From f813322cbdf5d6c2b0d97068f0a6e3f07eb41eeb Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Wed, 21 Oct 2020 15:21:51 +0200 Subject: [PATCH] mp3: done 1-3 --- mp2/Project_2_Maggioni_Claudio/ex2.m | 28 +- mp2/ex2_2_spy.tex | 2535 +- mp2/ex2_3_spy.tex | 5542 +- mp2/project_2_Maggioni_Claudio.pdf | Bin 1397192 -> 1813276 bytes mp2/project_2_Maggioni_Claudio.tex | 2 +- mp3/Makefile | 15 + .../datasets/2d_meshes/3elt.mat | Bin 0 -> 118304 bytes .../datasets/2d_meshes/Toy_meshes.mat | Bin 0 -> 52399 bytes .../datasets/2d_meshes/airfoil1.mat | Bin 0 -> 105639 bytes .../datasets/2d_meshes/barth4.mat | Bin 0 -> 147001 bytes .../datasets/2d_meshes/crack.mat | Bin 0 -> 167451 bytes .../datasets/2d_meshes/mesh3e1.mat | Bin 0 -> 4530 bytes .../datasets/Mesh_generation/grid3d.m | 31 + .../datasets/Mesh_generation/grid3dt.m | 38 + .../datasets/Mesh_generation/grid5.m | 21 + .../datasets/Mesh_generation/grid5rec.m | 20 + .../datasets/Mesh_generation/grid5recRotate.m | 25 + .../datasets/Mesh_generation/grid7.m | 21 + .../datasets/Mesh_generation/grid9.m | 23 + .../datasets/Mesh_generation/gridt.m | 25 + .../external/metismex.m | 30 + .../external/metismex.mexa64 | Bin 0 -> 258976 bytes .../external/metismex.mexmaci64 | Bin 0 -> 235008 bytes .../external/metispart.m | 31 + .../external/other.m | 31 + .../src/Bench_bisection.m | 196 + .../src/Bench_metis.m | 20 + .../src/Bench_rec_bisection.m | 80 + mp3/Project_3_Maggioni_Claudio/src/Contents.m | 45 + .../src/Initialize_case.m | 18 + .../src/Visualization/gplotg.m | 49 + .../src/Visualization/gplotmap.m | 90 + .../src/Visualization/gplotpart.m | 76 + .../src/Visualization/highlight.m | 60 + .../src/Visualization/spypart.m | 39 + .../src/addpaths_GP.m | 8 + .../src/adjacency_to_incidence.m | 54 + .../src/bisection_coordinate.m | 33 + .../src/bisection_inertial.m | 55 + .../src/bisection_metis.m | 25 + .../src/bisection_spectral.m | 64 + .../src/blockdiags.m | 86 + mp3/Project_3_Maggioni_Claudio/src/cutsize.m | 26 + .../src/ex2_bisection_table.m | 32 + .../src/ex3_eppstein.pdf | Bin 0 -> 137975 bytes .../src/ex3_grid9.pdf | Bin 0 -> 196614 bytes .../src/ex3_plots.m | 14 + .../src/ex3_small.pdf | Bin 0 -> 96110 bytes mp3/Project_3_Maggioni_Claudio/src/other.m | 26 + .../src/partition.m | 39 + .../src/rec_bisection.m | 61 + .../src/setfilter.m | 15 + mp3/assignment.sty | 95 + mp3/ex3_eppstein.pdf | Bin 0 -> 132923 bytes mp3/ex3_eppstein.tex | 9993 +++ mp3/ex3_grid9.pdf | Bin 0 -> 196614 bytes mp3/ex3_grid9.tex | 60762 ++++++++++++++++ mp3/ex3_small.pdf | Bin 0 -> 91087 bytes mp3/ex3_small.tex | 2310 + mp3/project_3_Maggioni_Claudio.pdf | Bin 0 -> 560991 bytes mp3/project_3_Maggioni_Claudio.tex | 114 + mp3/usi_inf.pdf | Bin 0 -> 34678 bytes 62 files changed, 82874 insertions(+), 29 deletions(-) create mode 100644 mp3/Makefile create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/3elt.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/Toy_meshes.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/airfoil1.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/barth4.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/crack.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/2d_meshes/mesh3e1.mat create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid3d.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid3dt.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5rec.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5recRotate.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid7.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid9.m create mode 100755 mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/gridt.m create mode 100755 mp3/Project_3_Maggioni_Claudio/external/metismex.m create mode 100755 mp3/Project_3_Maggioni_Claudio/external/metismex.mexa64 create mode 100755 mp3/Project_3_Maggioni_Claudio/external/metismex.mexmaci64 create mode 100644 mp3/Project_3_Maggioni_Claudio/external/metispart.m create mode 100644 mp3/Project_3_Maggioni_Claudio/external/other.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Bench_bisection.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Bench_metis.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Bench_rec_bisection.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Contents.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Initialize_case.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotg.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotmap.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotpart.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Visualization/highlight.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/Visualization/spypart.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/addpaths_GP.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/adjacency_to_incidence.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/bisection_coordinate.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/bisection_inertial.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/bisection_metis.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/bisection_spectral.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/blockdiags.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/cutsize.m create mode 100644 mp3/Project_3_Maggioni_Claudio/src/ex2_bisection_table.m create mode 100644 mp3/Project_3_Maggioni_Claudio/src/ex3_eppstein.pdf create mode 100644 mp3/Project_3_Maggioni_Claudio/src/ex3_grid9.pdf create mode 100644 mp3/Project_3_Maggioni_Claudio/src/ex3_plots.m create mode 100644 mp3/Project_3_Maggioni_Claudio/src/ex3_small.pdf create mode 100755 mp3/Project_3_Maggioni_Claudio/src/other.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/partition.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/rec_bisection.m create mode 100755 mp3/Project_3_Maggioni_Claudio/src/setfilter.m create mode 100644 mp3/assignment.sty create mode 100644 mp3/ex3_eppstein.pdf create mode 100644 mp3/ex3_eppstein.tex create mode 100644 mp3/ex3_grid9.pdf create mode 100644 mp3/ex3_grid9.tex create mode 100644 mp3/ex3_small.pdf create mode 100644 mp3/ex3_small.tex create mode 100644 mp3/project_3_Maggioni_Claudio.pdf create mode 100644 mp3/project_3_Maggioni_Claudio.tex create mode 100644 mp3/usi_inf.pdf diff --git a/mp2/Project_2_Maggioni_Claudio/ex2.m b/mp2/Project_2_Maggioni_Claudio/ex2.m index 5c25ff0..e780e98 100644 --- a/mp2/Project_2_Maggioni_Claudio/ex2.m +++ b/mp2/Project_2_Maggioni_Claudio/ex2.m @@ -1,4 +1,18 @@ -n = 5; +A = createA(5); + +spy(A); +matlab2tikz('../ex2_2_spy.tex'); + +A = createA(100); + +figure; +subplot(1,2,1) +spy(A); +subplot(1,2,2) +spy(chol(A)); +matlab2tikz('../ex2_3_spy.tex'); + +function A = createA(n) A = sparse(n, n); A(1,:) = 1; A(end,:) = 1; @@ -7,15 +21,7 @@ A(:,end) = 1; % make sure diagonal is 0 A(1,1) = 0; -A(n,n) = 0; +A(end,end) = 0; A = A + diag(n:(2*n-1)); -spy(A); -matlab2tikz('../ex2_2_spy.tex'); - -figure; -subplot(1,2,1) -spy(A); -subplot(1,2,2) -spy(chol(A)); -matlab2tikz('../ex2_3_spy.tex'); \ No newline at end of file +end \ No newline at end of file diff --git a/mp2/ex2_2_spy.tex b/mp2/ex2_2_spy.tex index 9cf45e5..b7aef8f 100644 --- a/mp2/ex2_2_spy.tex +++ b/mp2/ex2_2_spy.tex @@ -9,9 +9,2522 @@ \begin{tikzpicture} \begin{axis}[% -width=4.754in, -height=4.754in, -at={(1.648in,0.642in)}, +width=2.603in, +height=2.603in, +at={(1.011in,1.717in)}, +scale only axis, +xmin=0, +xmax=501, +xlabel style={font=\color{white!15!black}}, +xlabel={nz = 2494}, +y dir=reverse, +ymin=0, +ymax=501, 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+}; +\end{axis} + +\begin{axis}[% +width=2.603in, +height=2.603in, +at={(4.436in,1.717in)}, scale only axis, xmin=0, xmax=6, @@ -45,4 +2558,20 @@ axis background/.style={fill=white} 5 5\\ }; \end{axis} + +\begin{axis}[% +width=7.778in, +height=5.833in, +at={(0in,0in)}, +scale only axis, +xmin=0, +xmax=1, +ymin=0, +ymax=1, +axis line style={draw=none}, +ticks=none, +axis x line*=bottom, +axis y line*=left +] +\end{axis} \end{tikzpicture}% \ No newline at end of file diff --git a/mp2/ex2_3_spy.tex b/mp2/ex2_3_spy.tex index 66fdba1..58c83c4 100644 --- a/mp2/ex2_3_spy.tex +++ b/mp2/ex2_3_spy.tex @@ -14,35 +14,510 @@ height=2.603in, at={(1.011in,1.717in)}, scale only axis, xmin=0, -xmax=6, +xmax=101, xlabel style={font=\color{white!15!black}}, -xlabel={nz = 19}, +xlabel={nz = 494}, y dir=reverse, ymin=0, -ymax=6, +ymax=101, axis background/.style={fill=white} ] -\addplot [color=mycolor1, only marks, mark size=2.3pt, mark=*, mark options={solid, mycolor1}, forget plot] +\addplot 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6 -1], -1:1, k, k); +I = speye (k, k); +aa = blockdiags ([-I a -I], -1:1, k, k); +II = speye(k^2,k^2); +A = blockdiags ([-II aa -II], -1:1, k, k); + +A = diag(diag(A)) - A; + + +xyz = zeros(k^3,3); +x = ones(k,1) * (1:k); +y = x'; + +j = 1; +for i = 1:k + xyz(j:j+k^2-1 , 1) = x(:); + xyz(j:j+k^2-1 , 2) = y(:); + xyz(j:j+k^2-1 , 3) = i * ones(k^2,1); + j = j + k^2; +end + + +end diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid3dt.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid3dt.m new file mode 100755 index 0000000..afb1cf1 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid3dt.m @@ -0,0 +1,38 @@ +function [A,xyz] = grid3dt(k) +% GRID3DT : Generate 3-dimensional tetrahedral finite element mesh. +% +% [A,xyz] = GRID3DT(k) returns a k^3-by-k^3 symmetric positive definite +% matrix A of the k-by-k-by-k grid, with cells divided into tetrahedra, +% and an array xyz of coordinates for the grid points. + +% 1-d mesh +a = blockdiags ([-1 14 -1], -1:1, k, k); + +% glue to laminate 1-d meshes into 2-d meshes +b = blockdiags ([-1 -1], [0 1], k, k); + +% 2-d mesh +aa = blockdiags ([b a b'], -1:1, k, k); + +% glue to laminate 2-d meshes into 3-d meshes +bb = blockdiags ([b b'], [0 1], k, k); + +% 3-d mesh +A = blockdiags ([bb aa bb'], -1:1, k, k); +A = diag(diag(A)) - A; + + +% xyz coordinates of nodes +xyz = zeros(k^3,3); +x = ones(k,1) * (1:k); +y = x'; +j = 1; +for i = 1:k + slab = j:j+k^2-1; + xyz(slab,1) = x(:); + xyz(slab,2) = y(:); + xyz(slab,3) = i * ones(k^2,1); + j = j + k^2; +end + +end diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5.m new file mode 100755 index 0000000..9057a46 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5.m @@ -0,0 +1,21 @@ +function [A,xy] = grid5(k) +% GRID5 : Generate 5-point finite difference mesh. +% +% [A,xy] = GRID5(k) returns a k^2-by-k^2 symmetric positive definite +% matrix A with the structure of the k-by-k 5-point grid, +% and an array xy of coordinates for the grid points. + +a = blockdiags ([-1 4 -1], -1:1, k, k); +I = speye (k, k); +A = blockdiags ([-I a -I], -1:1, k, k); +A = diag(diag(A)) - A; + + +xy = zeros(k^2,2); +x = ones(k,1) * (1:k); +y = x'; +xy(:,1) = x(:); +xy(:,2) = y(:); + + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5rec.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5rec.m new file mode 100755 index 0000000..5c6d7af --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5rec.m @@ -0,0 +1,20 @@ +function [A,xy] = grid5rec(k1, k2) +% GRID5lrec : Generate 5-point finite difference on a rectangular mesh. +% +% [A,xy] = GRID5REC(k) returns a k1*k2-by-k1*k2 symmetric positive definite +% matrix A with the structure of the k-by-k 5-point grid, +% and an array xy of coordinates for the grid points. + +a = blockdiags ([-1 4 -1], -1:1, k1, k1); +I = speye (k1, k1); +A = blockdiags ([-I a -I], -1:1, k2, k2); +A = diag(diag(A)) - A; + + +[x,y] = meshgrid(1:k2, 1:k1); +xy = zeros(k1*k2,2); +xy(:,1) = x(:); +xy(:,2) = y(:); + + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5recRotate.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5recRotate.m new file mode 100755 index 0000000..0ac6413 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid5recRotate.m @@ -0,0 +1,25 @@ +function [A,xy] = grid5recRotate(k1, k2, angle) +% GRID5lrec : Generate 5-point finite difference on a rectangular mesh. +% +% [A,xy] = GRID5REC(k) returns a k1*k2-by-k1*k2 symmetric positive definite +% matrix A with the structure of the k-by-k 5-point grid, +% and an array xy of coordinates for the grid points. + +a = blockdiags ([-1 4 -1], -1:1, k1, k1); +I = speye (k1, k1); +A = blockdiags ([-I a -I], -1:1, k2, k2); +A = diag(diag(A)) - A; + + +[x,y] = meshgrid(1:k2, 1:k1); + +rad = angle*pi/180; + +newX = x * cos(rad) - y * sin(rad); +newY = y * cos(rad) + x * sin(rad); + +xy = zeros(k1*k2,2); +xy(:,1) = newX(:); +xy(:,2) = newY(:); + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid7.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid7.m new file mode 100755 index 0000000..a3f51dc --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid7.m @@ -0,0 +1,21 @@ +function [A,xy] = grid7(k) +% GRID7 : Generate 7-point finite difference mesh. +% +% [A,xy] = GRID7(k) returns a k^2-by-k^2 symmetric positive definite +% matrix A with the structure of the k-by-k 7-point grid, +% and an array xy of coordinates for the grid points. + +a = blockdiags ([-1 6 -1], -1:1, k, k); +b = blockdiags ([-1 -1], [0 1], k, k); +A = blockdiags ([b a b'], -1:1, k, k); +A = diag(diag(A)) - A; + + +xy = zeros(k^2,2); +x = ones(k,1) * (1:k); +y = x'; +xy(:,1) = x(:); +xy(:,2) = y(:); + + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid9.m b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid9.m new file mode 100755 index 0000000..3143d1d --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/grid9.m @@ -0,0 +1,23 @@ +function [A,xy] = grid9(k) +% GRID9 : Generate 9-point finite difference mesh. +% +% [A,xy] = GRID9(k) returns a k^2-by-k^2 symmetric positive definite +% matrix A with the structure of the k-by-k 9-point grid, +% and an array xy of coordinates for the grid points. + + +a = blockdiags ([-4 20 -4], -1:1, k, k); +b = blockdiags ([-1 -4 -1], -1:1, k, k); +A = blockdiags ([b a b], -1:1, k, k); +A = diag(diag(A)) - A; + +xy = zeros(k^2,2); +x = ones(k,1) * (1:k); +y = x'; + + +xy(:,1) = x(:); +xy(:,2) = y(:); + + +end diff --git a/mp3/Project_3_Maggioni_Claudio/datasets/Mesh_generation/gridt.m 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metispart(A,xy); +% METISPART : Partition a graph using Metis default method. +% +% p = metispart(A) returns a list of the vertices on one side of +% a partition obtained by Metis 4.0 applied to the graph of A. +% +% Optional arguments: +% metispart(A,xy) draws a picture of the partitioned graph, +% using the rows of xy as vertex coordinates. +% [p1,p2] = metispart(...) also returns the list of vertices +% on the other side of the partition. +% +% See also METISMEX (which accepts all the Metis options), +% GEOPART, GSPART, SPECPART, METISDICE, METISND. +% +% John Gilbert 3 Jul 01 +% Copyright (c) 1990-2001 by Xerox Corporation. All rights reserved. +% HELP COPYRIGHT for complete copyright and licensing notice. + +if nargin < 2 + xy = 0; +end; +picture = max(size(xy)) > 1; + +map = metismex('PartGraphRecursive',A,2); +[p1,p2] = other(map); + +if picture + gplotpart(A,xy,p1); + title('Metis Partition') +end; \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/external/other.m b/mp3/Project_3_Maggioni_Claudio/external/other.m new file mode 100644 index 0000000..d667d33 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/external/other.m @@ -0,0 +1,31 @@ +function [out1,out2] = other(in1,in2); +% OTHER : Find the other part of a partition, or +% convert a partition to the other representation. +% +% part2 = other(part1,n) : part1 is a list of subscripts in 1:n; +% part2 becomes the complementary list. +% part2 = other(part1,A) : Same, except n is taken as max(size(A)). +% [part2,p] = other(part1,n) : Also returns 0/1 partition vector of length n. +% [part2,p] = other(part1,A) : Same. +% [part1,part2] = other(p) : Converts 0/1 vector to lists of subscripts. +% +% John Gilbert, 1993. +% Copyright (c) 1990-1996 by Xerox Corporation. All rights reserved. +% HELP COPYRIGHT for complete copyright and licensing notice. + +% Modified 3 Jul 01 by JRG to fix a bug in the "map" case + +if nargin == 1 + + out1 = find(in1 == 0); + out2 = find(in1 ~= 0); + +else + + if max(size(in2)) > 1 + in2 = max(size(in2)); + end; + out2 = full(spones(sparse(1,in1,1,1,in2))); + out1 = find(out2 == 0); + +end; diff --git a/mp3/Project_3_Maggioni_Claudio/src/Bench_bisection.m b/mp3/Project_3_Maggioni_Claudio/src/Bench_bisection.m new file mode 100755 index 0000000..0fda582 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Bench_bisection.m @@ -0,0 +1,196 @@ +function Bench_bisection() +% Compare various graph bisection algorithms +% +% D.P & O.S for Numerical Computing at USI + +% add the necessaty paths +addpaths_GP; + +warning('off','all'); +picture = 1; + +if nargin < 1 + whichdemos = [1 2 3]; +end; + +format compact; + +disp(' *********************************************') +disp(' *** Graph bisection benchmark ***'); +disp(' *********************************************') +disp(' '); +disp(' The file "Toy_meshes.mat" contains sample meshes with coordinates.'); +disp(' '); + +% load meshes +load Toy_meshes; +whos; + + +for nmesh = 1:7 + close all; clf reset; + + if (nmesh==1) + disp(' '); + disp(' Function "grid5rec" produces a rectangular grid:'); + disp(' '); + disp('[A,xy] = grid5rec(8,80);'); + disp(' '); + [W,coords] = grid5rec(8, 80); + end + if (nmesh==2) + disp(' '); + disp(' Function "grid5rec" produces a rectangular grid:'); + disp(' '); + disp('[A,xy] = grid5rec(80,8);'); + disp(' '); + [W,coords] = grid5rec(80, 8); + end + if (nmesh==3) + disp(' '); + disp(' Function "grid5recRotate" produces a rotated rectangular grid:'); + disp(' '); + disp('[A,xy] = grid5recRotate(80,8, -45);'); + disp(' '); + [W,coords] = grid5recRotate(80, 8, -45); + end + if (nmesh==4) + disp(' '); + disp(' Function "gridt" produces a triangular grid:'); + disp(' '); + disp(' (See also grid5, grid7, grid9, grid3d, grid3dt.)'); + disp(' '); + disp('[A,xy] = gridt(40);'); + disp(' '); + [W,coords] = gridt(20); + end + if (nmesh==5) + disp(' '); + disp(' Function "gridt" produces a triangular grid:'); + disp(' '); + disp(' (See also grid5, grid7, grid9, grid3d, grid3dt.)'); + disp(' '); + disp('[A,xy] = gridt(30);'); + disp(' '); + [W,coords] = grid9(30); + end + if (nmesh==6) + W = Smallmesh; + coords = Smallmesh_coords; + end + if (nmesh==7) + W = Tapir; + coords = Tapir_coords; + end + if (nmesh==8) + W = Eppstein; + coords = Eppstein_coords; + end + + + disp(' '); + disp(' *********************************************') + disp(' *** Various Bisection Methods *** '); + disp(' *********************************************') + disp(' '); + disp(' '); + + + if (nmesh==1) + disp('An initial rectangular grid5rec(8,80) mesh'); + end + if (nmesh==2) + disp('An initial rectangular grid5rec(80,8) mesh'); + end + if (nmesh==3) + disp('An initial rectangular grid5rec(80,8) mesh, rotated by 45 degree'); + end + if (nmesh==4) + disp(' gridt(20) mesh'); + end + if (nmesh==5) + disp(' gridt9(20) mesh'); + end + if (nmesh==6) + disp(' Small mesh '); + end + if (nmesh==7) + disp(' "Tapir" is a test of a no-obtuse-angles mesh generation algorithm'); + disp(' due to Bern, Mitchell, and Ruppert. '); + end + if (nmesh==8) + disp(' Eppstein mesh'); + end + + figure(1) + disp('gplotg(Tmesh,Tmeshxy);'); + disp(' '); + gplotg(W,coords); + nvtx = size(W,1); + nedge = (nnz(W)-nvtx)/2; + xlabel([int2str(nvtx) ' vertices, ' int2str(nedge) ' edges'],'visible','on'); + + disp(' Hit space to continue ...'); + pause; + + disp(' 1. Coordinate bisection of a mesh. '); + disp(' p = coordpart(A,xy) returns a list of the vertices '); + disp(' on one side of a partition obtained by bisection '); + disp(' perpendicular to a coordinate axis. We try every '); + disp(' coordinate axis and return the best cut. Input W is '); + disp(' the adjacency matrix of the mesh; each row of xy is '); + disp(' the coordinates of a point in d-space. '); + + figure(2) + [p1,p2] = bisection_coordinate(W,coords,picture); + [cut_coord] = cutsize(W,p1); + disp('Space to continue ...'); + pause; + + figure(3) + disp(' '); + disp(' 2. A multilevel method from the "Metis 5.0.2" package.'); + disp(' This will only work if you have Metis and its Matlab interface.'); + disp(' '); + [p1,p2] = bisection_metis(W,coords,picture); + [cut_metis] = cutsize(W,p1); + disp(' '); + disp(' Hit space to continue ...'); + pause; + + + disp(' '); + disp(' 3. Spectral partitioning, which uses the second eigenvector of'); + disp(' the Laplacian matrix of the graph, also known as the "Fiedler vector".'); + disp(' '); + figure(6) + [p1,p2] = bisection_spectral(W,coords,picture); + [cut_spectral] = cutsize(W,p1); + disp(' '); + disp(' Hit space to continue ...'); + pause; + + figure(7) + disp(' '); + disp(' 4. Inertial partitioning, which uses the coordinates to find'); + disp(' a separating line in the plane.'); + disp(' '); + [p1,p2] = bisection_inertial(W,coords,picture); + [cut_inertial] = cutsize(W,p1); + disp(' '); + disp(' Hit space to continue ...'); + pause; + + close all; + format; + + disp(' '); + disp(' *****************************************************************') + disp(' *** Bisection Benchmark *** '); + disp(' *** D.P. & O.S. for Numerical Computing, USI Lugano *** '); + disp(' *****************************************************************') + disp(' '); + disp(' '); + + +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/Bench_metis.m b/mp3/Project_3_Maggioni_Claudio/src/Bench_metis.m new file mode 100755 index 0000000..b8b3faa --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Bench_metis.m @@ -0,0 +1,20 @@ +function [cut_recursive,cut_kway] = Bench_metis(picture) +% Compare recursive bisection and direct k-way partitioning, +% as implemented in the Metis 5.0.2 library. + +% Add necessary paths +addpaths_GP; + +% Graphs in question +% load 'airfoil1.mat' ; +% load 'crack.mat'; + +% Steps +% 1. Initialize the cases +% 2. Call metismex to +% a) Recursively partition the graphs in 16 and 32 subsets. +% b) Perform direct k-way partitioning of the graphs in 16 and 32 subsets. +% 3. Visualize the results for 32 partitions + + +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/Bench_rec_bisection.m b/mp3/Project_3_Maggioni_Claudio/src/Bench_rec_bisection.m new file mode 100755 index 0000000..31b6beb --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Bench_rec_bisection.m @@ -0,0 +1,80 @@ +% Benchmark for recursively partitioning meshes, based on various +% bisection approaches +% +% D.P & O.S for Numerical Computing in USI + + + +% add necessary paths +addpaths_GP; +nlevels_a = 3; +nlevels_b = 4; + +fprintf(' *********************************************\n') +fprintf(' *** Recursive graph bisection benchmark ***\n'); +fprintf(' *********************************************\n') + +% load cases +cases = { + 'airfoil1.mat'; + '3elt.mat'; + 'barth4.mat'; + 'mesh3e1.mat'; + 'crack.mat'; + }; + +nc = length(cases); +maxlen = 0; +for c = 1:nc + if length(cases{c}) > maxlen + maxlen = length(cases{c}); + end +end + +for c = 1:nc + fprintf('.'); + sparse_matrices(c) = load(cases{c}); +end + + +fprintf('\n\n Report Cases Nodes Edges\n'); +fprintf(repmat('-', 1, 40)); +fprintf('\n'); +for c = 1:nc + spacers = repmat('.', 1, maxlen+3-length(cases{c})); + [params] = Initialize_case(sparse_matrices(c)); + fprintf('%s %s %10d %10d\n', cases{c}, spacers,params.numberOfVertices,params.numberOfEdges); +end + +%% Create results table +fprintf('\n%7s %16s %20s %16s %16s\n','Bisection','Spectral','Metis 5.0.2','Coordinate','Inertial'); +fprintf('%10s %10d %6d %10d %6d %10d %6d %10d %6d\n','Partitions',8,16,8,16,8,16,8,16); +fprintf(repmat('-', 1, 100)); +fprintf('\n'); + + +for c = 1:nc + spacers = repmat('.', 1, maxlen+3-length(cases{c})); + fprintf('%s %s', cases{c}, spacers); + sparse_matrix = load(cases{c}); + + + % Recursively bisect the loaded graphs in 8 and 16 subgraphs. + % Steps + % 1. Initialize the problem + [params] = Initialize_case(sparse_matrices(c)); + W = params.Adj; + coords = params.coords; + % 2. Recursive routines + % i. Spectral + % ii. Metis + % iii. Coordinate + % iv. Inertial + % 3. Calculate number of cut edges + % 4. Visualize the partitioning result + + + fprintf('%6d %6d %10d %6d %10d %6d %10d %6d\n',0,0,... + 0,0,0,0,0,0); + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/Contents.m b/mp3/Project_3_Maggioni_Claudio/src/Contents.m new file mode 100755 index 0000000..43bbf8f --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Contents.m @@ -0,0 +1,45 @@ +% Graph Partitioning Toolbox. +% +% Datasets +% 2d_meshes - Graphs from aeronautics applications +% Mesh_generation - Simple 2d and 3d mesh generation routines +% +% External +% metismex - Interface between Metis 5.0.2 and Matlab +% +% Bisection methods. +% bisection_spectral - Spectral bisection +% bisection_coordinate - Coordinate bisection. +% bisection_inertial - Inertial bisection. +% bisection_metis - Multilevel method from Metis. +% partition - Partition points by a plane. +% +% Multiway partitions. +% rec_bisection - Use any bisection method to get a multiway partition. +% +% +% Benchmark scripts +% Bench_bisection - Compare various graph bisection algorithms +% Bench_eigen_plot - Visualize information from the eigenspectrum +% of the graph Laplacian +% Bench_metis_comparison - Compare recursive bisection and direct k-way partitioning, +% as implemented in the Metis 5.0.2 library. +% Bench_rec_bisection - Benchmark for recursively partitioning meshes, based on various +% bisection approaches +% +% +% Visualization and graphics. +% gplotpart - Draw a 2-way partition. +% gplotmap - Draw a multiway partition. +% highlight - Draw a mesh with some vertices highlighted. +% gplotg - Draw a 2D or 3D mesh (replaces Matlab's gplot). +% etreeplotg - Draw an elimination tree (replaces Matlab's etreeplot). +% spypart - Matrix spy plot with partition boundaries. +% +% Miscellaneous. +% cutsize - Find or count edges cut by a partition. +% other - Other side of a partition. +% adjacency_to_incidence - create the incidence matrix of the graph +% Contents - contents of the partitioning toolbox +% blockdiags - Create matrix with specified block diagonals. + diff --git a/mp3/Project_3_Maggioni_Claudio/src/Initialize_case.m b/mp3/Project_3_Maggioni_Claudio/src/Initialize_case.m new file mode 100755 index 0000000..ee07a1f --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Initialize_case.m @@ -0,0 +1,18 @@ +function [params] = Initialize_case(SM_case) + +% Adjacency +params.Adj = SM_case.Problem.A; +% Incidence +params.Inc = adjacency_to_incidence(params.Adj); +% Edges +params.numberOfEdges = size(params.Inc,1); +% Vertices +params.numberOfVertices = size(params.Inc,2); +% coords (if exist) +if isfield(SM_case, 'Problem') && isfield(SM_case.Problem, 'aux') && ... + isfield(SM_case.Problem.aux, 'coord') + params.coords = SM_case.Problem.aux.coord; +end + + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotg.m b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotg.m new file mode 100755 index 0000000..0fbbf70 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotg.m @@ -0,0 +1,49 @@ +function handle = gplotg(A,xy,lc) +% GPLOTG : Plot a "graph theoretic" graph. +% +% handle = gplotg(A,xy) plots the graph specified by A and xy. +% A graph, G, is a set of nodes numbered from 1 to n, +% and a set of connections, or edges, between them. +% In order to plot G, two matrices are needed. +% The adjacency matrix, A, has a(i,j) nonzero if and +% only if node i is connected to node j. The coordinates +% array, xy, is an n-by-2 or n-by-3 matrix with the position +% for node i in the i-th row, xy(i,:) = [x(i) y(i)], +% or xy(i,:) = [x(i) y(i) z(i)]. +% +% gplotg(A,xy,lc) uses line type and color instead of the +% default, 'r-'. For example, lc = 'g:'. See PLOT. + + + +if nargin < 3 + lc = 'r-'; +end + +[i, j] = find(A); +[~, p] = sort(max(i,j)); +i = i(p); +j = j(p); + +% Create a long, NaN-seperated list of line segments, +% rather than individual segments. + +X = [ xy(i,1) xy(j,1) NaN*ones(size(i))]'; +Y = [ xy(i,2) xy(j,2) NaN*ones(size(i))]'; +X = X(:); +Y = Y(:); + +if size(xy,2) == 2 + h = plot (X, Y, lc); +else + set(gca,'drawmode','fast'); + Z = [ xy(i,3) xy(j,3) NaN*ones(size(i))]'; + Z = Z(:); + h = plot3 (X, Y, Z, lc); +end + +axis equal; +axis off; +if nargout >= 1 + handle = h; +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotmap.m b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotmap.m new file mode 100755 index 0000000..11fefd6 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotmap.m @@ -0,0 +1,90 @@ +function handle = gplotmap(A,xy,map,pcolor,ecolor) +% GPLOTMAP : Plot a partitioned graph in 2 or 3 dimensions. +% +% gplotmap(A,xy,map) plots the n-vertex graph specified +% by the n by n adjacency (or Laplacian) matrix A +% and the n by 2 or 3 matrix of coordinates xy. +% Argument map is an n-vector of part numbers, which +% assigns each vertex to a part. +% +% By default, edges that join different parts are omitted, and +% the picture shows each part in a different color. The call +% +% gplotmap(A,xy,map,pcolor,ecolor) +% +% uses color "pcolor" for the parts and "ecolor" for the +% edges between the parts. If "pcolor" has multiple rows, +% each part gets the color of one row (in cyclic order). + + +[n,n] = size(A); +if nargin < 3, + map = zeros(1,n); +end; +if nargin < 4 + pcolor = []; +end; +if nargin < 5, + ecolor = []; +end; + +parts = setfilter(map); +nparts = length(parts); +if length(pcolor) == 0, + pcolor = hsv(nparts); + pcolor = pcolor(randperm(nparts),:); +end; + +clf reset +colordef(gcf,'black') +if size(xy,2) == 2 + axis([min(xy(:,1)) max(xy(:,1)) min(xy(:,2)) max(xy(:,2))]); +else + axis([min(xy(:,1)) max(xy(:,1)) min(xy(:,2)) max(xy(:,2)) ... + min(xy(:,3)) max(xy(:,3))]); +end; + +axis equal; +axis off; + +hold on + +% Count and plot the separating edges. +[i,j] = find(A); +f = find(map(i) > map(j)); + +%comm volume +volume = 0; +for r=1:size(A,1) + idx = find(i == r); + volume = volume + length(unique(map(j(idx)))) - 1; +end + +if length(f) + xlabel( [int2str(length(f)) ' cut edges on ' int2str(nparts) ' partitions, '... + 'communication volume ' int2str(volume) ],'visible','on'); + if length(ecolor) + cut = sparse(i(f),j(f),1,n,n); + set(gplotg(cut,xy,'-'),'color',ecolor); + end; +else + xlabel('0 cut edges','visible','on'); +end; + +% Plot each piece. +ncolor = 1; +for partnumber = parts; + c = pcolor(ncolor,:); + ncolor = rem(ncolor, size(pcolor,1)) + 1; + part = find(map == partnumber); + set(gplotg(A(part,part),xy(part,:),'-'),'color',c); + if n < 500, + if size(xy,2) == 2 + set(plot(xy(part,1),xy(part,2),'o'),'color',c); + else + set(plot3(xy(part,1),xy(part,2),xy(part,3),'o'),'color',c); + end; + end; +end; + +hold off diff --git a/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotpart.m b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotpart.m new file mode 100755 index 0000000..bae6e71 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Visualization/gplotpart.m @@ -0,0 +1,76 @@ +function handle = gplotpart(A,xy,part1,color1,color2,color3) +% GPLOTPART : Plot a partitioned graph in 2 or 3 dimensions. +% +% gplotpart(A,xy,part1) plots the n-vertex graph specified +% by the n by n adjacency (or Laplacian) matrix A +% and the n by 2 or 3 matrix of coordinates xy. +% Argument part1 is a vector of vertex names (integers 1:n); +% the subgraphs induced by part1 and its complement are plotted +% in different colors, with the edges joining them in a third color. +% Three more optional arguments give the three colors. + + +if nargin < 3 + part1 = []; +end +if nargin < 4 + color1 = 'yellow'; +end +if nargin < 5 + color2 = 'cyan'; +end +if nargin < 6 + color3 = 'red'; +end + +[n,~] = size(A); +part1 = part1(:); +part2 = 1:n; +part2(part1)=zeros(size(part1)); +part2 = find(part2); +part2 = part2(:); +cut = spaugment(A(part1,part2),1); +cutxy = xy([part1; part2],:); + +clf reset +%colordef(gcf,'black') +if size(xy,2) == 2 + axis([min(xy(:,1)) max(xy(:,1)) min(xy(:,2)) max(xy(:,2))]); +else + axis([min(xy(:,1)) max(xy(:,1)) min(xy(:,2)) max(xy(:,2)) ... + min(xy(:,3)) max(xy(:,3))]); +end + +axis equal; +axis off; + +hold on +if ~isempty(part1) && ~isempty(part2) + set(gplotg(cut,cutxy,'-'),'color',color3); + xlabel([int2str(cutsize(A,part1)) ' cut edges'],'visible','on'); +else + xlabel('0 cut edges','visible','on'); +end + +if ~isempty(part1) + set(gplotg(A(part1,part1),xy(part1,:),'-'),'color',color1); + if n < 500 + if size(xy,2) == 2 + set(plot(xy(part1,1),xy(part1,2),'o'),'color',color1); + else + set(plot3(xy(part1,1),xy(part1,2),xy(part1,3),'o'),'color',color1); + end + end +end + +if ~isempty(part2) + set(gplotg(A(part2,part2),xy(part2,:),'-'),'color',color2); + if n < 500 + if size(xy,2) == 2 + set(plot(xy(part2,1),xy(part2,2),'o'),'color',color2); + else + set(plot3(xy(part2,1),xy(part2,2),xy(part2,3),'o'),'color',color2); + end + end +end +hold off diff --git a/mp3/Project_3_Maggioni_Claudio/src/Visualization/highlight.m b/mp3/Project_3_Maggioni_Claudio/src/Visualization/highlight.m new file mode 100755 index 0000000..16c23b8 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Visualization/highlight.m @@ -0,0 +1,60 @@ +function highlight(A,xy,sep,highcolor,meshcolor,dotsize) +% HIGHLIGHT : Plot a mesh with subgraph highlighted. +% +% highlight(A,xy,sep) plots a picture of the mesh A with coordinates xy, +% highlighting the subgraph induced by the vertices in sep. +% Optional fourth and fifth arguments are color/linetype of highlight and mesh. +% Optional sixth argument is size of highlighting dot. + + + +if nargin < 4 + highcolor = 'w-'; +end +if nargin < 5 + meshcolor = 'r-'; +end +if nargin < 6 + dotsize = 15; +end + +[n,n] = size(A); +[i,j] = find(A); + +% Plot main graph with vertices before edges. + +[ignore, p] = sort(max(i,j)); +i = i(p); +j = j(p); + +% Create a long, NaN-seperated list of line segments, +% rather than individual segments. + +X = [ xy(i,1) xy(j,1) NaN*ones(size(i))]'; +Y = [ xy(i,2) xy(j,2) NaN*ones(size(i))]'; +X = X(:); +Y = Y(:); + +xymin = min(xy); +xymax = max(xy); +plot (X, Y, meshcolor,'erasemode','none'); +axis([xymin(1) xymax(1) xymin(2) xymax(2)]); +axis('equal'); +axis('off'); +hold on; + +% Highlight sep set. + +B = A(sep,sep); +xB = xy(sep,1); +yB = xy(sep,2); +[i,j] = find(B); +X = [xB(i) xB(j)]'; +Y = [yB(i) yB(j)]'; +plot (X, Y, highcolor,'erasemode','none'); +if n < 1200 + handle = plot(xB,yB,[highcolor(1) '.'],'erasemode','none'); + set(handle,'markersize',dotsize) +end + +hold off; diff --git a/mp3/Project_3_Maggioni_Claudio/src/Visualization/spypart.m b/mp3/Project_3_Maggioni_Claudio/src/Visualization/spypart.m new file mode 100755 index 0000000..e5ab720 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/Visualization/spypart.m @@ -0,0 +1,39 @@ +function spypart(S, rp, cp); +%SPYPART Spy plot with partitioning. +% SPYPART(S,rp,cp) plots the sparsity pattern of a matrix S, +% with lines marking a block partition described by +% rp (rows) and cp (columns). +% If S is square, cp may be omitted and defaults to rp. +% +% Partitions are specified as in the output of DMPERM: +% There are length(rp)-1 row blocks, of sizes diff(rp), with rp(1)=1. + + +if (nargin < 3), cp = rp; end + +clf +colordef(gcf,'black') +[m,n] = size(S); +if max(m,n) > 100 + spy(S,3) +else + spy(S,5) +end +hold on + +if length(rp) > 2 + k = length(rp)-2; + X = [zeros(1,k); n+ones(1,k)]; + Y = rp(2:k+1) - 0.5; + Y = [Y; Y]; + plot(X,Y,'w-') +end +if length(cp) > 2 + k = length(cp)-2; + X = cp(2:k+1) - .5; + X = [X; X]; + Y = [zeros(1,k); m+ones(1,k)]; + plot(X,Y,'w-') +end +axis('ij') +hold off \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/addpaths_GP.m b/mp3/Project_3_Maggioni_Claudio/src/addpaths_GP.m new file mode 100755 index 0000000..d39555d --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/addpaths_GP.m @@ -0,0 +1,8 @@ +% add necessary paths for the Partitioning Toolbox + +addpath ../datasets/ +addpath ../datasets/Mesh_generation/ +addpath ../datasets/2d_meshes +addpath ../datasets/Roads/ +addpath ../external/ +addpath Visualization/ diff --git a/mp3/Project_3_Maggioni_Claudio/src/adjacency_to_incidence.m b/mp3/Project_3_Maggioni_Claudio/src/adjacency_to_incidence.m new file mode 100755 index 0000000..eb217ed --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/adjacency_to_incidence.m @@ -0,0 +1,54 @@ +function [Inc] = adjacency_to_incidence(Adj) + +A = Adj; +for i = 1:size(A,1) + A(i,i) = 0; +end + +Inc = adj_2_inc(A); +% Inc = adj2inc(A); +Inc = Inc'; +end + +function Ic = adj_2_inc(A) + +% adjacency2incidence - convert an adjacency matrix to an incidence matrix +% +% Ic = adjacency2incidence(A); +% +% A(i,j) = 1 iff (i,j) is an edge of the graph. +% For each edge number k of the graph linking (i,j) +% Ic(i,k)=1 and Ic(j,k)=-1 +% +% Ic is a sparse matrix. +% Ic is also known as the graph gradient. +% +% Copyright (c) 2006 Gabriel Peyre + +%% compute list of edges +[i,j,s] = find(sparse(A)); +I = find(i<=j); +i = i(I); +j = j(I); +% number of edges +n = length(i); +% number of vertices +nverts = size(A,1); + +%% build sparse matrix +s = [ones(n,1); -ones(n,1)]; +is = [(1:n)'; (1:n)']; +js = [i(:); j(:)]; +Ic = sparse(is,js,s,n,nverts); +Ic = Ic'; + +% fix self-linking problem (0) +a = find(i==j); +if not(isempty(a)) + for t=a' + Ic(i(t),t) = 1; + end +end + +end + diff --git a/mp3/Project_3_Maggioni_Claudio/src/bisection_coordinate.m b/mp3/Project_3_Maggioni_Claudio/src/bisection_coordinate.m new file mode 100755 index 0000000..3d9665c --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/bisection_coordinate.m @@ -0,0 +1,33 @@ +function [part1,part2] = bisection_coordinate(A,xy,picture) +% COORDPART : Coordinate bisection partition of a mesh. +% +% [part1,part2] = bisection_coordinate(A,xy,picture) returns a list of the vertices on one side of a partition +% obtained by bisection perpendicular to a coordinate axis. We try every +% coordinate axis and return the best cut. +% Input A is the adjacency matrix of the mesh; +% each row of xy is the coordinates of a point in d-space. +% +% coordpart(A,xy,1) also draws a picture. + +d = size(xy,2); +best_cut = inf; +for dim = 1:d + v = zeros(d,1); + v(dim) = 1; + [p1,p2] = partition(xy,v); + this_cut = cutsize(A,p1); + if this_cut < best_cut + best_cut = this_cut; + part1 = p1; + part2 = p2; + end +end + +if picture + clf reset + gplotpart(A,xy,part1); + title('Coordinate bisection') +end + + +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/bisection_inertial.m b/mp3/Project_3_Maggioni_Claudio/src/bisection_inertial.m new file mode 100755 index 0000000..4ef46f4 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/bisection_inertial.m @@ -0,0 +1,55 @@ +function [part1,part2] = bisection_inertial(A,xy,picture) +% INERTPART : Inertial partition of a graph. +% +% p = inertpart(A,xy) returns a list of the vertices on one side of a partition +% obtained by bisection with a line or plane normal to a moment of inertia +% of the vertices, considered as points in Euclidean space. +% Input A is the adjacency matrix of the mesh (used only for the picture!); +% each row of xy is the coordinates of a point in d-space. +% +% inertpart(A,xy,1) also draws a picture. +% +% See also PARTITION + + +%disp(' '); +%disp(' Numerical Computing @ USI Lugano: '); +%disp(' Implement inertial bisection'); +%disp(' '); + + +% Steps +% 1. Calculate the center of mass. +m = sum(xy, 1) / size(xy, 1); +xm = m(1); +ym = m(2); + +% 2. Construct the matrix M. +% (Consult the pdf of the assignment for the creation of M) +h = [(xy(:, 1) - xm) (xy(:, 2) - ym)]; +X1 = sum(h(:, 1) .^ 2); +X2 = sum(h(:, 1) .* h(:, 2)); +X3 = sum(h(:, 2) .^ 2); + +M = [X1 X2; X2 X3]; + +% 3. Calculate the smallest eigenvector of M. +[u, ~] = eigs(M,1,'smallestabs'); +a = u(1); +b = u(2); + +% 4. Find the line L on which the center of mass lies. +S = -b * (xy(:, 1) - xm) + a * (xy(:, 2) - ym); +Sm = median(S); + +% 5. Partition the points around the line L. +% (you may use the function partition.m) +part1 = find(S < Sm); +part2 = find(S >= Sm); + +if picture == 1 + gplotpart(A,xy,part1); + title('Inertial bisection using the Fiedler Eigenvector'); +end + +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/bisection_metis.m b/mp3/Project_3_Maggioni_Claudio/src/bisection_metis.m new file mode 100755 index 0000000..06ed45b --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/bisection_metis.m @@ -0,0 +1,25 @@ +function [part1,part2] = bisection_metis(A,xy,picture) +% METISPART : Partition a graph using Metis default method. +% +% p = metispart(A) returns a list of the vertices on one side of +% a partition obtained by Metis 4.0 applied to the graph of A. +% +% Optional arguments: +% metispart(A,xy) draws a picture of the partitioned graph, +% using the rows of xy as vertex coordinates. +% [p1,p2] = metispart(...) also returns the list of vertices +% on the other side of the partition. +% +% See also METISMEX (which accepts all the Metis options), + + +map = metismex('PartGraphRecursive',A,2); +[part1,part2] = other(map); + +if picture + gplotpart(A,xy,part1); + title('Metis bisection') +end + + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/bisection_spectral.m b/mp3/Project_3_Maggioni_Claudio/src/bisection_spectral.m new file mode 100755 index 0000000..df7a92f --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/bisection_spectral.m @@ -0,0 +1,64 @@ +function [part1,part2] = bisection_spectral(A,xy,picture) +% bisection_spectral : Spectral partition of a graph. +% +% [part1,part2] = bisection_spectral(A) returns a partition of the n vertices +% of A into two lists part1 and part2 according to the +% spectral bisection algorithm of Simon et al: +% Label the vertices with the components of the Fiedler vector +% (the second eigenvector of the Laplacian matrix) and partition +% them around the median value or 0. + + + +%disp(' '); +%disp(' Numerical Computing @ USI Lugano: '); +%disp(' Implement inertial bisection'); +%disp(' '); + +% Steps + +% 1. Construct the Laplacian. +[i, j, ~] = find(triu(A)); +G = graph(i, j); +L = laplacian(G); + +% 2. Calculate its eigensdecomposition. +[V, ~] = eigs(L,2,'smallestabs'); +W = V(:,2); + +% 3. Label the vertices with the components of the Fiedler vector. +[values, indexes] = sort(W); + +% 4. Partition them around their median value, or 0. +cut = size(values(values < 0), 1); +part1 = indexes(1:cut); +part2 = indexes(cut+1:end); + +if picture == 1 + gplotpart(A,xy,part1); + title('Spectral bisection using the Fiedler Eigenvector'); +elseif picture == 2 + drawgraph(A, xy, W, part1, part2) + % 3d plot + title('Spectral bisection using the Fiedler Eigenvector'); +end +end + +function drawgraph(A, xy, W, part1, part2) + scatter3(xy(:, 1), xy(:, 2), W, 30, W, 'filled') + colorbar + + hold on + + gplot(A(part1, part1), xy(part1, :), '-k'); + + set(gca,'ColorOrder', ones(size(part2, 2), 3) .* 0.3) + gplot(A(part2, part2), xy(part2, :)); + + P1P2 = zeros(size(A)); + P1P2(part1, part2) = A(part1, part2); + P1P2(part2, part1) = A(part2, part1); + gplot(P1P2, xy, '-r'); + + hold off +end diff --git a/mp3/Project_3_Maggioni_Claudio/src/blockdiags.m b/mp3/Project_3_Maggioni_Claudio/src/blockdiags.m new file mode 100755 index 0000000..f6a633f --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/blockdiags.m @@ -0,0 +1,86 @@ +function A = blockdiags(B,d,m,n) +% BLOCKDIAGS : Create sparse block diagonal matrices. +% +% A = blockdiags(B,d,m,n). +% +% Blockdiags, which generalizes the function "spdiags", +% produces a sparse matrix with specified block diagonals. +% +% A is an m*k-by-n*k matrix, or an m-by-n matrix of k-by-k blocks. +% The nonzero blocks of A are located on p block diagonals. +% B is a min(m,n)*k-by-p*k matrix whose k-by-k block columns +% are the block diagonals of A. +% (Alternatively, B is k-by-p*k, and then A is block Toeplitz.) +% d is a vector of p integers in the range -m+1 : n-1, +% specifying which block diagonals in A are to be nonzero. +% The values of p and k are determined from the dimensions of B and d. +% +% For k=1 this is exactly the same as A = spdiags(B,d,m,n); see spdiags +% for examples of use. For k>1 this is conceptually the same as spdiags, +% but k-by-k blocks replace matrix elements everywhere. +% +% For example, the following code sets A to the n^2-by-n^2 matrix of +% the Laplacian on an n-by-n square grid; the matrix is block tridiagonal, +% and the nonzero blocks themselves are tridiagonal or the identity. +% +% a = blockdiags ([-1 4 -1], -1:1, n, n); +% I = speye (n, n); +% A = blockdiags ([-I a -I], -1:1, n, n); + + +if nargin ~= 4 + error ('Usage: A = blockdiags(B,d,m,n)'); +end + +k = length(d); +[nrB,ncB] = size(B); +p = ncB/k; + +% Check for reasonable input. + +if any(size(m)>1) | any(size(n)>1) + error ('blockdiags(B,d,m,n): m or n not scalar'); +end +if min(size(d)~=1) + error ('blockdiags(B,d,m,n): d not a vector'); +end +if any(rem(size(B),p)) + error ('blockdiags(B,d,m,n): block size does not divide size of B'); +end + + +% Process A in compact form. + +[i,j,a] = find(B); +nzB = length(a); + +% Duplicate rows of B if A is to be block Toeplitz. + +if nrB == p + nzA = m*nzB; + range = [0:nzA-1]'; + r = 1 + rem(range,nzB); + a = a(r); + i = i(r); + j = j(r); + a = a(:); + i = i(:) + p * floor(range/nzB); + j = j(:); +end + +% Rows of B are rows of A. Shift columns appropriately. + +dd = d(1+floor((j-1)/p)); +dd = dd(:); +j = 1 + rem(j-1,p) + p * (dd + floor((i-1)/p)); + +% Clip columns that shifted off the matrix. + +f = find(j>=1 & j<=n*p); +a = a(f); +i = i(f); +j = j(f); + +% Put A back into sparse form. + +A = sparse (i,j,a,m*p,n*p); diff --git a/mp3/Project_3_Maggioni_Claudio/src/cutsize.m b/mp3/Project_3_Maggioni_Claudio/src/cutsize.m new file mode 100755 index 0000000..1bb87fc --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/cutsize.m @@ -0,0 +1,26 @@ +function [ne,edges] = cutsize(A,map); +% CUTSIZE : Edges cut by a vertex partition. +% +% ne = cutsize(A,part) : part is a list of vertices of A; +% ne is the number of edges crossing the cut. +% [ne,edges] = cutsize(A,part): Same, edges is the crossing edges as rows [i j]. +% +% "part" may also be a map, with one entry per vertex. + + +% Take input in either order unless A is 1 by 1. +if size(map,1) == size(map,2) & size(map,1) > 1 + t = A; + A = map; + map = t; +end; + +% Convert input to a map in any case. +if length(map) ~= length(A) | max(map) == length(A) + [ignore,map] = other(map,A); +end; + +[i,j] = find(A); +f = find(map(i) > map(j)); +ne = length(f); +edges = [i(f) j(f)]; diff --git a/mp3/Project_3_Maggioni_Claudio/src/ex2_bisection_table.m b/mp3/Project_3_Maggioni_Claudio/src/ex2_bisection_table.m new file mode 100644 index 0000000..c6b32ee --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/ex2_bisection_table.m @@ -0,0 +1,32 @@ +load Toy_meshes; +addpaths_GP; + +[W1,c1] = grid5rec(10,100); +[W2,c2] = grid5rec(100,10); +[W3,c3] = grid5recRotate(100,10,-45); +[W4,c4] = gridt(40); +[W5,c5] = grid9(30); + +W = {W1, W2, W3, W4, W5, Smallmesh, Tapir, Eppstein}; +c = {c1, c2, c3, c4, c5, Smallmesh_coords, Tapir_coords, Eppstein_coords}; + +fprintf('implementation\tgrid5rec(10,100)\tgrid5rec(100,10)\tgrid5recRotate(100,10,-45)\tgridt(40)\tgrid9(30)\tSmallmesh\tTapir\tEppstein\n'); +fprintf('coordinate\t'); +runtest(@(a, b) bisection_coordinate(a, b, 0)); +fprintf('metis \t'); +runtest(@(a, b) bisection_metis(a, b, 0)); +fprintf('spectral\t'); +runtest(@(a, b) bisection_spectral(a, b, 0)); +fprintf('inertial\t'); +runtest(@(a, b) bisection_inertial(a, b, 0)); + +function runtest(implementation) + global W + global c + for i=1:8 + out = implementation(W{i}, c{i}); + ne = cutsize(W{i},out); + fprintf('%d\t', ne); + end + fprintf('\n'); +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/ex3_eppstein.pdf b/mp3/Project_3_Maggioni_Claudio/src/ex3_eppstein.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d29abdbc98c69853030619b4f8623cde86b1c7f0 GIT binary patch literal 137975 zcmeGDWn5HY*EWm~iXtci3I;VuD2=q_fPe!ct#o&HNP|ctBZzdTgtVa2g3{d$(%m%- z@SZd3bzjf@{QqCx !H?wE2z2aEMigQaRFaC^$os|oZE-$~}GaftDV=5bcb38#o zJT`fI8$%}pBYP@(QCmF&QzNQpa`IHlM)nS7Hr7-etX!;*S(Ic&<&`AmUh-1`G2%7` zPF6; zi~Wp{NGgOX-RE-nP0?oxL_dA#=~L5XY5aURZiZ7cGD^=OMn9jFe0imDYI2Hty_N4E 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find(out2 == 0); + +end; diff --git a/mp3/Project_3_Maggioni_Claudio/src/partition.m b/mp3/Project_3_Maggioni_Claudio/src/partition.m new file mode 100755 index 0000000..06319a3 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/partition.m @@ -0,0 +1,39 @@ +function [a,b] = partition(xyz,v) +% PARTITION : Partition points by a plane. +% +% [a,b] = partition(xyz,v): +% Each row of xyz is an input point in d-space. +% Input v is a vector, giving a direction normal to the partitioning plane. +% +% The output is two vectors of integers, +% the indices of the points on each side of the plane. +% Points on the plane are assigned to balance the cut. + + +[n,d] = size(xyz); + +v = v(:); % Make v a column vector + +if length(v) ~= d + error('v must be a d-vector') +end; + +dotprod = xyz * v; +split = median(dotprod); +a = find(dotprod < split); +b = find(dotprod > split); +c = find(dotprod == split); +nc = length(c); +if nc + na = length(a); + nca = max([ceil(n/2)-na, 0]); + nca = min(nca,nc); + if nca > 0 + a = [a; c(1:nca)]; + end; + if nca < nc + b = [b; c(nca+1:nc)]; + end; +end; +a = a'; +b = b'; diff --git a/mp3/Project_3_Maggioni_Claudio/src/rec_bisection.m b/mp3/Project_3_Maggioni_Claudio/src/rec_bisection.m new file mode 100755 index 0000000..83c29f1 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/rec_bisection.m @@ -0,0 +1,61 @@ +function [map,sepij,sepA] = rec_bisection(method,levels,A,varargin) +% DICE Separate a graph recursively. +% +% [map,sepij,sepA] = DICE(method,levels,A,arg...) partitions the +% mesh or graph A recursively by a specified method. 'method' is the name +% of the 2-way edge separator function to call. levels is the number of +% levels of partitioning. A is the adjacency matrix of the graph. arg2, +% arg3, arg4 are optional additional arguments to the 2-way function. arg2 +% is special: If it has the same number of rows as A, then the recursive +% calls use the appropriate subset of arg2 as well as of A. This is useful +% if the partitioner uses xy coords. map is a vector of integers from 0 to +% 2^levels-1, indexed by vertex, giving the partition number for each +% vertex. sepij is a list of separating edges; each row is an edge [i j]. +% sepA is the adjacency matrix of the graph less the separating edges. +% +% For example, dice('geopart',7,A,xy,0,ntries) splits A into 2^7=128 +% parts, using a call at each stage that looks like +% geopart(A,xy,0,ntries). + +minpoints = 8; % Don't separate pieces smaller than this. + +nargcall = nargin - 2; + +n = size(A,1); + +if n < minpoints || levels < 1 + map = zeros(1,n); +else + % Call the partitioner to split the graph, giving one part. + [p1,p2] = feval(method, A, varargin{:}); + + % Call recursively. + vararginp1 = varargin; + vararginp2 = varargin; + if nargcall >= 2 + if size(varargin{1},1) == n + vararginp1{1} = varargin{1}(p1,:); + vararginp2{1} = varargin{1}(p2,:); + end + end + + mapa = rec_bisection(method,levels-1,A(p1,p1),vararginp1{:}); + mapb = rec_bisection(method,levels-1,A(p2,p2),vararginp2{:}); + + % Set up the whole map. + map = zeros(1,n); + mapb = mapb + max(mapa) + 1; + map(p1) = mapa; + map(p2) = mapb; +end + +% Set up the separating edge list and separated graph. +if nargout >= 2 + [i,j] = find(A); + f = find(map(i) > map(j)); + sepij = [i(f) j(f)]; + f = find(map(i) == map(j)); + sepA = sparse(i(f), j(f), 1, n, n); +end + +end \ No newline at end of file diff --git a/mp3/Project_3_Maggioni_Claudio/src/setfilter.m b/mp3/Project_3_Maggioni_Claudio/src/setfilter.m new file mode 100755 index 0000000..e740816 --- /dev/null +++ b/mp3/Project_3_Maggioni_Claudio/src/setfilter.m @@ -0,0 +1,15 @@ +function B = setfilter(A) +% SETFILTER : Sort and remove duplicates. +% +% B = setfilter(A) returns a row vector with one occurrence +% of each different element of A, in sorted order. + + +if length(A) == 0 + B = []; + return; +end; + +B = sort(A(:)); +B(find(diff(B)==0)) = []; +B = B.'; diff --git a/mp3/assignment.sty b/mp3/assignment.sty new file mode 100644 index 0000000..54ff904 --- /dev/null +++ b/mp3/assignment.sty @@ -0,0 +1,95 @@ +\usepackage{ifthen} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usepackage{graphicx} +\usepackage{hyperref} + +\pagestyle{plain} +\voffset -5mm +\oddsidemargin 0mm +\evensidemargin -11mm +\marginparwidth 2cm +\marginparsep 0pt +\topmargin 0mm +\headheight 0pt +\headsep 0pt +\topskip 0pt +\textheight 255mm +\textwidth 165mm + +\newcommand{\duedate} {} +\newcommand{\setduedate}[1]{% +\renewcommand\duedate {Due date:~ #1}} +\newcommand\isassignment {false} +\newcommand{\setassignment}{\renewcommand\isassignment {true}} +\newcommand{\ifassignment}[1]{\ifthenelse{\boolean{\isassignment}}{#1}{}} +\newcommand{\ifnotassignment}[1]{\ifthenelse{\boolean{\isassignment}}{}{#1}} + +\newcommand{\assignmentpolicy}{ +\begin{table}[h] +\begin{center} +\scalebox{0.8} {% +\begin{tabular}{|p{0.02cm}p{16cm}|} +\hline +&\\ +\multicolumn{2}{|c|}{\Large\textbf{Numerical Computing 2020 --- Submission Instructions}}\\ +\multicolumn{2}{|c|}{\large\textbf{(Please, notice that following instructions are mandatory: }}\\ +\multicolumn{2}{|c|}{\large\textbf{submissions that don't comply with, won't be considered)}}\\ +&\\ +\textbullet & Assignments must be submitted to \href{https://www.icorsi.ch/course/view.php?id=10018}{iCorsi} (i.e. in electronic format).\\ +\textbullet & Provide both executable package and sources (e.g. C/C++ files, Matlab). +If you are using libraries, please add them in the file. Sources must be organized in directories called:\\ +\multicolumn{2}{|c|}{\textit{Project\_number\_lastname\_firstname}}\\ +& and the file must be called:\\ +\multicolumn{2}{|c|}{\textit{project\_number\_lastname\_firstname.zip}}\\ +\multicolumn{2}{|c|}{\textit{project\_number\_lastname\_firstname.pdf}}\\ +\textbullet & The TAs will grade your project by reviewing your project write-up, and looking at the implementation + you attempted, and benchmarking your code's performance.\\ + +\textbullet & You are allowed to discuss all questions with anyone you like; however: (i) your submission must list anyone you discussed problems with and (ii) you must write up your submission independently.\\ +\hline +\end{tabular} +} +\end{center} +\end{table} +} +\newcommand{\punkte}[1]{\hspace{1ex}\emph{\mdseries\hfill(#1~\ifcase#1{Points}\or{Points}\else{Points}\fi)}} + + +\newcommand\serieheader[6]{ +\thispagestyle{empty}% +\begin{flushleft} +\includegraphics[width=0.4\textwidth]{usi_inf} +\end{flushleft} + \noindent% + {\large\ignorespaces{\textbf{#1}}\hspace{\fill}\ignorespaces{ \textbf{#2}}}\\ \\% + {\large\ignorespaces #3 \hspace{\fill}\ignorespaces #4}\\ + \noindent% + \bigskip + \hrule\par\bigskip\noindent% + \bigskip {\ignorespaces {\Large{\textbf{#5}}} + \hspace{\fill}\ignorespaces \large \ifthenelse{\boolean{\isassignment}}{\duedate}{#6}} + \hrule\par\bigskip\noindent% \linebreak + } + +\makeatletter +\def\enumerateMod{\ifnum \@enumdepth >3 \@toodeep\else + \advance\@enumdepth \@ne + \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list + {\csname label\@enumctr\endcsname}{\usecounter + {\@enumctr}%%%? the following differs from "enumerate" + \topsep0pt% + \partopsep0pt% + \itemsep0pt% + \def\makelabel##1{\hss\llap{##1}}}\fi} +\let\endenumerateMod =\endlist +\makeatother + + + + +\usepackage{textcomp} + + + + diff --git a/mp3/ex3_eppstein.pdf b/mp3/ex3_eppstein.pdf new file mode 100644 index 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+\documentclass[unicode,11pt,a4paper,oneside,numbers=endperiod,openany]{scrartcl} +\usepackage{graphicx} +\usepackage{subcaption} +\usepackage{amsmath} +\input{assignment.sty} +\usepackage{pgfplots} +\pgfplotsset{compat=newest} +\usetikzlibrary{plotmarks} +\usetikzlibrary{arrows.meta} +\usepgfplotslibrary{patchplots} +\usepackage{grffile} +\usepackage{amsmath} +\usepackage{subcaption} + +\hyphenation{PageRank} +\hyphenation{PageRanks} + +\begin{document} + + +\setassignment +\setduedate{Wednesday, 4 November 2020, 11:55 PM} + +\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 3}{} +\newline + +\assignmentpolicy + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Install METIS 5.0.2, and the corresponding Matlab mex interface} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Implement various graph partitioning algorithms \punkte{60}} +I summarize the various benchmark results in table \ref{table:bisection}. Please note that this table +can be generated at will with the script \texttt{ex2\_bisection\_table.m}. + +\section{Visualize the Fiedler eigenvector\punkte{10}} + +In figure \ref{fig:run1} there are graph outputs respectively from \textit{Grid9}, \textit{Small}, and \textit{Eppstein}. + +\begin{figure}[h] +\begin{subfigure}{0.5\textwidth} +\centering +\includegraphics[trim=50 200 50 200,clip,width=\textwidth]{ex3_grid9.pdf} + \caption{Plot for \textit{Grid9}} +\end{subfigure} +\begin{subfigure}{0.5\textwidth} +\centering +\includegraphics[trim=50 200 50 200,clip,width=\textwidth]{ex3_small.pdf} + \caption{Plot for \textit{Small}} +\end{subfigure} +\begin{subfigure}{0.5\textwidth} +\includegraphics[trim=50 200 50 200,clip,width=\textwidth]{ex3_eppstein.pdf} + \caption{Plot for \textit{Eppstein}} +\end{subfigure} +\caption{Graph outputs for the 3 adjacency matrices.} +\label{fig:run1} +\end{figure} + +\section{Recursively bisecting meshes \punkte{20}} + +Summarize your results in table \ref{table:Rec_bisection}. + +\section{Compare recursive bisection to direct $k$-way partitioning\punkte{10}} + +Summarize your results in table \ref{table:Compare_Metis}. + +\begin{table}[h] + \caption{Bisection results} + \centering + \begin{tabular}{|l|r|r|r|r|} \hline\hline + Mesh & Coordinate & Metis 5.0.2 & Spectral & Inertial \\ \hline + grid5rect(10,100)& 10 & 10 & 10 & 10 \\ + grid5rect(100,10)& 10 & 10 & 10 & 10 \\ + grid5recRotate(100,10,-45)& 18 & 10 & 10 & 10 \\ + gridt(40) & 58 & 58 & 58 & 58 \\ + grid9(30) & 88 & 92 & 104 & 88 \\ + Smallmesh & 25 & 13 & 12 & 30 \\ + Tapir & 55 & 34 & 18 & 49 \\ + Eppstein & 42 & 48 & 45 & 45 \\ \hline \hline + \end{tabular} + \label{table:bisection} +\end{table} + + +\begin{table}[h] + \caption{Edge-cut results for recursive bi-partitioning.} + \centering + \begin{tabular}{|l|r|r|r|r|r|} \hline\hline + Case & Spectral & Metis 5.0.2 & Coordinate & Inertial \\ \hline + mesh3e1 & & & & \\ + airfoil1 & & & & \\ + 3elt & & & & \\ + barth4 & & & & \\ + crack & & & & \\ \hline \hline + \end{tabular} + \label{table:Rec_bisection} +\end{table} + + +\begin{table}[h] +\caption{Comparing the number of cut edges for recursive bisection and direct multiway partitioning in Metis 5.0.2.} +\centering +\begin{tabular}{|l|r|r|r|r|} \hline\hline +Partitions & crack & airfoil1 \\ \hline + 16 & & \\ + 32 & & \\ \hline \hline +\end{tabular} +\label{table:Compare_Metis} +\end{table} + + + +\end{document} diff --git a/mp3/usi_inf.pdf b/mp3/usi_inf.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b89da51e65359ae5afbaa96a79ad018fb0c2b8f7 GIT binary patch literal 34678 zcmbrl2RK~a)&M+eLXhY|qLUzqE^2h68!e0yWiXggM<-SEVDuin6K%AJBzh2x7A=YB zHBl1;{~3An-uHgrz2E;o&wq|{W}UUy-mC7t*V$*|(oDoBKY`p-ORS=B0 z%_@1`JRpvso`E!{lMv%Ib;+Wx1fZ_SCErRr1 zTxC~MeNvq1{-XTLh&Uo#n<7Gg%!<#R0={7tG=%wj;WP$Z0tT%4S3QZpr~u3S z%Og0G{1NcCNB%Mzsh|PM#_QaV=M9kx>R2NHrvi2)6;y@0c)>gXhY2bx87jkUUF~2v zubIH@00jYcNCnkla0f@gEqBCl&UA70FmSWBg|Q0Z^aRYM2>0^TgLx>qBHUb&IR8nK 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