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Speeding up KDE heatmaps with Barnes-Hut quadtrees
license: gpl-3.0
height: 500
scrolling: no
border: yes

Performance comparison of a brute force kernel density estimation (KDE) heatmap vs. a Barnes-Hut quadtree KDE heatmap.

This example creates an array of node positions at every tick of a force-directed algorithm. Then it generates two heatmaps from all of the collected node positions.

The Barnes-Hut approximation is markedly faster and looks the same when there are many points. When there is a very small number of points, the extra cost of constructing the quadtree makes the Barnes-Hut heatmap slower. But after just a few ticks, there are enough points that the Barnes-Hut heatmap is much, much faster.

In data visualization, the Barnes-Hut approximation is often used for speeding up force-directed graph layouts, but it has many other potential uses, such as this example here.

See also:

<!DOCTYPE html>
<meta charset="utf-8">
<style>
body {
background: #fff;
font: 10px sans-serif;
margin: 0;
}
.container {
width: 945px;
height: 315px;
}
.container div {
display: inline-block;
width: 315px;
height: 315px;
overflow: hidden;
}
.force svg {
background: rgba(0,0,0,0);
position: absolute;
top: 0;
left: 0;
}
p {
color: black;
}
.container div p {
font-size: 12px;
font-weight: bold;
text-align: center;
margin: 0;
position: relative;
bottom: 25px;
}
.brute p {
color: steelblue;
}
.barneshut p {
color: maroon;
}
svg {
background: #fff;
}
.node {
fill: #555;
stroke: #fff;
stroke-width: 1px;
}
.link {
stroke: #555;
stroke-opacity: 0.6;
}
.point {
fill: black;
}
</style>
<body>
<div class="container"><div class="force"><canvas></canvas><svg></svg><p>Graph layout node positions</p></div><div class="brute"><canvas></canvas><p>Brute force heatmap</p></div><div class="barneshut"><canvas></canvas><p>Barnes-Hut heatmap</p></div></div>
<script src="https://d3js.org/d3.v4.min.js"></script>
<script>
var kdeWidth = 315;
var kdeHeight = 315;
var chartMargin = {top: 10, right: 20, bottom: 30, left: 50};
var chartWidth = 945 - chartMargin.left - chartMargin.right;
var chartHeight = 175 - chartMargin.top - chartMargin.bottom;
var computeTimes = [
{name: 'Brute force', values: []},
{name: 'Barnes-Hut', values: []}
];
var gridSize = 9;
var xyPoints = [];
var qt = d3.quadtree()
.extent([[0, 0], [kdeWidth, kdeHeight]])
.visitAfter(accumulate);
var h, h2, iqr, xPoints;
// Array of grid cell points. Each point is the center of the grid cell.
var grid = d3.merge(d3.range(0, kdeHeight/gridSize).map(function(i) {
return d3.range(0, kdeWidth/gridSize).map(function(j) { return [j*gridSize + gridSize/2, i*gridSize + gridSize/2] });
}));
var x = d3.scaleLinear().rangeRound([0, chartWidth]);
var y = d3.scaleLinear().rangeRound([chartHeight, 0]);
var outerScale = d3.scalePow()
.exponent(0.4)
.domain([0,1])
.range([0,1]);
var heatmapColor = d3.scaleLinear()
.clamp(true)
.domain([0, 0.1111111111111111, 0.2222222222222222, 0.3333333333333333, 0.4444444444444444, 0.5555555555555555, 0.6666666666666666, 0.7777777777777777, 0.8888888888888888, 1])
.range(["#fff", "#dadada", "#b9b9b9", "#9b9b9b", "#7f7e7e", "#646363", "#4a4a4a", "#313030", "#161616", "#000"]);
var force = d3.forceSimulation()
.alphaDecay(0.03)
.force('link', d3.forceLink().id(function(d, i) { return i; }).distance(10))
.force('charge', d3.forceManyBody().strength(-40))
.force('x', d3.forceX(kdeWidth/2))
.force('y', d3.forceY(kdeHeight/2))
.force('center', d3.forceCenter(kdeWidth/2, kdeHeight/2));
var tickCount = 0;
var forceSvg = d3.select(".force svg")
.attr("width", kdeWidth)
.attr("height", kdeHeight)
.append("g");
var pointsContext = d3.select('.force canvas')
.attr("width", kdeWidth)
.attr("height", kdeHeight)
.node()
.getContext('2d');
var bruteCanvas = d3.select(".brute canvas")
.attr('id', 'brute')
.attr("width", kdeWidth)
.attr("height", kdeHeight);
var barneshutCanvas = d3.select(".barneshut canvas")
.attr('id', 'barneshut')
.attr("width", kdeWidth)
.attr("height", kdeHeight);
var chartSvg = d3.select("body").append("svg")
.attr("width", chartWidth + chartMargin.left + chartMargin.right)
.attr("height", chartHeight + chartMargin.top + chartMargin.bottom)
.append('g')
.attr("transform", "translate(" + chartMargin.left + "," + chartMargin.top + ")");
var line = d3.line()
.x(function (d) { return x(d.numPoints); })
.y(function (d) { return y(d.time); });
var bruteContext = bruteCanvas.node().getContext("2d");
var barneshutContext = barneshutCanvas.node().getContext("2d");
d3.json("miserables.json", function(error, graph) {
if (error) throw error;
var link = forceSvg.selectAll(".link")
.data(graph.links)
.enter().append("line")
.attr("class", "link")
.style("stroke-width", function(d) { return Math.sqrt(d.value); });
var node = forceSvg.selectAll(".node")
.data(graph.nodes)
.enter().append("circle")
.attr("class", "node")
.attr("r", 3);
point = forceSvg.selectAll('.point')
.data(xyPoints)
.enter().append('circle')
.attr('class', 'point');
node.append("title")
.text(function(d) { return d.name; });
force.nodes(graph.nodes)
.on("end", function(event) {
computeDensities();
updateChart();
})
.on('tick', function(event) {
link.attr("x1", function(d) { d.source.x = boundedCoords(d.source.x, kdeWidth); return d.source.x; })
.attr("y1", function(d) { d.source.y = boundedCoords(d.source.y, kdeHeight); return d.source.y; })
.attr("x2", function(d) { d.target.x = boundedCoords(d.target.x, kdeWidth); return d.target.x; })
.attr("y2", function(d) { d.target.y = boundedCoords(d.target.y, kdeHeight); return d.target.y; });
node.attr("cx", function(d) { d.x = boundedCoords(d.x, kdeWidth); return d.x; })
.attr("cy", function(d) { d.y = boundedCoords(d.y, kdeHeight); return d.y; });
// Track new node positions
node.each(function(d) {
xyPoints.push([d.x,d.y]);
pointsContext.fillStyle = '#c9c9c9';
pointsContext.beginPath();
pointsContext.arc(d.x, d.y, 3, 0, 2*Math.PI);
pointsContext.fill();
});
if (++tickCount == 50) {
force.stop();
}
computeDensities();
updateChart();
});
force.force('link')
.links(graph.links);
});
function updateChart () {
x.domain([0, xyPoints.length]);
y.domain([0, d3.max(computeTimes, function (d) { return d3.max(d.values.map(function(v) { return v.time; })); })]);
chartSvg.selectAll('g').remove();
chartSvg.append('g')
.attr('transform', 'translate(0,' + chartHeight + ')')
.call(d3.axisBottom(x))
.append("text")
.attr("fill", "#000")
.attr('transform', 'translate(' + chartMargin.left + ',' + 0 + ')')
.attr("y", 19)
.attr("dy", "0.71em")
.attr("text-anchor", "begin")
.text("Number of points");
chartSvg.append('g')
.call(d3.axisLeft(y))
.append("text")
.attr("fill", "#000")
.attr("transform", "rotate(-90)")
.attr("y", 6)
.attr("dy", "0.71em")
.attr("text-anchor", "end")
.text("Compute time (ms)");
var lineG = chartSvg.selectAll('g.line-g')
.data(computeTimes)
.enter().append('g')
.attr('class', 'line-g');
lineG.append('path')
.attr("fill", "none")
.attr("stroke", function (d) { return d.name === 'Brute force' ? 'steelblue' : 'maroon'; })
.attr("stroke-linejoin", "round")
.attr("stroke-linecap", "round")
.attr("stroke-width", 1.5)
.attr("d", function (d) { return line(d.values); });
lineG.append("text")
.datum(function(d) { return {name: d.name, value: d.values[d.values.length - 1]}; })
.attr("transform", function(d) { return "translate(" + x(d.value.numPoints) + "," + y(d.value.time) + ")"; })
.attr("x", 3)
.attr('y', -6)
.attr("dy", "0.35em")
.attr('text-anchor', 'end')
.style("font", "10px sans-serif")
.text(function(d) { return d.name; });
}
function computeDensities () {
// Density at each (x,y) coordinate in the grid
var densities = [];
var startTime;
// Update bandwidth
// Use same bandwidth for each dimension
xPoints = xyPoints.map(function(d) { return d[0] }).sort(function(a,b) { return a - b });
iqr = d3.quantile(xPoints, 0.75) - d3.quantile(xPoints, 0.25);
h = 1.06 * Math.min(d3.deviation(xPoints), iqr / 1.34) * Math.pow(xyPoints.length, -0.2);
h2 = h*h;
// Compute KDE for each (x,y) pair and update the color scale
startTime = new Date();
densities = grid.map(function(point) { return kdeBrute(point); });
computeTimes[0].values.push({numPoints: xyPoints.length, time: new Date() - startTime})
outerScale.domain([0, d3.max(densities)]);
drawHeatmap(bruteContext, densities);
startTime = new Date();
qt = d3.quadtree(xyPoints)
.extent([[0, 0], [kdeWidth, kdeHeight]])
.visitAfter(accumulate);
densities = grid.map(function(point) { return kdeBarneshut(point); });
computeTimes[1].values.push({numPoints: xyPoints.length, time: new Date() - startTime})
outerScale.domain([0, d3.max(densities)]);
drawHeatmap(barneshutContext, densities);
}
function drawHeatmap(context, densities) {
// Draw the grid
grid.forEach(function(point, idx) {
context.beginPath();
context.fillStyle = heatmapColor(outerScale(densities[idx]));
// Subtract to get the corner of the grid cell
context.rect(point[0] - gridSize/2, point[1] - gridSize/2, gridSize, gridSize);
context.fill();
context.closePath();
});
}
function boundedCoords(pos, maxPos) {
return Math.min(maxPos, Math.max(0, pos));
}
function accumulate(quad) {
// Sum of x/y coordinates
var sx = 0;
var sy = 0;
var count = 0, q;
// This quadrant has children.
if (quad.length) {
var i = -1;
var c;
while (++i < 4) {
c = quad[i];
if (c == null) continue;
count += c.count;
sx += c.sx;
sy += c.sy;
}
}
// This is a leaf node.
else {
q = quad;
do {
count += 1;
sx += q.data[0];
sy += q.data[1];
} while (q = q.next);
}
quad.sx = sx;
quad.sy = sy;
quad.count = count;
}
// Use same bandwidth for each dimension
function kdeBrute(gridPoint) {
return d3.mean(xyPoints, function(p) { return gaussian(norm(p, gridPoint) / h) }) / h2;
}
function kdeBarneshut(gridPoint) {
//return d3.mean(xyPoints, function(p) { return gaussian(norm(p, gridPoint) / h) }) / h2;
var estimate = 0;
var theta2 = 0.64;
function recurse(quadNode, x1, y1, x2, y2) {
// Average x/y coordinates in this quadrant.
var p = [quadNode.sx/quadNode.count, quadNode.sy/quadNode.count];
var dx = p[0] - gridPoint[0];
var dy = p[1] - gridPoint[1];
var dw = x2 - x1;
var dist = dx * dx + dy * dy;
// Barnes-Hut: If the quadrant size is small relative to its
// distance from the grid point, then aggregate the points in
// that quadrant and treat them as a single large point (or many
// individual points all in the same location).
if (dw * dw / theta2 < dist && !(gridPoint[0] >= x1 && gridPoint[0] < x2 && gridPoint[1] >= y1 && gridPoint[1] < y2)) {
estimate += quadNode.count * gaussian(norm(p, gridPoint) / h);
return true;
}
if (quadNode.point) {
estimate += gaussian(norm(quadNode.point, gridPoint) / h);
}
// Stop recursing if there are no more nodes.
return !quadNode.count;
}
qt.visit(recurse);
return estimate / xyPoints.length;
}
// Norm of 2D arrays/vectors
function norm(v1, v2) {
return Math.sqrt((v1[0] - v2[0]) * (v1[0] - v2[0]) + (v1[1] - v2[1]) * (v1[1] - v2[1]));
}
function gaussian(x) {
// sqrt(2 * PI) is approximately 2.5066
return Math.exp(-x * x / 2) / 2.5066;
}
</script>
{
"nodes":[
{"name":"Myriel","group":1},
{"name":"Napoleon","group":1},
{"name":"Mlle.Baptistine","group":1},
{"name":"Mme.Magloire","group":1},
{"name":"CountessdeLo","group":1},
{"name":"Geborand","group":1},
{"name":"Champtercier","group":1},
{"name":"Cravatte","group":1},
{"name":"Count","group":1},
{"name":"OldMan","group":1},
{"name":"Labarre","group":2},
{"name":"Valjean","group":2},
{"name":"Marguerite","group":3},
{"name":"Mme.deR","group":2},
{"name":"Isabeau","group":2},
{"name":"Gervais","group":2},
{"name":"Tholomyes","group":3},
{"name":"Listolier","group":3},
{"name":"Fameuil","group":3},
{"name":"Blacheville","group":3},
{"name":"Favourite","group":3},
{"name":"Dahlia","group":3},
{"name":"Zephine","group":3},
{"name":"Fantine","group":3},
{"name":"Mme.Thenardier","group":4},
{"name":"Thenardier","group":4},
{"name":"Cosette","group":5},
{"name":"Javert","group":4},
{"name":"Fauchelevent","group":0},
{"name":"Bamatabois","group":2},
{"name":"Perpetue","group":3},
{"name":"Simplice","group":2},
{"name":"Scaufflaire","group":2},
{"name":"Woman1","group":2},
{"name":"Judge","group":2},
{"name":"Champmathieu","group":2},
{"name":"Brevet","group":2},
{"name":"Chenildieu","group":2},
{"name":"Cochepaille","group":2},
{"name":"Pontmercy","group":4},
{"name":"Boulatruelle","group":6},
{"name":"Eponine","group":4},
{"name":"Anzelma","group":4},
{"name":"Woman2","group":5},
{"name":"MotherInnocent","group":0},
{"name":"Gribier","group":0},
{"name":"Jondrette","group":7},
{"name":"Mme.Burgon","group":7},
{"name":"Gavroche","group":8},
{"name":"Gillenormand","group":5},
{"name":"Magnon","group":5},
{"name":"Mlle.Gillenormand","group":5},
{"name":"Mme.Pontmercy","group":5},
{"name":"Mlle.Vaubois","group":5},
{"name":"Lt.Gillenormand","group":5},
{"name":"Marius","group":8},
{"name":"BaronessT","group":5},
{"name":"Mabeuf","group":8},
{"name":"Enjolras","group":8},
{"name":"Combeferre","group":8},
{"name":"Prouvaire","group":8},
{"name":"Feuilly","group":8},
{"name":"Courfeyrac","group":8},
{"name":"Bahorel","group":8},
{"name":"Bossuet","group":8},
{"name":"Joly","group":8},
{"name":"Grantaire","group":8},
{"name":"MotherPlutarch","group":9},
{"name":"Gueulemer","group":4},
{"name":"Babet","group":4},
{"name":"Claquesous","group":4},
{"name":"Montparnasse","group":4},
{"name":"Toussaint","group":5},
{"name":"Child1","group":10},
{"name":"Child2","group":10},
{"name":"Brujon","group":4},
{"name":"Mme.Hucheloup","group":8}
],
"links":[
{"source":1,"target":0,"value":1},
{"source":2,"target":0,"value":8},
{"source":3,"target":0,"value":10},
{"source":3,"target":2,"value":6},
{"source":4,"target":0,"value":1},
{"source":5,"target":0,"value":1},
{"source":6,"target":0,"value":1},
{"source":7,"target":0,"value":1},
{"source":8,"target":0,"value":2},
{"source":9,"target":0,"value":1},
{"source":11,"target":10,"value":1},
{"source":11,"target":3,"value":3},
{"source":11,"target":2,"value":3},
{"source":11,"target":0,"value":5},
{"source":12,"target":11,"value":1},
{"source":13,"target":11,"value":1},
{"source":14,"target":11,"value":1},
{"source":15,"target":11,"value":1},
{"source":17,"target":16,"value":4},
{"source":18,"target":16,"value":4},
{"source":18,"target":17,"value":4},
{"source":19,"target":16,"value":4},
{"source":19,"target":17,"value":4},
{"source":19,"target":18,"value":4},
{"source":20,"target":16,"value":3},
{"source":20,"target":17,"value":3},
{"source":20,"target":18,"value":3},
{"source":20,"target":19,"value":4},
{"source":21,"target":16,"value":3},
{"source":21,"target":17,"value":3},
{"source":21,"target":18,"value":3},
{"source":21,"target":19,"value":3},
{"source":21,"target":20,"value":5},
{"source":22,"target":16,"value":3},
{"source":22,"target":17,"value":3},
{"source":22,"target":18,"value":3},
{"source":22,"target":19,"value":3},
{"source":22,"target":20,"value":4},
{"source":22,"target":21,"value":4},
{"source":23,"target":16,"value":3},
{"source":23,"target":17,"value":3},
{"source":23,"target":18,"value":3},
{"source":23,"target":19,"value":3},
{"source":23,"target":20,"value":4},
{"source":23,"target":21,"value":4},
{"source":23,"target":22,"value":4},
{"source":23,"target":12,"value":2},
{"source":23,"target":11,"value":9},
{"source":24,"target":23,"value":2},
{"source":24,"target":11,"value":7},
{"source":25,"target":24,"value":13},
{"source":25,"target":23,"value":1},
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{"source":26,"target":11,"value":31},
{"source":26,"target":16,"value":1},
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{"source":27,"target":11,"value":17},
{"source":27,"target":23,"value":5},
{"source":27,"target":25,"value":5},
{"source":27,"target":24,"value":1},
{"source":27,"target":26,"value":1},
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