## File: assign_lisa.w

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This demonstration program takes a matrix of numbers constructed by the {\sc GB\_\,LISA} module and chooses at most one number from each row and column in such a way as to maximize the sum of the numbers chosen. It also reports the number of mems'' (memory references) expended during its computations, so that the algorithm it uses can be compared with alternative procedures. The matrix has $m$ rows and $n$ columns. If $m\le n$, one number will be chosen in each row; if $m\ge n$, one number will be chosen in each column. The numbers in the matrix are brightness levels (pixel values) in a digitized version of the Mona Lisa. Of course the author does not pretend that the location of highlights'' in da Vinci's painting, one per row and one per column, has any application to art appreciation. However, this program does seem to have pedagogic value, because the relation between pixel values and shades of gray allows us to visualize the data underlying this special case of the assignment problem; ordinary matrices of numeric data are much harder to perceive. The nonrandom nature of pixels in a work of art might also have similarities to the organic'' properties of data in real-world applications. This program is optionally able to produce an encapsulated PostScript file from which the solution can be displayed graphically, with halftone shading. @ As explained in {\sc GB\_\,LISA}, the subroutine call |lisa(m,n,d,m0,m1,n0,n1,d0,d1,@[@t\\{area}@>@])| constructs an $m\times n$ matrix of integers between $0$ and~$d$, inclusive, based on the brightness levels in a rectangular region of a digitized Mona Lisa, where |m0|, |m1|, |n0|, and |n1| define that region. The raw data is obtained as a sum of |(m1-m0)(n1-n0)| pixel values between $0$ and~$255$, then scaled in such a way that sums |<=d0| are mapped to zero, sums |>=d1| are mapped to~$d$, and intermediate sums are mapped linearly to intermediate values. Default values |m1=360|, |n1=250|, |m=m1-m0|, |n=n1-n0|, |d=255|, and |d1=255(m1-m0)(n1-n0)| are substituted if any of the parameters |m|, |n|, |d|, |m1|, |n1|, or |d1| are zero. The user can specify the nine parameters |(m,n,d,m0,m1,n0,n1,d0,d1)| on the command line, at least in a \UNIX/ implementation, thereby obtaining a variety of special effects; the relevant command-line options are \.{m=}\, \.{m0=}\, and so on, with no spaces before or after the \.= signs that separate parameter names from parameter values. Additional options are also provided: \.{-s} (use only Mona Lisa's $16\times32$ smile''); \.{-e} (use only her $20\times50$ eyes); \.{-c} (complement black/white); \.{-p} (print the matrix and solution); \.{-P} (produce a PostScript file \.{lisa.eps} for graphic output); \.{-h} (use a heuristic that applies only when $m=n$); and \.{-v} or \.{-V} (print verbose or Very verbose commentary about the algorithm's performance). @^UNIX dependencies@> Here is the overall layout of this \CEE/ program: @p #include "gb_graph.h" /* the GraphBase data structures */ #include "gb_lisa.h" /* the |lisa| routine */ @h@# @@; main(argc,argv) int argc; /* the number of command-line arguments */ char *argv[]; /* an array of strings containing those arguments */ {@+@@;@# @; mtx=lisa(m,n,d,m0,m1,n0,n1,d0,d1,working_storage); if (mtx==NULL) { fprintf(stderr,"Sorry, can't create the matrix! (error code %ld)\n", panic_code); return -1; } printf("Assignment problem for %s%s\n",lisa_id,(compl?", complemented":"")); sscanf(lisa_id,"lisa(%lu,%lu,%lu",&m,&n,&d); /* adjust for defaults */ if (m!=n) heur=0; if (printing) @; if (PostScript) @; mems=0; @; if (printing) @; if (PostScript) @; printf("Solved in %ld mems%s.\n",mems, (heur?" with square-matrix heuristic":"")); return 0; /* normal exit */ } @ @= Area working_storage; /* where to put the input data and auxiliary arrays */ long *mtx; /* input data for the assignment problem */ long mems; /* the number of memory references counted while solving the problem */ @ The following local variables are related to the command-line options: @= unsigned long m=0,n=0; /* number of rows and columns desired */ unsigned long d=0; /* number of pixel values desired, minus~1 */ unsigned long m0=0,m1=0; /* input will be from rows $[|m0|\,.\,.\,|m1|)$ */ unsigned long n0=0,n1=0; /* and from columns $[|n0|\,.\,.\,|n1|)$ */ unsigned long d0=0,d1=0; /* lower and upper threshold of raw pixel scores */ long compl=0; /* should the input values be complemented? */ long heur=0; /* should the square-matrix heuristic be used? */ long printing=0; /* should the input matrix and solution be printed? */ long PostScript=0; /* should an encapsulated PostScript file be produced? */ @ @= while (--argc) { @^UNIX dependencies@> if (sscanf(argv[argc],"m=%lu",&m)==1) ; else if (sscanf(argv[argc],"n=%lu",&n)==1) ; else if (sscanf(argv[argc],"d=%lu",&d)==1) ; else if (sscanf(argv[argc],"m0=%lu",&m0)==1) ; else if (sscanf(argv[argc],"m1=%lu",&m1)==1) ; else if (sscanf(argv[argc],"n0=%lu",&n0)==1) ; else if (sscanf(argv[argc],"n1=%lu",&n1)==1) ; else if (sscanf(argv[argc],"d0=%lu",&d0)==1) ; else if (sscanf(argv[argc],"d1=%lu",&d1)==1) ; else if (strcmp(argv[argc],"-s")==0) { smile; /* sets |m0|, |m1|, |n0|, |n1| */ d1=100000; /* makes the pixels brighter */ } else if (strcmp(argv[argc],"-e")==0) { eyes; d1=200000; } else if (strcmp(argv[argc],"-c")==0) compl=1; else if (strcmp(argv[argc],"-h")==0) heur=1; else if (strcmp(argv[argc],"-v")==0) verbose=1; else if (strcmp(argv[argc],"-V")==0) verbose=2; /* terrifically verbose */ else if (strcmp(argv[argc],"-p")==0) printing=1; else if (strcmp(argv[argc],"-P")==0) PostScript=1; else { fprintf(stderr, "Usage: %s [param=value] [-s] [-c] [-h] [-v] [-p] [-P]\n",argv[0]); return -2; } } @ @= for (k=0;k @:Konig}{K\H{o}nig, D\'enes@> [{\sl Matematikai \'es Fizikai Lapok\/ \bf38} (1931), 16--28, 116--119] that the maximum number of independent zeroes in a matrix is equal to the minimum number of rows and/or columns that are needed to cover'' every zero. In other words, if we can find $p$ independent zeroes but not~$p+1$, then there is a way to choose $p$ lines in such a way that every zero of the matrix is included in at least one of the chosen lines, where a line'' is either a row or a column. Their proof was constructive, and it leads to a useful computer algorithm. Given a set of $p$ independent zeroes of a matrix, let us write $r_k\ddash c_l$ or $c_l\ddash r_k$ and say that $r_k$ is matched with $c_l$ if $a_{kl}$ is one of these $p$ special zeroes, while we continue to write $r_k\dash c_l$ or $c_l\dash r_k$ if $a_{kl}$ is one of the nonspecial zeroes. A given set of $p$ special zeroes defines a choice of $p$ lines in the following way: Column~$c$ is chosen if and only if it is reachable by a path of the form $$r^{(0)}\dash c^{(1)}\ddash r^{(1)}\dash c^{(2)}\ddash\cdots \dash c^{(q)}\ddash r^{(q)}\,,\eqno(*)$$ where $r^{(0)}$ is unmatched, $q\ge1$, and $c=c^{(q)}$. Row~$r$ is chosen if and only if it is matched with a column that is not chosen. Thus exactly $p$ lines are chosen. We can now prove that the chosen lines cover all the zeroes, unless there is a way to find $p+1$ independent zeroes. For if $c\ddash r$, either $c$ or $r$ has been chosen. And if $c\dash r$, one of the following cases must arise. (1)~If $r$ and~$c$ are both unmatched, we can increase~$p$ by matching them to each other. (2)~If $r$ is unmatched and $c\ddash r'$, then $c$ has been chosen, so the zero has been covered. (3)~If $r$ is matched to $c'\ne c$, then either $r$ has been chosen or $c'$ has been chosen. In the latter case, there is a path of the form $$r^{(0)}\dash c^{(1)}\ddash r^{(1)}\dash c^{(2)}\ddash\cdots\ddash r^{(q-1)}\dash c'\ddash r\dash c\,,$$ where $r^{(0)}$ is unmatched and $q\ge1$. If $c$ is matched, it has therefore been chosen; otherwise we can increase $p$ by redefining the matching to include $$r^{(0)}\ddash c^{(1)}\dash r^{(1)}\ddash c^{(2)}\dash\cdots\dash r^{(q-1)}\ddash c'\dash r\ddash c\,.$$ @ Now suppose $A$ is a {\sl nonnegative\/} matrix, of size $n\times n$. Cover the zeroes of~$A$ with a minimum number of lines, $p$, using the algorithm of Egerv\'ary and K\H{o}nig. If $p0$. Then we can subtract $\delta$ from each unchosen row and add $\delta$ to each chosen column. The net effect is to subtract~$\delta$ from all uncovered elements and to add~$\delta$ to all doubly covered elements, while leaving all singly covered elements unchanged. This transformation causes a new zero to appear, while preserving $p$ independent zeroes of the previous matrix (since they were each \vadjust{\goodbreak}% covered only once). If we repeat the Egerv\'ary-K\H{o}nig construction with the same $p$ independent zeroes, we find that either $p$~is no longer maximum or at least one more column has been chosen. (The new zero $r\dash c$ occurs in a row~$r$ that was either unmatched or matched to a previously chosen column, because row~$r$ was not chosen.) Therefore if we repeat the process, we must eventually be able to increase $p$ until finally $p=n$. This will solve the assignment problem, proving the remarkable claim made earlier. @ If the given matrix $A$ has $m$ rows and $n>m$ columns, we can extend it artificially until it is square, by setting $a_{kl}=0$ for all $m\le k= { for (l=0; l= register long k; /* the current row of interest */ register long l; /* the current column of interest */ register long j; /* another interesting column */ register long s; /* the current matrix element of interest */ @* Algorithmic details. The algorithm sketched above is quite simple, except that we did not discuss how to determine the chosen columns~$c^{(q)}$that are reachable by paths of the stated form$(*)$. It is easy to find all such columns by constructing an unordered forest whose nodes are rows, beginning with all unmatched rows~$r^{(0)}$and adding a row~$r$for which$c\ddash r$when$c$is adjacent to a row already in the forest. Our data structure, which is based on suggestions of Papadimitriou and @^Papadimitriou, Christos Harilaos@> @^Steiglitz, Kenneth@> Steiglitz [{\sl Combinatorial Optimization\/} (Prentice-Hall, 1982),$\mathchar"278$11.1], will use several arrays. If row~$r$is matched with column~$c$, we will have |col_mate[r]=c| and |row_mate[c]=r|; if row~$r$is unmatched, |col_mate[r]| will be |-1|, and if column~$c$is unmatched, |row_mate[c]| will be |-1|. If column~$c$has a mate and is also reachable in a path of the form$(*)$, we will have$|parent_row|[c]=r'$for some$r'$in the forest. Otherwise column~$c$is not chosen, and we will have |parent_row[c]=-1|. The rows in the current forest will be called |unchosen_row[0]| through |unchosen_row[t-1]|, where |t| is the current total number of nodes. The amount$\sigma_k$subtracted from row$k$is called |row_dec[k]|; the amount$\tau_{\,l}$added to column~$l$is called |col_inc[l]|. To compute the minimum uncovered element efficiently, we maintain a quantity called |slack[l]|, which represents the minimum uncovered element in each column. More precisely, if column~$l$is not chosen, |slack[l]| is the minimum of$a_{kl} -\sigma_k+\tau_{\,l}$for$k\in\{|unchosen_row|[0],\ldots,\allowbreak |unchosen_row|[q-1]\}$, where$q\le t$is the number of rows in the forest that we have explored so far. We also remember |slack_row[l]|, the number of a row where the stated minimum occurs. Column$l$is chosen if and only if |parent_row[l]>=0|. We will arrange things so that we also have |slack[l]=0| in every chosen column. @= long* col_mate; /* the column matching a given row, or$-1$*/ long* row_mate; /* the row matching a given column, or$-1$*/ long* parent_row; /* ancestor of a given column's mate, or$-1$*/ long* unchosen_row; /* node in the forest */ long t; /* total number of nodes in the forest */ long q; /* total number of explored nodes in the forest */ long* row_dec; /*$\sigma_k$, the amount subtracted from a given row */ long* col_inc; /*$\tau_{\,l}$, the amount added to a given column */ long* slack; /* minimum uncovered entry seen in a given column */ long* slack_row; /* where the |slack| in a given column can be found */ long unmatched; /* this many rows have yet to be matched */ @ @= col_mate=gb_typed_alloc(m,long,working_storage); row_mate=gb_typed_alloc(n,long,working_storage); parent_row=gb_typed_alloc(n,long,working_storage); unchosen_row=gb_typed_alloc(m,long,working_storage); row_dec=gb_typed_alloc(m,long,working_storage); col_inc=gb_typed_alloc(n,long,working_storage); slack=gb_typed_alloc(n,long,working_storage); slack_row=gb_typed_alloc(n,long,working_storage); if (gb_trouble_code) { fprintf(stderr,"Sorry, out of memory!\n"); return -3; } @ The algorithm operates in stages, where each stage terminates when we are able to increase the number of matched elements. The first stage is different from the others; it simply goes through the matrix and looks for zeroes, matching as many rows and columns as it can. This stage also initializes table entries that will be useful in later stages. @d INF 0x7fffffff /* infinity (or darn near) */ @= t=0; /* the forest starts out empty */ for (l=0; l1) printf(" matching col %ld==row %ld\n",l,k); goto row_done; } o,col_mate[k]=-1; if (verbose>1) printf(" node %ld: unmatched row %ld\n",t,k); o,unchosen_row[t++]=k; row_done:; } @ If a subsequent stage has not succeeded in matching every row, we prepare for a new stage by reinitializing the forest as follows. @= t=0; for (l=0; l1) printf(" node %ld: unmatched row %ld\n",t,k); o,unchosen_row[t++]=k; } @ Here, then, is the algorithm's overall control structure. There are at most$m$stages, and each stage does$O(mn)$operations, so the total running time is$O(m^2n)$. @= @; if (t==0) goto done; unmatched=t; while(1) { if (verbose) printf(" After %ld mems I've matched %ld rows.\n",mems,m-t); q=0; while(1) { while (q; q++; } @; } breakthru: @; if(--unmatched==0) goto done; @; } done: @; @ @= { o,k=unchosen_row[q]; o,s=row_dec[k]; for (l=0; l1) printf(" node %ld: row %ld==col %ld--row %ld\n", t,row_mate[l],l,k); oo,unchosen_row[t++]=row_mate[l]; }@+else { o,slack[l]=del; o,slack_row[l]=k; } } } } @ At this point, column$l$is unmatched, and row$k$is in the forest. By following parent links in the forest, we can rematch rows and columns so that a previously unmatched row~$r^{(0)}$gets a mate. @= if (verbose) printf(" Breakthrough at node %ld of %ld!\n",q,t); while (1) { o,j=col_mate[k]; o,col_mate[k]=l; o,row_mate[l]=k; if (verbose>1) printf(" rematching col %ld==row %ld\n",l,k); if (j<0) break; o,k=parent_row[j]; l=j; } @ If we get to this point, we have explored the entire forest; none of the unchosen rows has led to a breakthrough. An unchosen column with smallest |slack| will allow us to make further progress. @= s=INF; for (l=0; l; }@+else oo,col_inc[l]+=s; @ There might be several columns tied for smallest slack. If any of them leads to a breakthrough, we are very happy; but we must finish the loop on~|l| before going to |breakthru|, because the |col_inc| variables need to be maintained for the next stage. Within column |l|, there might be several rows that produce the same slack; we have remembered only one of them, |slack_row[l]|. Fortunately, one is sufficient for our purposes. Either we have a breakthrough or we choose column~|l|, regardless of which row or rows led us to consider that column. @= { o,k=slack_row[l]; if (verbose>1) printf(" Decreasing uncovered elements by %ld produces zero at [%ld,%ld]\n", s,k,l); if (o,row_mate[l]<0) { for (j=l+1; j1) printf(" node %ld: row %ld==col %ld--row %ld\n", t,row_mate[l],l,k); oo,unchosen_row[t++]=row_mate[l]; } } @ The code in the present section is redundant, unless cosmic radiation has caused the hardware to malfunction. But there is some reassurance whenever we find that mathematics still appears to be consistent, so the author could not resist writing these few unnecessary lines, which verify that the assignment problem has indeed been solved optimally. (We don't count the mems.) @^discussion of \\{mems}@> @= for (k=0;km) { fprintf(stderr,"Oops, I adjusted too many columns!\n"); return-666; /* can't happen */ } @* Interfacing. A few nitty-gritty details still need to be handled: Our algorithm is not symmetric between rows and columns, and it works only for$m\le n$; so we will transpose the matrix when$m>n\$. Furthermore, our algorithm minimizes, but we actually want it to maximize (except when |compl| is nonzero). Hence, we want to make the following transformations to the data before processing it with the algorithm developed above. @= if (m>n) @@; else transposed=0; @; if (compl==0) for (k=0; k; @; @ @= { if (verbose>1) printf("Temporarily transposing rows and columns...\n"); tmtx=gb_typed_alloc(m*n,long,working_storage); if (tmtx==NULL) { fprintf(stderr,"Sorry, out of memory!\n");@+return -4; } for (k=0; k= long* tmtx; /* the transpose of |mtx| */ long transposed; /* has the data been transposed? */ @ @= { printf("The following entries produce an optimum assignment:\n"); for (k=0; k associated \.{epsf.tex} macros, one can say $$\.{\\epsfxsize=10cm \\epsfbox\{lisa.eps\}}$$ within a \TEX/ document and the illustration will be typeset in a box that is 10 centimeters wide. The conventions of PostScript allow the illustration to be scaled to any size. Best results are probably obtained if each pixel is at least one millimeter wide (about 1/25 inch) when printed. The illustration is formed by first painting'' the input data as a rectangle of pixels, with up to 256 shades of gray. Then the solution pixels are framed in black, with a white trim just inside the black edges to help make the frame visible in already-dark places. The frames are created by painting over the original image; the center of each solution pixel retains its original color. Encapsulated PostScript files have a simple format that is recognized by many software packages and printing devices. We use a subset of PostScript that should be easy to convert to other languages if necessary. @= { eps_file=fopen("lisa.eps","w"); if (!eps_file) { fprintf(stderr,"Sorry, I can't open the file `lisa.eps'!\n"); PostScript=0; }@+else { fprintf(eps_file,"%%!PS-Adobe-3.0 EPSF-3.0\n"); /* \.{1.0} and \.{2.0} also OK */ fprintf(eps_file,"%%%%BoundingBox: -1 -1 %ld %ld\n",n+1,m+1); fprintf(eps_file,"/buffer %ld string def\n",n); fprintf(eps_file,"%ld %ld 8 [%ld 0 0 -%ld 0 %ld]\n",n,m,n,m,m); fprintf(eps_file,"{currentfile buffer readhexstring pop} bind\n"); fprintf(eps_file,"gsave %ld %ld scale image\n",n,m); for (k=0;k; fprintf(eps_file,"grestore\n"); } } @ @= FILE *eps_file; /* file for encapsulated PostScript output */ @ This program need not produce machine-independent output, so we can safely use floating-point arithmetic here. At most 64 characters (32 pixel-bytes) are output on each line. @= {@+register float conv=255.0/(float)d; register long x; for (l=0; l255?255L:x); if ((l&0x1f)==0x1f) fprintf(eps_file,"\n"); } if (n&0x1f) fprintf(eps_file,"\n"); } @ @= { fprintf(eps_file, "/bx {moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath\n"); fprintf(eps_file," gsave .3 setlinewidth 1 setgray clip stroke"); fprintf(eps_file," grestore stroke} bind def\n"); fprintf(eps_file," .1 setlinewidth\n"); for (k=0; k