diff --git a/presentations/Diskrete-Mathematik/Handout/Handout.tex b/presentations/Diskrete-Mathematik/Handout/Handout.tex index e7ac065..3a05327 100644 --- a/presentations/Diskrete-Mathematik/Handout/Handout.tex +++ b/presentations/Diskrete-Mathematik/Handout/Handout.tex @@ -125,10 +125,13 @@ $K \subseteq E \times E$ die Kantenmenge bezeichnet. \end{definition} -\begin{definition}{Inzidenz} -Sei $e \in E$ und $k = \Set{e_1, e_2} \in K$. +\begin{definition}{Grad einer Ecke} +Der \textbf{Grad} einer Ecke ist die Anzahl der Kanten, die von dieser Ecke +ausgehen. +\end{definition} -$e$ heißt \textbf{inzident} zu $k :\Leftrightarrow e = e_1$ oder $e = e_2$ +\begin{definition}{Isolierte Ecke} +Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}. \end{definition} \begin{definition}{Schlinge} @@ -137,6 +140,12 @@ Sei $G=(E, K)$ ein Graph und $k=\Set{e_1, e_2} \in K$ eine Kante. $k$ heißt \textbf{Schlinge} $:\Leftrightarrow e_1 = e_2$ \end{definition} +\begin{definition}{Inzidenz} +Sei $e \in E$ und $k = \Set{e_1, e_2} \in K$. + +$e$ heißt \textbf{inzident} zu $k :\Leftrightarrow e = e_1$ oder $e = e_2$ +\end{definition} + \begin{definition}{Vollständiger Graph} Sei $G = (E, K)$ ein Graph. @@ -189,7 +198,7 @@ Sei $G = (E, K)$ ein Graph und $A = (k_1, k_2 \dots, k_s)$ ein Kantenzug. A heißt \textbf{Weg} $:\Leftrightarrow \forall_{i, j \in 1, \dots, s}: i \neq j \Rightarrow k_i \neq k_j$ . \end{definition} -\begin{definition}{Einfacher Kreis} +\begin{definition}{Kreis} Sei $G = (E, K)$ ein Graph und $A = (k_1, k_2 \dots, k_s)$ ein Kantenzug. A heißt \textbf{Kreis} $:\Leftrightarrow A$ ist geschlossen und ein Weg. @@ -216,15 +225,6 @@ $G$ heißt \textbf{zusammenhängend} $:\Leftrightarrow \forall e_1, e_2 \in E: $ Es ex. ein Kantenzug, der $e_1$ und $e_2$ verbindet \end{definition} -\begin{definition}{Grad einer Ecke} -Der \textbf{Grad} einer Ecke ist die Anzahl der Kanten, die von dieser Ecke -ausgehen. -\end{definition} - -\begin{definition}{Isolierte Ecke} -Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}. -\end{definition} - \begin{definition}{Eulerscher Kreis} Sei $G$ ein Graph und $A$ ein Kreis in $G$. diff --git a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf index f22223b..bd38aa8 100644 Binary files a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf and b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf differ diff --git a/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex b/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex index 75b49ee..92e8976 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex @@ -34,7 +34,7 @@ Verwechslungsgefahr: Hamiltonkreis $\neq$ Eulerkreis \begin{block}{Hamiltonkreis} Sei $G$ ein Graph und $A$ ein Kreis in $G$. -$A$ heißt \textbf{Hamilton-Kreis} $:\Leftrightarrow \forall_{e \in E}: e \in A$. +$A$ heißt \textbf{Hamilton-Kreis} $:\Leftrightarrow \forall_{e \in E}: e \text{ ist genau ein mal in } A$. \end{block} \begin{block}{Eulerscher Kreis} diff --git a/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex b/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex index 2a4735e..0e211e9 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex @@ -63,10 +63,6 @@ Ein Kantenzug wird durch den Tupel $(e_0, \dots, e_s) \in E^{s+1}$ charakterisiert. \begin{gallery} - \galleryimage{walks/walk-1} - \galleryimage{walks/walk-2} - \galleryimage{walks/k-3-3-walk} - \galleryimage{walks/k-5-walk}\\ \galleryimage{walks/k-16-walk} \galleryimage{walks/star-graph-walk} \galleryimage{walks/tree-walk} diff --git a/presentations/Diskrete-Mathematik/isomorph-graph-example/README.md b/presentations/Diskrete-Mathematik/isomorph-graph-example/README.md new file mode 100644 index 0000000..715ac5f --- /dev/null +++ b/presentations/Diskrete-Mathematik/isomorph-graph-example/README.md @@ -0,0 +1,6 @@ +See [StackExchange](http://stackoverflow.com/q/7034/562769) for many +visualization options. The best I've seen are: + +* [Graph Dracula](http://www.graphdracula.net/) +* [jsPlumb](http://jsplumbtoolkit.com/) +* [graph-it](http://js-graph-it.sourceforge.net/) diff --git a/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graffle.js b/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graffle.js new file mode 100644 index 0000000..c0ad9eb --- /dev/null +++ b/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graffle.js @@ -0,0 +1,101 @@ +/** + * Originally grabbed from the official RaphaelJS Documentation + * http://raphaeljs.com/graffle.html + * Changes and comments by Philipp Strathausen http://blog.ameisenbar.de + * Licenced under the MIT licence. + */ + +/** + * Usage: + * connect two shapes + * parameters: + * source shape [or connection for redrawing], + * target shape, + * style with { fg : linecolor, bg : background color, directed: boolean } + * returns: + * connection { draw = function() } + */ +Raphael.fn.connection = function (obj1, obj2, style) { + var selfRef = this; + /* create and return new connection */ + var color = style.fg || "#000"; + var edge = {/* + from : obj1, + to : obj2, + style : style,*/ + draw : function() { + /* get bounding boxes of target and source */ + var bb1 = obj1.getBBox(); + var bb2 = obj2.getBBox(); + bb1.height = bb1.height + 15; /* respect the label TODO do it in a cleaner way */ + bb2.height = bb2.height + 15; + var off1 = 1; + var off2 = 2; + /* coordinates for potential connection coordinates from/to the objects */ + var p = [ + {x: bb1.x + bb1.width / 2, y: bb1.y - off1}, /* NORTH 1 */ + {x: bb1.x + bb1.width / 2, y: bb1.y + bb1.height + off1}, /* SOUTH 1 */ + {x: bb1.x - off1, y: bb1.y + bb1.height / 2}, /* WEST 1 */ + {x: bb1.x + bb1.width + off1, y: bb1.y + bb1.height / 2}, /* EAST 1 */ + {x: bb2.x + bb2.width / 2, y: bb2.y - off2}, /* NORTH 2 */ + {x: bb2.x + bb2.width / 2, y: bb2.y + bb2.height + off2}, /* SOUTH 2 */ + {x: bb2.x - off2, y: bb2.y + bb2.height / 2}, /* WEST 2 */ + {x: bb2.x + bb2.width + off2, y: bb2.y + bb2.height / 2} /* EAST 2 */ + ]; + + /* distances between objects and according coordinates connection */ + var d = {}, dis = []; + + /* + * find out the best connection coordinates by trying all possible ways + */ + /* loop the first object's connection coordinates */ + for (var i = 0; i < 4; i++) { + /* loop the seond object's connection coordinates */ + for (var j = 4; j < 8; j++) { + var dx = Math.abs(p[i].x - p[j].x), + dy = Math.abs(p[i].y - p[j].y); + if ((i == j - 4) || (((i != 3 && j != 6) || p[i].x < p[j].x) && ((i != 2 && j != 7) || p[i].x > p[j].x) && ((i != 0 && j != 5) || p[i].y > p[j].y) && ((i != 1 && j != 4) || p[i].y < p[j].y))) { + dis.push(dx + dy); + d[dis[dis.length - 1]] = [i, j]; + } + } + } + var res = dis.length == 0 ? res = [0, 4] : d[Math.min.apply(Math, dis)]; + /* bezier path */ + var x1 = p[res[0]].x, + y1 = p[res[0]].y, + x4 = p[res[1]].x, + y4 = p[res[1]].y, + dx = Math.max(Math.abs(x1 - x4) / 2, 10), + dy = Math.max(Math.abs(y1 - y4) / 2, 10), + x2 = [x1, x1, x1 - dx, x1 + dx][res[0]].toFixed(3), + y2 = [y1 - dy, y1 + dy, y1, y1][res[0]].toFixed(3), + x3 = [0, 0, 0, 0, x4, x4, x4 - dx, x4 + dx][res[1]].toFixed(3), + y3 = [0, 0, 0, 0, y1 + dy, y1 - dy, y4, y4][res[1]].toFixed(3); + /* assemble path and arrow */ + var path = ["M", x1.toFixed(3), y1.toFixed(3), "C", x2, y2, x3, y3, x4.toFixed(3), y4.toFixed(3)].join(","); + /* arrow */ + if(style && style.directed) { + /* magnitude, length of the last path vector */ + var mag = Math.sqrt((y4 - y3) * (y4 - y3) + (x4 - x3) * (x4 - x3)); + /* vector normalisation to specified length */ + var norm = function(x,l){return (-x*(l||5)/mag);}; + /* calculate array coordinates (two lines orthogonal to the path vector) */ + var arr = [ + {x:(norm(x4-x3)+norm(y4-y3)+x4).toFixed(3), y:(norm(y4-y3)+norm(x4-x3)+y4).toFixed(3)}, + {x:(norm(x4-x3)-norm(y4-y3)+x4).toFixed(3), y:(norm(y4-y3)-norm(x4-x3)+y4).toFixed(3)} + ]; + path = path + ",M"+arr[0].x+","+arr[0].y+",L"+x4+","+y4+",L"+arr[1].x+","+arr[1].y; + } + + // applying path + edge.fg && edge.fg.attr({path:path}) + || (edge.fg = selfRef.path(path).attr({stroke: color, fill: "none"}).toBack()); + edge.bg && edge.bg.attr({path:path}) + || style && style.bg && (edge.bg = style.bg.split && selfRef.path(path).attr({stroke: style.bg.split("|")[0], fill: "none", "stroke-width": style.bg.split("|")[1] || 3}).toBack()); + } + } + edge.draw(); + return edge; +}; diff --git a/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graph.js b/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graph.js new file mode 100644 index 0000000..268c283 --- /dev/null +++ b/presentations/Diskrete-Mathematik/isomorph-graph-example/graph1-js/graph.js @@ -0,0 +1,303 @@ +/* Graph JavaScript framework, version 0.0.1 + * (c) 2006 Aslak Hellesoy + * (c) 2006 Dave Hoover + * + * Ported from Graph::Layouter::Spring in + * http://search.cpan.org/~pasky/Graph-Layderer-0.02/ + * The algorithm is based on a spring-style layouter of a Java-based social + * network tracker PieSpy written by Paul Mutton Epaul@jibble.orgE. + * + * Adopted by Philipp Strathausen to support Raphael JS + * for rendering, dragging and much more. See http://blog.ameisenbar.de + * + * Graph is freely distributable under the terms of an MIT-style license. + * For details, see the Graph web site: http://dev.buildpatternd.com/trac + * + * Links: + * + * Demo of the original applet: + * http://redsquirrel.com/dave/work/webdep/ + * + * Mirrored original source code at snipplr: + * http://snipplr.com/view/1950/graph-javascript-framework-version-001/ + * + * Original usage example: + * http://ajaxian.com/archives/new-javascriptcanvas-graph-library + * +/*--------------------------------------------------------------------------*/ + +/* + * Graph + */ +var Graph = function() { + this.nodes = []; + this.edges = []; +}; +Graph.prototype = { + addNode: function(id, content) { + /* testing if node is already existing in the graph */ + var new_node = this.nodes[id]; + if(new_node == undefined) { + new_node = new Graph.Node(id, content||{"id":id}); + this.nodes[id] = new_node; + this.nodes.push(new_node); // TODO get rid of the array + } + return new_node; + }, + + addEdge: function(source, target, style) { + var s = this.addNode(source); + var t = this.addNode(target); + var color; + var colorbg; + var directed; + if(style) { color = style.color; colorbg = style.colorbg; directed = style.directed } + var edge = { source: s, target: t, color: color, colorbg: colorbg, directed: directed }; + this.edges.push(edge); + } +}; + +/* + * Node + */ +Graph.Node = function(id, value){ + this.id = id; + this.content = value; +}; +Graph.Node.prototype = { +}; +Graph.Renderer = {}; +Graph.Renderer.Raphael = function(element, graph, width, height) { + this.width = width||400; + this.height = height||400; + var selfRef = this; + this.r = Raphael(element, this.width, this.height); + this.radius = 40; /* max dimension of a node */ + this.graph = graph; + this.mouse_in = false; + + /* + * Dragging + */ + this.isDrag = false; + this.dragger = function (e) { + this.dx = e.clientX; + this.dy = e.clientY; + selfRef.isDrag = this; + this.animate({"fill-opacity": .2}, 500); + e.preventDefault && e.preventDefault(); + }; + + document.onmousemove = function (e) { + e = e || window.event; + if (selfRef.isDrag) { + var newX = e.clientX - selfRef.isDrag.dx + (selfRef.isDrag.attrs.cx == null ? (selfRef.isDrag.attrs.x + selfRef.isDrag.attrs.width / 2) : selfRef.isDrag.attrs.cx); + var newY = e.clientY - selfRef.isDrag.dy + (selfRef.isDrag.attrs.cy == null ? (selfRef.isDrag.attrs.y + selfRef.isDrag.attrs.height / 2) : selfRef.isDrag.attrs.cy); + /* prevent shapes from being dragged out of the canvas */ + var clientX = e.clientX - (newX < 20 ? newX - 20 : newX > selfRef.width - 20 ? newX - selfRef.width + 20 : 0); + var clientY = e.clientY - (newY < 20 ? newY - 20 : newY > selfRef.height - 20 ? newY - selfRef.height + 20 : 0); + selfRef.isDrag.translate(clientX - selfRef.isDrag.dx, clientY - selfRef.isDrag.dy); + selfRef.isDrag.label.translate(clientX - selfRef.isDrag.dx, clientY - selfRef.isDrag.dy); + for (var i in selfRef.graph.edges) { + selfRef.graph.edges[i].connection.draw(); + } + //selfRef.r.safari(); + selfRef.isDrag.dx = clientX; + selfRef.isDrag.dy = clientY; + } + }; + document.onmouseup = function () { + selfRef.isDrag && selfRef.isDrag.animate({"fill-opacity": 0}, 500); + selfRef.isDrag = false; + }; +}; + +/* + * Renderer using RaphaelJS + */ +Graph.Renderer.Raphael.prototype = { + translate: function(point) { + return [ + (point[0] - this.graph.layoutMinX) * this.factorX + this.radius, + (point[1] - this.graph.layoutMinY) * this.factorY + this.radius + ]; + }, + + rotate: function(point, length, angle) { + var dx = length * Math.cos(angle); + var dy = length * Math.sin(angle); + return [point[0]+dx, point[1]+dy]; + }, + + draw: function() { + this.factorX = (width - 2 * this.radius) / (this.graph.layoutMaxX - this.graph.layoutMinX); + this.factorY = (height - 2 * this.radius) / (this.graph.layoutMaxY - this.graph.layoutMinY); + for (var i = 0; i < this.graph.nodes.length; i++) { + this.drawNode(this.graph.nodes[i]); + } + for (var i = 0; i < this.graph.edges.length; i++) { + this.drawEdge(this.graph.edges[i]); + } + }, + drawNode: function(node) { + var point = this.translate([node.layoutPosX, node.layoutPosY]); + node.point = point; + + /* if node has already been drawn, move the nodes */ + if(node.shape) { +// console.log(node.shape.attrs ); + var opoint = [ node.shape.attrs.cx || node.shape.attrs.x + node.shape.attrs.width / 2 , node.shape.attrs.cy || node.shape.attrs.y + node.shape.attrs.height / 2 + 15 ]; + node.shape.translate(point[0]-opoint[0], point[1]-opoint[1]); + node.shape.label.translate(point[0]-opoint[0], point[1]-opoint[1]); + this.r.safari(); + return; + } + var shape; + if(node.content.getShape) { + shape = node.content.getShape(this.r, point[0], point[1]); + shape.attr({"fill-opacity": 0}); + } else { + shape = this.r.ellipse(point[0], point[1], 30, 20); + var color = Raphael.getColor(); + shape.attr({fill: color, stroke: color, "fill-opacity": 0, "stroke-width": 2}) + } + shape.mousedown(this.dragger); + shape.node.style.cursor = "move"; + shape.label = this.r.text(point[0], point[1] + 30, node.content.label || node.id); // Beware: operator || also considers values like -1, 0, ... + node.shape = shape; + }, + drawEdge: function(edge) { + /* if edge already has been drawn, only refresh the edge */ + edge.connection && edge.connection.draw(); + if(!edge.connection) + edge.connection = this.r.connection(edge.source.shape, edge.target.shape, { fg: edge.color, bg: edge.colorbg, directed: edge.directed }); + } +}; +Graph.Layout = {}; +Graph.Layout.Spring = function(graph) { + this.graph = graph; + this.iterations = 500; + this.maxRepulsiveForceDistance = 6; + this.k = 2; + this.c = 0.01; + this.maxVertexMovement = 0.5; + }; +Graph.Layout.Spring.prototype = { + layout: function() { + this.layoutPrepare(); + for (var i = 0; i < this.iterations; i++) { + this.layoutIteration(); + } + this.layoutCalcBounds(); + }, + + layoutPrepare: function() { + for (var i = 0; i < this.graph.nodes.length; i++) { + var node = this.graph.nodes[i]; + node.layoutPosX = 0; + node.layoutPosY = 0; + node.layoutForceX = 0; + node.layoutForceY = 0; + } + + }, + + layoutCalcBounds: function() { + var minx = Infinity, maxx = -Infinity, miny = Infinity, maxy = -Infinity; + + for (var i = 0; i < this.graph.nodes.length; i++) { + var x = this.graph.nodes[i].layoutPosX; + var y = this.graph.nodes[i].layoutPosY; + + if(x > maxx) maxx = x; + if(x < minx) minx = x; + if(y > maxy) maxy = y; + if(y < miny) miny = y; + } + + this.graph.layoutMinX = minx; + this.graph.layoutMaxX = maxx; + this.graph.layoutMinY = miny; + this.graph.layoutMaxY = maxy; + }, + + layoutIteration: function() { + // Forces on nodes due to node-node repulsions + for (var i = 0; i < this.graph.nodes.length; i++) { + var node1 = this.graph.nodes[i]; + for (var j = i + 1; j < this.graph.nodes.length; j++) { + var node2 = this.graph.nodes[j]; + this.layoutRepulsive(node1, node2); + } + } + // Forces on nodes due to edge attractions + for (var i = 0; i < this.graph.edges.length; i++) { + var edge = this.graph.edges[i]; + this.layoutAttractive(edge); + } + + // Move by the given force + for (var i = 0; i < this.graph.nodes.length; i++) { + var node = this.graph.nodes[i]; + var xmove = this.c * node.layoutForceX; + var ymove = this.c * node.layoutForceY; + + var max = this.maxVertexMovement; + if(xmove > max) xmove = max; + if(xmove < -max) xmove = -max; + if(ymove > max) ymove = max; + if(ymove < -max) ymove = -max; + + node.layoutPosX += xmove; + node.layoutPosY += ymove; + node.layoutForceX = 0; + node.layoutForceY = 0; + } + }, + + layoutRepulsive: function(node1, node2) { + var dx = node2.layoutPosX - node1.layoutPosX; + var dy = node2.layoutPosY - node1.layoutPosY; + var d2 = dx * dx + dy * dy; + if(d2 < 0.01) { + dx = 0.1 * Math.random() + 0.1; + dy = 0.1 * Math.random() + 0.1; + var d2 = dx * dx + dy * dy; + } + var d = Math.sqrt(d2); + if(d < this.maxRepulsiveForceDistance) { + var repulsiveForce = this.k * this.k / d; + node2.layoutForceX += repulsiveForce * dx / d; + node2.layoutForceY += repulsiveForce * dy / d; + node1.layoutForceX -= repulsiveForce * dx / d; + node1.layoutForceY -= repulsiveForce * dy / d; + } + }, + + layoutAttractive: function(edge) { + var node1 = edge.source; + var node2 = edge.target; + + var dx = node2.layoutPosX - node1.layoutPosX; + var dy = node2.layoutPosY - node1.layoutPosY; + var d2 = dx * dx + dy * dy; + if(d2 < 0.01) { + dx = 0.1 * Math.random() + 0.1; + dy = 0.1 * Math.random() + 0.1; + var d2 = dx * dx + dy * dy; + } + var d = Math.sqrt(d2); + if(d > this.maxRepulsiveForceDistance) { + d = this.maxRepulsiveForceDistance; + d2 = d * d; + } + var attractiveForce = (d2 - this.k * this.k) / this.k; 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