var p = d3.select("#quicksort-static"); d1, .attr("class", "point"); n = array.length, function recurse(left, right) { active = 0; i0 = frontier[i]; cells[i0] |= d0, cells[i1] |= d1; var distance = 0, Popular / About. g This visualization doesn’t show the state of the stack, so it can appear to jump around arbitrarily due to the nature of recursion. var cellSize = 6, }, (function() { } !function(){function t(t){return function(e,i){e=d3.hsl(e),i=d3.hsl(i);var r=(e.h+120)*a,h=(i.h+120)*a-r,s=e.s,l=i.s-s,o=e.l,u=i.l-o;return isNaN(l)&&(l=0,s=isNaN(s)?i.s:s),isNaN(h)&&(h=0,r=isNaN(r)?i.h:r),function(a){var e=r+h*a,i=Math.pow(o+u*a,t),c=(s+l*a)*i*(1-i);return"#"+n(i+c*(-.14861*Math.cos(e)+1.78277*Math.sin(e)))+n(i+c*(-.29227*Math.cos(e)-.90649*Math.sin(e)))+n(i+c*1.97294*Math.cos(e))}}}function n(t){var n=(t=0>=t?0:t>=1?255:0|255*t).toString(16);return 16>t? .selectAll("line") distance, .size([height, width]); stroke: #000; d3.timer(function() { })(), No patterns are visible in this matrix, other than a small amount of noise due to empirical measurement. i1 = Math.min(i + 3, gridWidth), If we then remove redundant information across key frames, we use space more efficiently. YouTube user andrut’s audibilizations are interesting. This is why you shouldn’t wear a finely-striped shirt on camera: the stripes resonate with the grid of pixels in the camera’s sensor and cause Moiré patterns. function merge(left, right, end) { }); return { Pierre Guilleminot’s Fourier series explanation is great, as are Lucas V. Barbosa’s Fourier transform time and frequency domains and an explanation of Simpson’s paradox by Lewis Lehe & Victor Powell; also see Powell’s animations of the central limit theorem and conditional probabilities. for (var i = 0; i < n0; ++i) { else if (d0 === W) fillEast(i1), x1 = x0 - 1, y1 = y0, d1 = E; The algorithm splits the array into two parts: the right side of the array (in black) is the shuffled section, while the left side of the array (in gray) contains elements remaining to be shuffled. Wilson’s. } array[i] = parent; p1 = points[i]; .range([-45, 45]); context.save(); stroke-width: 8px; .attr("y2", -height); Here’s how it works: d.x0 = (i / cellWidth | 0) * (cellSize + cellSpacing); queue(1) } text-anchor: start; for (var i = left; i < right; ++i) if (array[i] < v) swap(i, left++); } var x = d3.scale.ordinal() quadtree.add(bestCandidate); } queue.push(s); } for (var i = left; i < right; ++i) if (array[i] <= v) swap(i, left++); .append("g") function transform(d) { if (cells[i0] & S && !visited[i1 = i0 + width]) visited[i1] = true, frontier1.push(i1); .attr("width", width + margin.left + margin.right) var v = array[pivot]; heap.push = function(value) { GoogleAnalyticsObject = "ga", ga = function() { ga.q.push(arguments); }, ga.q = [], ga.l = +new Date; .attr("y2", -height); ++matrix[i][j]; image.data[i0 + 0] = color.r; merge(i, Math.min(i + m, n), Math.min(i + (m << 1), n)); .on("click", click); .attr("class", "exclusion"); else fillSouth(i0); dy = closest[1] - y, case "partition": { .clamp(true); cells = new Array(cellWidth * cellHeight); // each cell’s edge bits .x(function(d) { return x(d.index); }) function sample(x, y) { In effect, this is breadth-first quicksort, where the partition operation on both left and right proceeds in parallel. The name D3 stands for Data-Driven Documents. return done && p.classed("animation--playing", false); i = Math.random() * m-- | 0; I’m a fan of Processing and three.js. g.append("circle") This algorithm functions visibly differently than the other two: it builds incrementally from existing samples, rather than scattering new samples randomly throughout the sample area. .interpolate(interpolateLineStraight) cells, .attr("height", x.rangeBand()) path.push("M", x0, ",", y1 - rowHeight / 2, "L", p1); grid[gridWidth * (y / cellSize | 0) + (x / cellSize | 0)] = s; i1 = Math.random() * 4 | 0; height = 360; } .range([-45, 45]); } var i = Math.random() * queueSize | 0, .interrupt(); y0 = i0 / cellWidth | 0, var i = index % cellWidth, j = index / cellWidth | 0; var svg = p.append("svg") var z = d3.scale.log() width = 720, array = [10,89,72,53,61,31,27,11,21,77,65,84,82,63,49,76,67,28,26,57,85,75,45,51,36,44,20,81,7,13,41,9,3,71,74,64,86,23,39,4,79,78,8,59,29,55,19,16,60,32,87,37,30,12,42,83,50,35,69,5,15,22,56,25,6,34,38,58,33,68,24,17,80,48,70,46,0,40,1,43,54,52,14,66,47,62,2,18,73,88], // d3.shuffle(d3.range(n)) context.fillRect(i * cellSize + (i + 1) * cellSpacing, (j + 1) * (cellSize + cellSpacing), cellSize, cellSpacing); float: left; As before, the pivots for each partition operation are highlighted in red. .text(function(d, i) { return i; }); .transition(); if (x1 < cellWidth - 1 && cells[i1 + 1] == null) fillEast(i1), frontier.push({index: i1, direction: E}); .style("opacity", null); else if (d0 === S) x1 = x0, y1 = y0 + 1, d1 = N; } .enter().append("line") p.classed("animation--playing", true); .attr("x", 6 - x(0)) image.data[i0 + 2] = color.b; .domain(d3.range(n)) childIndex, // Also erase this walk’s history to not interfere with subsequent walks. function swap(i, j) { child, frontier.push({index: start, direction: N, weight: Math.random()}); Thus, notice that in all swap operations, only values darker than the pivot move right, and only values lighter than the pivot move left. return function() { All rights reserved – Chartio, 548 Market St Suite 19064 San Francisco, California 94104 • Email Us • Terms of Service • Privacy var canvas = p.append("canvas") While animations are improved by controls to pause and scrub time, static displays that show everything at once can be even more effective. The resulting array is often barely shuffled, as shown by the strong green diagonal in this matrix. .attr("x2", width); else if (i1 === 1) { if (y0 >= height - 1) continue; ++y0, i1 = i0 + width; } .attr("class", "candidate"); .data(array) .on("click", click); var m = i1 - i0, t, i, j; }; // If this cell is part of the maze, we’re done walking. var numSamples = 0; You can see that typically there is only one, and rarely more than a few, active branches at any particular depth. margin = {left: Math.floor((outerWidth - cellWidth * cellSize - (cellWidth + 1) * cellSpacing) / 2) + cellSpacing + cellSize / 2 + .5, top: Math.floor((outerHeight - cellHeight * cellSize - (cellHeight + 1) * cellSpacing) / 2) + cellSpacing + cellSize / 2 + .5}, timerActive = 0; swapped[i] = swapped[j]; .attr("cy", y) return s; .style("stroke", function(d) { return color(d.value); }) stroke-width: 1.5px; .text("▶ Play"); if (i1 === 0) { if (y0 <= 0) continue; --y0, i1 = i0 - cellWidth; } var x = d3.scale.ordinal() .interrupt() This is the default sorting algorithm in Java and Dart. Randomized Prim’s. if (x1 > 0 && cells[i1 - 1] == null) fillEast(i1 - 1), frontier.push({index: i1, direction: W}), ++m; Each candidate is chosen uniformly from the sampling area. To output an accurate answer, the calculator needs accurate inputs. } .data(arrays) var sample = uniformRandomSampler(width, height, numSamples), .defer(d3.json, "random-comparator-chrome-shuffle-bias.json") .attr("x", x(0) - 6) At each step, a new element is added either of the set of lesser values (in which case a swap occurs) or to the set of greater values (in which case no swap occurs). image.data[i0 + 3] = 255; Thank you for reading! i0 = frontier[i] << 2; array[i = j] = value; cellWidth = Math.floor((width - cellSpacing) / (cellSize + cellSpacing)), .range([-45, 45]); cellSpacing = 6, if ((edge = frontier.pop()) == null) return true; for (var j = 0; j < k; ++j) { Randomized depth-first traversal. return function(t) { A good shuffling algorithm is unbiased, where every ordering is equally likely. e = line1[j] = line0[i], function poissonDiscSampler(width, height, radius) { (This form is inspired by Aldo Cortesi’s sorting visualizations.) var gConnection = svg.append("g") .attr("y2", -height); previous[i1] = NaN; queue = []; cellWidth : d0 === W ? Chief Technology Officer Mike Bostock created D3.js, the popular open source library for data visualization, and was previously a Graphics Editor at The New York Times. var gLine = svg.append("g") height = (swaps.length + .5) * rowHeight; Since these mazes are spanning trees, we can also use specialized tree visualizations to show structure. var arcEmptyAnnulus = d3.svg.arc() .classed("animation--playing", true); p.append("button") i; if (++numSamples > numSamplesMax) return; Four new papers at UIST 2011, plus four more at VisWeek 2011. .attr("class", "search") }); var p = d3.select("#random-comparator-firefox-shuffle-bias"); var j = ((i + 1) >> 1) - 1, .datum(function() { return this.src; }) var p = d3.select("#random-traversal-color-flood"); .attr("width", width) var frontier1 = [], var cells = generateMaze(width, height), // each cell’s edge bits If a candidate is acceptable, it is added as a new sample, and a new active sample is randomly selected. Now, everyone can be connected to data in real time, as browsers support extremely fast rendering and interactive displays. .enter().append("path") This deception demonstrates that visualization is not a magic wand. return !frontier1.length; context.clearRect(0, 0, width, height); frontier, .defer(beforeVisible, p.node()) for (var i = 0; i < n0; ++i) { }); worker.addEventListener("message", function(event) { context.translate( if (y1 > 0 && cells[i1 - cellWidth] == null) fillSouth(i1 - cellWidth), frontier.push({index: i1, direction: N, weight: Math.random()}); .transition() (function selectActive() { context.translate( if (actions.length) nextTransition(); visited, The area-as-color Voronoi diagram shows Poisson-disc sampling’s improvement over best-candidate, with no dark-blue or light-yellow cells: The Starry Night under Poisson-disc sampling retains the greatest amount of detail and the least noise. return whenBoundsVisible(function() { .enter().append("line") } .style("fill", z); .append("g") .attr("height", height + margin + margin) The slope of the chart shows the associated variable’s importance: the greater the slope, the more the decision depends on that variable. links.forEach(function(d) { d.index = j; April 28, 2017A Better Way to Code December 9, 2016Command-Line Cartography March 9, 2016What Makes Software Good? width : d0 === W ? var width = 960, var n = 90, Both algorithms thus appear to grow organically, like a fungus. Detail has been lost because densely-packed samples (small cells) are underutilized. return !links[0].target.__transition__ && p.classed("animation--playing", false); nodes.forEach(function(d) { if (d > bestDistance) { var p = d3.select("#mergesort-static"); d[1] = x0 + t * dx; Yet the techniques discussed here apply to a broader space of problems: mathematical formulas, dynamical systems, processes, etc. beforeVisible(p.node(), function() { On the other hand, if you want to reduce the data before you send it out, you will be doing work on the server side (for example, there are lots of tools built on Hadoop and MongoDB’s Mapreduce functionality). svg.append("circle") while ((i0 = previous[i1]) !== i1) { .each(function() { this.parentNode.appendChild(this); }) searchAnnulus.interrupt(); return root; p.append("button") } var a = d3.scale.linear() Concentric circles of color, like a tie-dye shirt, reveal that random traversal produces many branching paths. d0 = edge.direction, .attr("class", "line--inactive") function generateMaze(width, height) { } var line = gLine.selectAll("line") .attr("transform", transform) context.save(); ‍ Mike worked at the The New York Times for a while and is now independently working on D3.js while working at Observable. // Explore the frontier until the tree spans the graph. x1, .domain(d3.range(n)) var canvas = p.append("canvas") array[i] = array[j]; Math.round((width - cellWidth * cellSize - (cellWidth + 1) * cellSpacing) / 2), Candidate samples within distance r from an existing sample are rejected; this “exclusion zone” is shown in gray, along with a black line connecting the rejected candidate to the nearby existing sample.
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