are strictly source < target; arrows separately specified by booleans
        var source = mousedown_node,
            target = mouseup_node;

        var link = networkGraph.addLink({source: source, target: target});
        textBox.attr('placeholder', "New Max Capacity");

        // select new link
        selected_link = link;
        selected_node = null;
        restart();
      });

    // remove old nodes
    circle.exit().remove();

    // set the graph in motion
    force.start();
  }

  // ========== Listeners ===========

  // ========== Mouse Listeners ===========
  function mousedown() {
    svg.classed('active', true);
    
    if(d3.event.which !== 1) {return}
    if(d3.event.altKey || mousedown_node || mousedown_link) {return}

    // insert new node at point
    var point = d3.mouse(this),
        node = networkGraph.addNode(point);

    textBox.attr('placeholder', "New Node Name");

    selected_link = null;
    selected_node = node;
    restart();
  }

  function mousemove() {
    if (! mousedown_node) {return}

    drag_line.attr('d', 'M' + mousedown_node.x + ',' + mousedown_node.y + 'L' + d3.mouse(this)[0] + ',' + d3.mouse(this)[1]);
    restart();
  }

  function mouseup() {
    if (mousedown_node) {
      drag_line
        .classed('hidden', true)
        .style('marker-end', '');
    }

    svg.classed('active', false);

    // clear mouse event vars
    resetMouseVars();
  }

  // ========== Key Listeners ===========
  function keydown() {
    if (d3.event.altKey) {
      circle.call(drag);
      svg.classed('drag', true);
    }
  }
  function keyup() {
    circle
      .on('mousedown.drag', null)
      .on('touchstart.drag', null);
    svg.classed('drag', false);
  }
  function fKey() {
    if (! selected_node) {return}

    if (! selected_node.fixed) {
      selected_node.fixed = true;
    } else {
      selected_node.fixed = false;
    }
    restart();
  }
  function rmObj() {
    // Backspace should not navigate back
    d3.event.preventDefault();

    if (selected_node) {
      networkGraph.deleteNode(selected_node.id);
    } else if(selected_link) {
      networkGraph.deleteLinks(function(l) {return l === selected_link; });
    }
    selected_link = null;
    selected_node = null;

    resetFlow();
    restart();
  }

  d3.select(window)
      .onKey('backspace/del', rmObj)
      .onKey('f', fKey);

  // app starts here
  svg.on('mousedown', mousedown)
    .on('mousemove', mousemove)
    .on('mouseup', mouseup);
  d3.select(window)
    .on('keydown', keydown)
    .on('keyup', keyup);
  restart();



  // JSONify graph
  var transBtnContainer = body.append('div')
      .attr('class', 'transbtncontainer');

  var transcribeButton = transBtnContainer.append('button')
      .attr('class', 'transcribe')
      .attr("name", "solve")
      .attr("type", "button")
      .text("Transcribe")
      .on('click', function(){
        transcribeArea[0][0].value = networkGraph.toJSON();
      });

  var renderButton = transBtnContainer.append('button')
      .attr('class', 'render')
      .attr("name", "solve")
      .attr("type", "button")
      .text("Render")
      .on('click', renderGraph);

  window.autoBoxResize = function () {
    var preHeight = this.style.height;
    this.style.height = 0;
    var height = this.scrollHeight;
    this.style.height = preHeight;
    d3.select(this)
      .transition()
        .attr('style', 'height:' + height + 'px');
  };

  var transcribeArea = body.append('textarea')
      .attr('class', 'transcribe')
      .on('keyup', autoBoxResize)
      .on('mousedown', autoBoxResize);

  function renderGraph () {
    // Flush

    networkGraph = Graph.fromJSON(transcribeArea[0][0].value);
    restart();
  }
}

makeEditor();

flowview.js

function flowView(){
  "use strict";

  var margin = {top: 20, right: 10, bottom: 20, left: 10};
  var width = 960 - margin.left - margin.right,
      height = 800 - margin.top - margin.bottom,
      colors = d3.scale.category20(),
      percentile = width/10;

  var index = 0;
  // Build a log viewer
  d3.select('#app-container').remove();
  var body = d3.select('body')
      .append("div")
    .attr('id', 'app-container');

  var navContainer = body.append('div')
      .attr('class', 'navcontainer');

  var editor = navContainer.append('button')
      .attr('id', 'editor')
      .attr("name", "editor")
      .attr("type", "button")
      .attr("style", "display:block; margin:auto;")
      .text("Editor")
      .on('click', function() {
        var loc = window.location.href,
            editor = loc.split('flowview')[0];
        window.history.pushState(null,null,editor);
        makeEditor();
      });

  var back = navContainer.append('button')
      .attr('id', 'back')
      .attr("name", "back")
      .attr("type", "button")
      .text("<")
      .on('click', function() {showStep(--index);});

  var forward = navContainer.append('button')
      .attr('id', 'forward')
      .attr("name", "forward")
      .attr("type", "button")
      .text(">")
      .on('click', function() {showStep(++index);});


  var svg = body.append('svg')
      .attr('class', 'steps')
      .attr('width', width)
      .attr('height', height);

  // Make defs for links (arrows, etc)
  var defs = svg.append('svg:defs');
  defs.append('svg:marker')
      .attr('id', 'end-arrow-flow')
      .attr('viewBox', '0 -5 10 10')
      .attr('refX', 10)
      .attr('markerWidth', 10)
      .attr('markerHeight', 10)
      .attr('markerUnits', "userSpaceOnUse")
      .attr('orient', 'auto')
    .append('svg:path')
      .attr('d', 'M0,-5L10,0L0,5');

  defs.append('svg:marker')
      .attr('id', 'end-arrow-res')
      .attr('viewBox', '0 -5 10 10')
      .attr('refX', 10)
      .attr('refY', -0.5)
      .attr('markerWidth', 10)
      .attr('markerHeight', 10)
      .attr('markerUnits', "userSpaceOnUse")
      .attr('orient', 'auto')
    .append('svg:path')
      .attr('d', 'M0,-5L10,0L0,5');

  // Make flow and residual graphs. (non-interactive)
  var flowG = svg.append('g')
      .attr('class', 'flow');
  var residualG = svg.append('g')
      .attr('class', 'residual')
      .attr('transform', 'translate(0,'+ (height/2) + ')');
  
  // === initialize flow and residual

  function chargeF (d) {
    // Source and sink should have more charge
    if (d.name === 's' || d.name === 't') {
      return -5000;
    }
    return -4000;
  }

  function forceTick() {
    flowPath.selectAll("path").attr("d", linkArc);
    flowCircle.attr("transform", transform);

    function transform(d) {
      return "translate(" + d.x + "," + d.y + ")";
    }
    function linkArc(d) {
      var dx = d.target.x - d.source.x,
          dy = d.target.y - d.source.y,
          dist = Math.sqrt(dx * dx + dy * dy),
          normX = dx / dist,
          normY = dy / dist,
          sourcePadding = 10,
          targetPadding = 10,
          sourceX = d.source.x + (sourcePadding * normX),
          sourceY = d.source.y + (sourcePadding * normY),
          targetX = d.target.x - (targetPadding * normX),
          targetY = d.target.y - (targetPadding * normY);
      return 'M' + sourceX + ',' + sourceY + 'L' + targetX + ',' + targetY;
    }
  }
  var flowForce = d3.layout.force()
      .nodes([])
      .links([])
      .size([width, height/2])
      .linkDistance(150)
      .gravity(0.1)
      .charge(chargeF)
      .on("tick", forceTick)
      .start();

  var flowPath = flowG.append("g").selectAll("g");
  var flowCircle = flowG.append("g").selectAll("g");

  function resTick() {
    resPath.selectAll("path").attr("d", linkArc);
    resCircle.attr("transform", transform);

    function transform(d) {
      return "translate(" + d.x + "," + d.y + ")";
    }
    function linkArc(d) {
      var dx = d.target.x - d.source.x,
          dy = d.target.y - d.source.y,
          dist = Math.sqrt(dx * dx + dy * dy),
          dr = dist*2,
          normX = dx / dist,
          normY = dy / dist,
          sourcePadding = 10,
          targetPadding = 10,
          sourceX = d.source.x + (sourcePadding * normX),
          sourceY = d.source.y + (sourcePadding * normY),
          targetX = d.target.x - (targetPadding * normX),
          targetY = d.target.y - (targetPadding * normY);
      return "M" + sourceX + "," + sourceY + "A" + dr + "," + dr + " 0 0,1 " + targetX + "," + targetY;
    }
    // resPath.select('tspan')
    //     .filter(function(d) { return d.source.x > d.target.x })
    //   .attr('rotate', 180)
    //   .attr('style', 'text-anchor: start; dominant-baseline:text-before-edge;');
  }
  
  var resForce = d3.layout.force()
      .nodes([])
      .links([])
      .size([width, height/2])
      .linkDistance(200)
      .gravity(0.1)
      .charge(chargeF)
      .on("tick", resTick)
      .start();
  var resPath = residualG.append("g").selectAll("g");
  var resCircle = residualG.append("g").selectAll("g");


  function showStep(index) {
    var logItem = log[index],
        path = logItem.path,
        flow = new Graph(logItem.flow.vertices, logItem.flow.edges),
        residual = new Residual(flow, logItem.residual);

    back.attr("disabled", function(){ return index <= 0 ? true : null; });
    forward.attr("disabled", function(){ return index >= log.length-1 ? true : null; });

    // Flow
    flowForce.nodes(flow.nodes())
        .links(flow.links());

    // Build svg of flow and residual.
    flowPath = flowPath.data(flow.links(), function(l) { return l.source.id.toString() + l.target.id.toString(); });
    var flowLinkEnter = flowPath.enter().append("g");

    flowLinkEnter
        .attr("class", "link")
      .append("path")
        .attr("class", "link")
        .attr("marker-end", "url(#end-arrow-flow)")
        .attr('id', function(d) {return 'flow' + d.source.id.toString() + '-' + d.target.id.toString();});

    flowLinkEnter.append('text')
      .append('textPath')
        .attr('xlink:href', function (d) {
          return "#flow" + d.source.id.toString() + '-' + d.target.id.toString();
        })
        .attr('startOffset', '50%')
      .append('tspan')
        .attr('dy', -5);

    flowPath
        .classed('sending', function(d) {
          if (! path) {return false}
          var sourceIndex = path.indexOf(d.source.id),
              targetIndex = path.indexOf(d.target.id);
          return sourceIndex >=0 && targetIndex >= 0 && sourceIndex === targetIndex-1;
        })
        .classed('resending', function(d) {
          if (! path) {return false}
          var sourceIndex = path.indexOf(d.source.id),
              targetIndex = path.indexOf(d.target.id);
          return sourceIndex >=0 && targetIndex >= 0 && sourceIndex === targetIndex+1;
        });


    flowPath.select('text')
      .select('tspan')
        .text(function(d) {
          if (d.flow) {return d.flow + " / " + d.capacity;}
          else {return 0 + " / " + d.capacity;}
        });

    flowCircle = flowCircle.data(flow.nodes(), function(n){ return n.id; });
    var flowNodeEnter = flowCircle.enter().append("g")
        .attr('class', 'node')
        .attr('transform', function(d) {
          if (!d.x || !d.y) {return ""}
          return 'translate(' + d.x + ',' + d.y + ')';
        });

    flowNodeEnter.append("circle")
        .attr('r', 12)
        .attr("fill", function(d){ return colors(d.id) })
        .style('stroke', function(d) { return d3.rgb(colors(d.id)).darker().toString(); });

    flowNodeEnter.append("text")
        .attr("x", 8)
        .attr('class', 'name')
        .attr('x', 0)
        .attr('y', 4)
        .text(function(d) { return d.name; });

    flowPath.exit().remove();
    flowCircle.exit().remove();

    // Residual
    resForce.nodes(residual.nodes())
        .links(residual.links());

    // Build svg of flow and residual.
    resPath = resPath.data(residual.links(), function(l) { return l.source.id.toString() + l.target.id.toString(); });
    var resLinkEnter = resPath.enter().append("g");
    
    resLinkEnter
        .attr("class", "link")
      .append("path")
        .attr("class", "link")
        .attr("marker-end", "url(#end-arrow-res)")
        .attr('id', function(d) {return 'res' + d.source.id.toString() + d.target.id.toString();});

    resLinkEnter.append('text')
        .attr("text-anchor", "middle")
      .append('textPath')
        .attr('xlink:href', function (d) {
          return "#res" + d.source.id.toString() + d.target.id.toString();
        })
        .attr('startOffset', '50%')
      .append('tspan');


    resPath
        .classed('sending', function(d) {
          if (! path) {return false}
          var sourceIndex = path.indexOf(d.source.id),
              targetIndex = path.indexOf(d.target.id);
          return sourceIndex >=0 && targetIndex >= 0 && sourceIndex === targetIndex+1;
        })
        .classed('resending', function(d) {
          if (! path) {return false}
          var sourceIndex = path.indexOf(d.source.id),
              targetIndex = path.indexOf(d.target.id);
          return sourceIndex >=0 && targetIndex >= 0 && sourceIndex === targetIndex-1;
        });

    resPath.select('text')
      .select('tspan')
        .text(function(d) {
          return d.flow || 0;
        });
    

    resCircle = resCircle.data(residual.nodes(), function(n){ return n.id; });
    var resNodeEnter = resCircle.enter().append("g")
        .attr('class', 'node')
        .attr('transform', function(d) {
          if (!d.x || !d.y) {return ""}
          return 'translate(' + d.x + ',' + d.y + ')';
        });
        
    resNodeEnter.append("circle")
        .attr('r', 12)
        .attr("fill", function(d){ return colors(d.id) })
        .style('stroke', function(d) { return d3.rgb(colors(d.id)).darker().toString(); });

    resNodeEnter.append("text")
        .attr('class', 'name')
        .attr('x', 0)
        .attr('y', 4)
        .text(function(d) { return d.name; });

    resPath.exit().remove();
    resCircle.exit().remove();

    resForce.stop();
    flowForce.stop();

  }

  showStep(index);
  resForce.start();
  flowForce.start();
}

fordfulkerson.js

function fordFulkerson(vs, es, cb) {
  "use strict";
  // Make a copy of the edges.
  es = es.slice(0).map(function(o) {
    return {
      source: o.source,
      target: o.target,
      capacity: o.capacity,
    };
  });
  var rG = initResidual(vs,es),
      path = null,
      log = [];

  // Initialize a residual graph structure.
  // returns a map of vetex id's to an array of their outgoing edges.
  function initResidual(vs,es) {
    var g = d3.map(),
        outgoing,
        incoming;

    vs.forEach(function(v) {
      g.set(v.id, []);
    });

    es.forEach(function(e) {
      var bEdge = {target: e.target.id, flow: e.capacity};
      var fEdge = {target: e.source.id, flow: 0};

      outgoing = g.get(e.source.id) || [];
      outgoing.push(bEdge);

      incoming = g.get(e.target.id) || [];
      incoming.push(fEdge);

      g.set(e.source.id, outgoing);
      g.set(e.target.id, incoming);
    });

    return g;
  }
  function findAugmentingP(rG) {
    // run dfs to find s-t path, return the path seq.
    var explored = d3.map();

    function dfs(path) {
      var u = path[path.length-1];
      explored.set(u, true);

      if (u === 1) { return path; }

      // Consider edges which have nonzero residual flow.
      var es = rG.get(u).filter(function(e){return e.flow>0;});

      if (!es || es.length === 0) { return false; }

      for (var i=0; i<es.length; i++) {
        var tgt = es[i].target;
        if (! explored.get(tgt)) {
          var newpath = path.slice(0);
          newpath.push(tgt);
          var p = dfs(newpath);
          
          if (p) { return p; }
        }
      }
    }
    return dfs([0]);
  }
  function augment(path, rG) {
    function bottleneck(path, rG) {
      // return the min residual capacity of any edge on the path.
      path = path.reverse();
      var u = path.pop(),
          min = parseFloat('Infinity');
      
      for (var v; v = path.pop();) {
        var edge = rG.get(u).filter(function(e) {return e.target === v})[0];
        if (edge.flow < min) {min = edge.flow;}

        u = v;
      }
      return min;
    }

    var b = bottleneck(path.slice(0), rG);
    var u = 0;
    path.forEach(function(v) {
      if (u === v) {return;}

      var bEdge = rG.get(u).filter(function(e) {return e.target === v})[0];
      var fEdge = rG.get(v).filter(function(e) {return e.target === u})[0];

      fEdge.flow += b;
      bEdge.flow -= b;

      u = v;
    });
  }
  // Returns [maxflow, edges]
  function makeFlow(rG, vs, es) {

    // Display the result flow
    es.forEach(function(l) {
      var edge = rG.get(l.target.id).filter(function(e) { return e.target === l.source.id})[0];
      l.flow = edge.flow;
    });

    // Display the max flow
    var sinkIncedent = rG.get(1);
    var total = 0;
    sinkIncedent.forEach(function (e) {
      total += e.flow;
    });

    return {
      maxflow: total,
      vertices: vs,
      edges: es
    };
  }
  function logState(rG, path) {
    var newRG = JSON.parse(JSON.stringify(rG));
    var newGraph = new Graph(vs, es);
    newGraph = Graph.fromJSON( newGraph.toJSON() );
    var newVs = newGraph.nodes(),
        newEs = newGraph.links();

    log.push({
      flow: makeFlow(rG, newVs, newEs),
      residual: newRG,
      path: path
    });
  }

  // === Main Loop ===

  logState(rG, path);

  while (path = findAugmentingP(rG)) {
    augment(path, rG);
    logState(rG, path);
  }

  console.debug("Residual Graph flow: ", rG);
  
  return [makeFlow(rG, vs, es), log];
}

graphs.js

(function(){
  "use strict";

  // Graph Class
  window.Graph = Graph;
  function Graph (nodes, links) {
    var self = this;
    this._nodes = nodes;
    this._lastLinkId = 0;
    this._links = [];
    links.forEach(function(l) { self.addLink(l) });
    // Get the max id so that the added nodes will have a unique id.
    this._lastNodeId = nodes.reduce(function( acc, x ){
      return x.id > acc ? x.id : acc;
    }, 0);

    return this;
  }
  
  Graph.prototype.toJSON = function () {

    // Make a new copy of links.
    var es = JSON.parse(JSON.stringify(this._links));
    es.map(function(e) {
      e.source = e.source.id;
      e.target = e.target.id;
      return e;
    });

    var graph = {
      vertices: this._nodes,
      edges: es,
    },
    graphJSON = JSON.stringify(graph, null, 2);

    return graphJSON;
  };

  Graph.fromJSON = function (JSONString) {
    var graphObj = JSON.parse(JSONString);

    var ns = graphObj.vertices;

    var ls = graphObj.edges.map(function(e){
      var sourceId = e.source,
          targetId = e.target;
      e.source = ns.filter(function(n) {
        return n.id === sourceId;
      })[0];
      e.target = ns.filter(function(n) {
        return n.id === targetId;
      })[0];
      return e;
    });

    return new Graph(ns, ls);
  };



  Graph.prototype.resetFlow = function () {
    this._links.forEach(function(l) {
      l.flow = 0;
    });
  };

  Graph.prototype.nodes = function () {
    return this._nodes;
  };

  Graph.prototype.links = function () {
    return this._links;
  };

  Graph.prototype.addNode = function (point) {
    var node =
      { id: ++this._lastNodeId
      , x: point[0]
      , y: point[1]
      };

    this._nodes.push(node);
    return node;
  };

  Graph.prototype.getNode = function (id) {
    if (! this._nodes) {return null}

    var res = this._nodes.filter(function (n) {
      return n.id === id;
    });

    if (res.length === 0) {return null}

    return res[0];
  };

  Graph.prototype.deleteNode = function (id) {
    this._nodes = this._nodes.filter(function(n) { return n.id !== id; });

    // Remove links incident to node
    this.deleteLinks(function(l) {
      return (l.source.id === id || l.target.id === id);
    });

  };

  Graph.prototype.addLink = function (link) {
    var source = link.source,
        target = link.target;

    var previous = this.getLink(source, target);

    if (previous) {return previous}

    link.id = Date.now().toString() + (++this._lastLinkId).toString();

    this._links.push( link );

    return link;
  };
  Graph.prototype.getLink = function (source, target) {
    if (!this._links) {return null}

    var res = this._links.filter(function(l) {
      return (l.source === source && l.target === target);
    });

    if (res.length === 0) {return null}

    return res[0];
  };

  Graph.prototype.deleteLinks = function (cb) {
    var self = this;
    var toSplice = this._links.filter(cb);

    toSplice.map(function(l) {
      self._links.splice(self._links.indexOf(l), 1);
    });
  };


  // Residual Sub-Class
  window.Residual = Residual;
  function Residual (graph, preres) {
    var self = this,
        ns = [],
        ls = [];

    var id;
    for (id in preres) {
      var es = preres[id];

      id = parseInt(id.charAt(1));
      ns.push({id: id});
      // No need to check for dupes.
      es.forEach(function (e) {
        ls.push({
          source: id,
          target: parseInt(e.target),
          flow: e.flow
        });
      });
    }

    ns = ns.map(function(n) {
      var node = graph.getNode(n.id);
      return JSON.parse( JSON.stringify(node) );
    });

    ls = ls.map(function(e){
      var sourceId = e.source,
          targetId = e.target;
      e.source = ns.filter(function(n) {
        return n.id === sourceId;
      })[0];
      e.target = ns.filter(function(n) {
        return n.id === targetId;
      })[0];
      return e;
    });

    return new Graph(ns, ls);
  }
})();

index.html

<!DOCTYPE html>
<html>
  <head>
    <meta charset="utf-8">
    <title>Network Flow Solver</title>
    <link rel="stylesheet" href="editor.css">
  </head>
  
  <body>

  <script src="d3.min.js" charset="utf-8"></script>
  <script src="keybinding.js"></script>
  <script src="graphs.js"></script>
  <script src="fordfulkerson.js"></script>
  <script src="flowview.js"></script>
  <script src="editor.js"></script>
</html>

keybinding.js

// How to use keybindings:
// d3.select('body').onKey('left', function(d,i) {debugger;})
(function() {
  "use strict";
  d3.selection.prototype.onKey = function(keyString, cb) {
    var keybinding = d3.keybinding.apply(this);
    return this.call(keybinding.on(keyString, cb));
  };
})();

/*
 * This code is licensed under the MIT license.
 *
 * Copyright © 2013, iD authors.
 *
 * Portions copyright © 2011, Keith Cirkel
 * See https://github.com/keithamus/jwerty
 *
 */
d3.keybinding = function(namespace) {
  "use strict";
  this.bindings = this.bindings || [];
  var bindings = this.bindings;

  function matches(binding, event) {
    for (var p in binding.event) {
      if (event[p] != binding.event[p])
        {return false}
    }

    return ( (! binding.capture) === (event.eventPhase !== Event.CAPTURING_PHASE) );
  }

  function capture() {
    for (var i = 0; i < bindings.length; i++) {
      var binding = bindings[i];
      if (matches(binding, d3.event)) {
        binding.callback();
      }
    }
  }

  function bubble() {
    var tagName = d3.select(d3.event.target).node().tagName;
    if (tagName == 'INPUT' || tagName == 'SELECT' || tagName == 'TEXTAREA') {
      return;
    }
    capture();
  }

  function keybinding(selection) {
    selection = selection || d3.select(document);
    selection.on('keydown.capture' + namespace, capture, true);
    selection.on('keydown.bubble' + namespace, bubble, false);
    return keybinding;
  }

  keybinding.off = function(selection) {
    selection = selection || d3.select(document);
    selection.on('keydown.capture' + namespace, null);
    selection.on('keydown.bubble' + namespace, null);
    return keybinding;
  };

  keybinding.on = function(code, callback, capture) {
    var codes = code.toLowerCase().split('/');

    codes.forEach(function(code) {
      var binding = {
        event: {
          keyCode: 0,
          shiftKey: false,
          ctrlKey: false,
          altKey: false,
          metaKey: false
        },
        capture: capture,
        callback: callback
      };

      code = code.match(/(?:(?:[^+])+|\+\+|^\+$)/g);

      for (var i = 0; i < code.length; i++) {
        // Normalise matching errors
        if (code[i] === '++') {code[i] = '+'}

        if (code[i] in d3.keybinding.modifierCodes) {
          binding.event[d3.keybinding.modifierProperties[d3.keybinding.modifierCodes[code[i]]]] = true;
        } else if (code[i] in d3.keybinding.keyCodes) {
          binding.event.keyCode = d3.keybinding.keyCodes[code[i]];
        }
      }

      bindings.push(binding);
    });

    return keybinding;
  };

  return keybinding;
};

(function () {
  "use strict";

  d3.keybinding.modifierCodes = {
    shift: 16,
    ctrl: 17,
    alt: 18, option: 18,
    meta: 91, cmd: 91, 'super': 91, win: 91
  };

  d3.keybinding.modifierProperties = {
    16: 'shiftKey',
    17: 'ctrlKey',
    18: 'altKey',
    91: 'metaKey'
  };

  d3.keybinding.keyCodes = {
    backspace: 8,
    tab: 9,
    'return': 13, enter: 13,
    'pause': 19, 'pause-break': 19,
    caps: 20, 'caps-lock': 20,
    esc: 27, escape: 27, 
    space: 32,
    pgup: 33, 'page-up': 33,
    pgdown: 34, 'page-down': 34,
    end: 35,
    home: 36,
    ins: 45, insert: 45,
    del: 46, 'delete': 46,
    left: 37, 'arrow-left': 37,
    up: 38, 'arrow-up': 38,
    right: 39, 'arrow-right': 39,
    down: 40, 'arrow-down': 40,
    '*': 106, star: 106, asterisk: 106, multiply: 106,
    '+': 107, 'plus': 107,
    '-': 109, minus: 109, subtract: 109,
    ';': 186, semicolon:186,
    '=': 187, 'equals': 187,
    ',': 188, comma: 188,
    '.': 190, period: 190, 'full-stop': 190,
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Makefile

SOURCE := more.md
TARGET := more.html
STYLE := more.css

all: 
	pandoc --mathml -t html -Ss -o $(TARGET) --css $(STYLE) $(SOURCE)

more.css

@import 'editor.css';

#container {
  width: 80%;
  min-width: 800px;
  margin: auto;
}
h1, h2, h3, h4, h5 {
  color: brown;
}
a {
  color: steelblue;
}

.svg-container {
  text-align: center;
}

/* Make the cursor imply that the svg is static */
svg {
  cursor: default !important;
}
svg circle,path {
  cursor: default !important;
}

more.html

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
  <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
  <meta http-equiv="Content-Style-Type" content="text/css" />
  <meta name="generator" content="pandoc" />
  <title></title>
  <style type="text/css">code{white-space: pre;}</style>
  <link rel="stylesheet" href="more.css" type="text/css" />
  <script type="text/javascript">/*<![CDATA[*/
  /* 
  March 19, 2004 MathHTML (c) Peter Jipsen http://www.chapman.edu/~jipsen
  Released under the GNU General Public License version 2 or later.
  See the GNU General Public License (at http://www.gnu.org/copyleft/gpl.html)
  for more details.
  */
  
  function convertMath(node) {// for Gecko
    if (node.nodeType==1) {
      var newnode = 
        document.createElementNS("http://www.w3.org/1998/Math/MathML",
          node.nodeName.toLowerCase());
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                       "\""+node.attributes[j].nodeValue+"\"";
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  if(typeof window.addEventListener != 'undefined'){
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  if(typeof window.attachEvent != 'undefined') {
    window.attachEvent('onload', convert);
  }
  /*]]>*/
  </script>
</head>
<body>
<div id="container">
<h1 id="network-flow">Network Flow</h1>
<h2 id="introduction">Introduction</h2>
<p>I was first introduced to network flow in CS201: Algorithms in my final quarter of my time as an undergraduate at UC Santa Cruz. The class was taught by Professor Dimitris Achilloptas. The course loosely followed the text <a href="http://books.google.com/books/about/Algorithm_Design.html?id=OiGhQgAACAAJ">Algorithm Design</a> by Jon Kleinberg and Eva Tardos. After covering the standard Greedy algorithms, Divide and Conquer, and Dynamic Programming, and with only a few weeks left of the quarter, we began studying Network Flow. I found the problems extremely challenging and inspiring, and even after graduating am still thinking about then.</p>
<h2 id="network-flow-preliminaries">Network Flow Preliminaries</h2>
<p>First and foremost, a Network Flow is defined on a Digraph or directed Graph.</p>
<p><em>Definition:</em> A <strong>Graph</strong> is defined by a set of vertices and edges <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></math>, such that for all <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math>, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>e</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow></math> where <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo>∈</mo><mi>V</mi></mrow></math>.</p>
<p><em>Definition:</em> In a <strong>Digraph</strong> we make the distinction that for all <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math>, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>e</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math>. This simply indicates that the edges have a direction.</p>
<p><em>Definition:</em> A <strong>Network</strong> is defined by extending a Digraph <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></math> by a specification of the capacity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>∀</mo><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math> and two special nodes the source <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math> and the sink <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi></mrow></math>.</p>
<p><em>Definition:</em> A <strong>Network Flow</strong> is an abstraction for material flowing throught the graph. Finding the solution to a network flow consists of determining the maximum flow which can be pushed throught the network from the source <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math> to the sink <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi></mrow></math>. As we’ll learn later, this is in fact equivalent to what is called the <em>minimum cut</em> of the Network.</p>
<p><em>Remark:</em> For our purposes we’ll be considering network flows with integer capacity only.</p>
<h2 id="considering-a-trivial-flow">Considering a Trivial Flow</h2>
<p>Consider the trivial flow.</p>
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</svg>
</div>

<p>In this flow there is a single edge which has capacity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow></math> from the source node <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math>, to the sink node <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi></mrow></math>. Clearly, given this flow we may have a maximum flow of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn></mrow></math> through the network.</p>
<p>The flow below illustrates a core concept in network flow, namely that the maximum flow is equal to the minimum cut. We’ll see what this means in a minute, but in this simple network, the flow is limited by the edge with the least capacity.</p>
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<svg height="160" width="580" class="editor">
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</svg>
</div>

<h2 id="maximum-flow-and-minimum-cut">Maximum Flow and Minimum Cut</h2>
<p>Lets begin with the definitions.</p>
<p><em>Definition:</em> The <strong>Maximum Flow</strong> of a network is the maximum units of flow which can propagate through the network from the source to the sink given the capacities of each edge.</p>
<p><em>Definition:</em> Technically speaking, a <strong>cut</strong> is a partitioning of the vertices in a graph. With respect to a network a <strong><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext mathvariant="bold">st</mtext></mrow></math>-cut</strong> is a partition <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">}</mo></mrow></math> such that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>∈</mo><mi>A</mi></mrow></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>∈</mo><mi>B</mi></mrow></math>. Consider the following example where the vertices in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi></mrow></math> are blue and the vertices in the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>B</mi></mrow></math> are brown.</p>
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</div>

<p><em>Definition:</em> The <strong>cut-set</strong> is the set of edges whose vertices lie in different sets of the partition.</p>
<p>Below, the brown edges are in the cut-set of the above cut.</p>
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<p><em>Definition:</em> With respect to a cut, a <strong>forward directed edge</strong> or <strong>forward edge</strong> is an edge which has flow from a vertex in the source’s partition to a vertex in the sink’s partition.</p>
<p><em>Definition:</em> Finally, the <strong>Minimum Cut</strong> is the cut for which the sum of the weights of the forward directed edges in the cut-set’s are minimuized. In other words, The Minimum Cut is the cut <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>P</mi></mrow></math> corresponding to the cut-set <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi></mrow></math> of forward edges such that for all valid cut-sets <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>K</mi></mrow></math>, we have:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo>∑</mo><mrow><mo>∀</mo><mi>e</mi><mo>∈</mo><mi>K</mi></mrow></munder><mrow><mi>c</mi><mi>a</mi><mi>p</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><mo>&gt;</mo><munder><mo>∑</mo><mrow><mo>∀</mo><mi>e</mi><mo>∈</mo><mi>C</mi></mrow></munder><mrow><mi>c</mi><mi>a</mi><mi>p</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mrow></math></p>
<p>Continuing the above example, let us find the minimum cut. Can you find it? How would you go about finding the cut? The most obvious way is to just apply the definition and consider all valid cuts, then take the one with the minimum sum of capacities. This has complexity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mrow><mo stretchy="false">∣</mo><mi>V</mi><mo stretchy="false">∣</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math>, which is very poor. I should note that this is a loose upper bound, for a more precise bound see: <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind">Stirling Numbers of the second kind</a>.</p>
<p>The following is the minimum cut for the given graph.</p>
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<p>For the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mi>t</mi></mrow></math>-cut we have <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>s</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow></math>, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>t</mi><mo stretchy="false">}</mo></mrow></math> and for the cut-set we have <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></math>. Notice the subtlety here: although the edge <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> is in the cut-set it is not a forward edge, hence we do not count its capacity. It follows that the value of the minimum cut is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>19</mn></mrow></math>.</p>
<p><em>Remark:</em> For a proof that the max-flow<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>≡</mo></mrow></math>the min-cut see the text listed in the resources or the acompanying slides.</p>
<h2 id="ford-fulkerson">Ford-Fulkerson</h2>
<p>As the name implies the Ford-Fulkerson algorithm was developed by Ford and Fulkerson. The algorithm is a means to solve for the max-flow / min-cut.</p>
<p><em>Definition:</em> The <strong>residual graph</strong> of a network flow is isomorphic to the original network. The verticies remain unchanged. The edges are mapped as follows: consider an edge <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> with capacity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi></mrow></math> and flow <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi></mrow></math>. In the residual graph, this information is represented as an edge <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math> with a value of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>−</mo><mi>f</mi></mrow></math> and an edge <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></math> with a value of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi></mrow></math>. (See the example below.)</p>
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<p>You may ask yourself why we bother with the arguably more complicated representation of the same information. The beauty of this representation is that we can simply do a search from the source for the sink to find a path to “push” more flow through. We call this leftover flow <strong>residual capacity</strong> and such paths <strong>augmenting paths</strong>. Can you spot an augmenting path in the above flow?</p>
<p>Yep, you guessed it! (Well, maybe you didn’t, but in fact there are two valid augmenting paths.)</p>
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<p>Consider the above graph, can you find another augmenting path? The answer is simple: no. In fact, we can’t even leave the source since there is no remaining capacity to vertex <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi></mrow></math>. This is an excellent certificate that we have pushed as much flow through the network as is possible.</p>
<p>Lets backtrack a bit. What is this algorithm anyway? Informally, Ford-Fulkerson goes like this:</p>
<ol>
<li>Map the network to a residual graph.</li>
<li>Begin a search for the sink at the source along non-zero edges. Two possible cases result.
<ol style="list-style-type: lower-roman">
<li>We cannot reach the sink, and hence have our certificate of having saturated the network.</li>
<li>We reach the sink.
<ul>
<li>Push as much flow as possible through the path.</li>
<li>Start a new search.</li>
</ul></li>
</ol></li>
</ol>
<p>Pretty simple stuff, and again for proof see the text or slides.</p>
<h2>Applications</h2>
<ul>
<li>TODO</li>
</ul>
<h2>Notes and References</h2>
<ul>
<li>The book I have is <a href="http://books.google.com/books/about/Algorithm_Design.html?id=OiGhQgAACAAJ">Algorithm Design</a> by Jon Kleinberg and Eva Tardos.</li>
<li>The slides which acompany the text are available online <a href="http://www.cs.princeton.edu/~wayne/kleinberg-tardos/">here</a>.</li>
</ul>
</div>
</body>
</html>

more.md

# Network Flow

Introduction

I was first introduced to network flow in CS201: Algorithms in my final quarter of my time as an undergraduate at UC Santa Cruz. The class was taught by Professor Dimitris Achilloptas. The course loosely followed the text Algorithm Design by Jon Kleinberg and Eva Tardos. After covering the standard Greedy algorithms, Divide and Conquer, and Dynamic Programming, and with only a few weeks left of the quarter, we began studying Network Flow. I found the problems extremely challenging and inspiring, and even after graduating am still thinking about then.

Network Flow Preliminaries

First and foremost, a Network Flow is defined on a Digraph or directed Graph.

Definition: A Graph is defined by a set of vertices and edges $G = (V, E)$, such that for all $e \in E$, $e = {v_1, v_2}$ where $v_1, v_2 \in V$.

Definition: In a Digraph we make the distinction that for all $e \in E$, $e = (v_1, v_2)$. This simply indicates that the edges have a direction.

Definition: A Network is defined by extending a Digraph $G=(V,E)$ by a specification of the capacity $c(e)$ $\forall e \in E$ and two special nodes the source $s$ and the sink $t$.

Definition: A Network Flow is an abstraction for material flowing throught the graph. Finding the solution to a network flow consists of determining the maximum flow which can be pushed throught the network from the source $s$ to the sink $t$. As we'll learn later, this is in fact equivalent to what is called the minimum cut of the Network.

Remark: For our purposes we'll be considering network flows with integer capacity only.

Considering a Trivial Flow

Consider the trivial flow.

0 / 1st

In this flow there is a single edge which has capacity $1$ from the source node $s$, to the sink node $t$. Clearly, given this flow we may have a maximum flow of $1$ through the network.

The flow below illustrates a core concept in network flow, namely that the maximum flow is equal to the minimum cut. We'll see what this means in a minute, but in this simple network, the flow is limited by the edge with the least capacity.

0 / 20 / 1sta

Maximum Flow and Minimum Cut

Lets begin with the definitions.

Definition: The Maximum Flow of a network is the maximum units of flow which can propagate through the network from the source to the sink given the capacities of each edge.

Definition: Technically speaking, a cut is a partitioning of the vertices in a graph. With respect to a network a $\textbf{st}$-cut is a partition ${A,B}$ such that $s \in A$ and $t \in B$. Consider the following example where the vertices in $A$ are blue and the vertices in the $B$ are brown.

0 / 100 / 100 / 20 / 40 / 80 / 90 / 100 / 60 / 10stabcd

Definition: The cut-set is the set of edges whose vertices lie in different sets of the partition.

Below, the brown edges are in the cut-set of the above cut.

0 / 100 / 100 / 20 / 40 / 80 / 90 / 100 / 60 / 10stabcd

Definition: With respect to a cut, a forward directed edge or forward edge is an edge which has flow from a vertex in the source's partition to a vertex in the sink's partition.

Definition: Finally, the Minimum Cut is the cut for which the sum of the weights of the forward directed edges in the cut-set's are minimuized. In other words, The Minimum Cut is the cut $P$ corresponding to the cut-set $C$ of forward edges such that for all valid cut-sets $K$, we have:

$$ \sum_{\forall e \in K}{ cap(e) } > \sum_{\forall e \in C}{ cap(e) } $$

Continuing the above example, let us find the minimum cut. Can you find it? How would you go about finding the cut? The most obvious way is to just apply the definition and consider all valid cuts, then take the one with the minimum sum of capacities. This has complexity $O(2^{|V|})$, which is very poor. I should note that this is a loose upper bound, for a more precise bound see: Stirling Numbers of the second kind.

The following is the minimum cut for the given graph.

0 / 100 / 100 / 20 / 40 / 80 / 90 / 100 / 60 / 10stabcd

For the $st$-cut we have $A={s,b}$, $B={a,c,d,t}$ and for the cut-set we have ${(s,a),(b,d),(a,b)}$. Notice the subtlety here: although the edge $(a,b)$ is in the cut-set it is not a forward edge, hence we do not count its capacity. It follows that the value of the minimum cut is $19$.

Remark: For a proof that the max-flow$\equiv$the min-cut see the text listed in the resources or the acompanying slides.

Ford-Fulkerson

As the name implies the Ford-Fulkerson algorithm was developed by Ford and Fulkerson. The algorithm is a means to solve for the max-flow / min-cut.

Definition: The residual graph of a network flow is isomorphic to the original network. The verticies remain unchanged. The edges are mapped as follows: consider an edge $(a,b)$ with capacity $c$ and flow $f$. In the residual graph, this information is represented as an edge $(a,b)$ with a value of $c-f$ and an edge $(b,a)$ with a value of $f$. (See the example below.)

4 / 104 / 40 / 84 / 100 / 60 / 10stxzy6404084600610stxzy

You may ask yourself why we bother with the arguably more complicated representation of the same information. The beauty of this representation is that we can simply do a search from the source for the sink to find a path to "push" more flow through. We call this leftover flow residual capacity and such paths augmenting paths. Can you spot an augmenting path in the above flow?

Yep, you guessed it! (Well, maybe you didn't, but in fact there are two valid augmenting paths.)

10 / 104 / 46 / 810 / 106 / 60 / 10stxzy010010024066010stxzy

Consider the above graph, can you find another augmenting path? The answer is simple: no. In fact, we can't even leave the source since there is no remaining capacity to vertex $x$. This is an excellent certificate that we have pushed as much flow through the network as is possible.

Lets backtrack a bit. What is this algorithm anyway? Informally, Ford-Fulkerson goes like this:

#. Map the network to a residual graph. #. Begin a search for the sink at the source along non-zero edges. Two possible cases result. i. We cannot reach the sink, and hence have our certificate of having saturated the network. #. We reach the sink. - Push as much flow as possible through the path. - Start a new search.

Pretty simple stuff, and again for proof see the text or slides.

Applications

Now we're talking! So far, I bet this has seemed specific to a network of pipes and fluid. Hopefully though, you can see how it would translate to other situations. (For example, cars moving from one place to another through a network of streets.) Let's take it one step further though for our first application problem, lets see if we can apply this to a quality control problem. (Credit again goes to Kleinberg and Tardos for this problem.)

• TODO

Notes and References

• The book I have is Algorithm Design by Jon Kleinberg and Eva Tardos.
• The slides which acompany the text are available online here.