This algorithm solves the assignment problem for rectangular matrices by finding an optimal one-to-one matching between rows and columns that minimizes the total cost. It begins by preprocessing the matrix to subtract the minimum value from each row and column. It then performs the Hungarian algorithm, labeling matched and unmatched rows and columns, to find the optimal assignment. The time complexity is O(n^3) for an n by n matrix.

1. Submittal of an algorithm for consideration for publication in
L.D. Fosdick
Communications of the A C M implies unrestricted use o f the
Algorithms Editor algorithm within a computer is permissible.
Algorithm 415 begin
switch switch := N E X T , L1, N E X T 1, M A R K ;
real min ;
integer array c[l:n], cb[l:m], lambda[l:m], mu[l:n], r[l:n],
Algorithm for the Assignment y[l:m];
integer cbl, cl, clO, i, j, k, 1, rl, rs, sw, imin, imax, flag;
Problem (Rectangular total := 0; imin := m i i m a x := n i
if n > m then go to JA;
Matrices) [H] imin : = n; i m a x := m;
for i : = 1 step 1 until n do
F. Bourgeois, a n d J . C . L a s s a l l e [ R e c d . 21 S e p t . 1970 begin
and 20 May 1971] C E R N , Geneva, Switzerland min := a[i, 1];
f o r j : = 2 step 1 until m do i f a [ i , j ] < rain then rain := a[i,j];
f o r j : = 1 step 1 until m do a[i,j] := a[i,j] -- mitt;
total := total d- mini
end;
i f m > n t h e n g o t o JB;
JA:
for j : = 1 step 1 until m do
begin
Key Words and Phrases: operations research, optimization rain : = a[1,j];
theory, assignment problem, rectangular matrices for i : = 2 step 1 until n do i f a [ i , j ] < min then rain := a[i,j];
CR Categories: 5.39, 5.40 for i : = 1 step 1 until n do a[i, j] : = a[i, j] -- min i
total := total ~ min i
end;
Description JB:
This ~lgorithm is a companion to [3] where the theoretical for i : = 1 step 1 until n do x[i] : = 0;
background is described. f o r j : = 1 step 1 u n t i l m do y[j] := 0;
for i : = 1 step 1 until n do
References begin
1. Silver, R. A n Algorithm for the assignment problem. C o m m . f o r j : = 1 step 1 until m do
A C M 3 (Nov. 1960), 605-606. begin
2. Munkres, J. Algorithms for the assignment and transportation ifa[i,j] ~0 Vx[i] ~0Vy[j] ~0thengotoJ1;
problems. J. S I A M 5 (Mar. 1957), 32-38. x[i] := j; y[jl := i;
3. Bourgeois, F. and Lassalle, J. C. A n extension o f the Munkres J1 :
algorithm for the assignment problem to rectangular matrices. end i
C o m m . A C M 15 (Dec. 1971), 802-804. end;
comment Start labeling;
Algorithm START:
procedure assignment (a, n, m, x, total) ; f l a g := n; rl := cl := O; rs := 1;
value a, n, m; integer n, m; for i : = 1 step 1 until n do
real total; array a; integer array x i begin
comment: a [i, j] is an n X m matrix, x [1 ], x[2], . . . , x[n] are assigned mu[i] : = 0;
integer values which minimize total := s u m ( i := l(1)n) of the if x[i] ~ 0 then go to 11;
elements a[i, x[i]]. If m > n the x[i] are distinct and are a subset rl := rl d- lir[rl] := i;mu[i] := --1;
o f the integers 1, 2 . . . . , m. If m = n the x[i] are a permutation f l a g : = f l a g -- 1;
of the integers 1, 2 . . . . . n. If m < n the set o f x[i] consists o f I1:
some permutation o f the integers 1, 2, . . . , m interspersed with end;
n - m zeros. The permutation and the positions of the zeros are if f l a g = i m & then go to F I N I ;
chosen in such a way as to minimize the above sum with the f o r j : = 1 step 1 until m do lambda[j] : = 0;
convention that a[i, o] is to be taken equal to zero. imin = comment Label and scan;
min(n, m) and i m a x = m a x ( n , m) must be such that: imin > O, LABEL:
i m a x > 1. i := r[rs]; rs := rs + 1;
This procedure is based on that o f Silver [1] which uses the f o r j : = 1 step 1 until m do
assignment algorithm of Munkres [2]. Silver's procedure has begin
been extended to handle the case n ~ m; if a[i,j] ~ 0 V lambda[j] ~ 0 then go to J2;
iambda [j] : = i; cl := cl -t- 1; c[cl] := j;
if y[j] = 0 then go to M A R K ;
Copyright O 1971, Association for Computing Machinery, Inc. rl := rl -b 1; r[rl] : = y[y]; mu[y[j]] := i;
General permission to republish, but not for profit, an algorithm J2:
is granted, provided that reference is made to this publication, to end;
its date of issue, and to the fact that reprinting privileges were if rs =< d then go to L A B E L i
granted by permission o f the Association for Computing Machinery. comment Renormalize;
sw : = 1;cl0 : = cl; cbl : = 0;
f o r j : = 1 step 1 until m do
begin
iflambda[j] ~ 0 then go to J3;
cbl := cbl -b 1; cb[cbl] := j;
805 Communications December 1971
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the A C M N u m b e r 12

2. J3: if y[cb[l]] = 0 then
end; begin
min := a[r[1], cb[1]]; j : = cb[l]; sw : = 2; g o t o L 1 ;
for k : = 1 step 1 until rl do end;
begin cl:=el+l; c[cl]:=cb[l]; r l : = r l + l ;
for I : = 1 step I until cbl do r[rl] : = y[cb[l]];
if air[k], cb[l]] < rain then rain := a[r[k], cb[l]]; LI:
end; end;
total := total + rain X (rl+cbl--imax); 13:
for i : = 1 step 1 until n do end;
begin go to switch[sw + 2];
ifmu[i] ~ 0 then go to 12; NEXT 1:
ifcl0 < 1 then go to 13; if clO = cl then go to L A B E L ;
for l : = 1 step 1 until clO do a[i, c[I]] : = a[i, c[l]] -t- min; for i : = c/0 + 1 step 1 until c / d o mu[y[c[i]]] : = c[i];
go to 13; go to L A B E L ;
/2: comment Mark new column and permute;
for I : = 1 step 1 until cbl do MARK:
begin y[j] : = i : = lambda[j];
a[i, cb[l]] := a[i, cb[l]] - rain; if x[i] = 0 then begin x[i] : = j; go to S T A R T ;
go to switch[sw]; end;
NEXT: k:=j; j:=x[i]; x [ i ] : = k; go to M A R K ;
if a[i, cb[l]] ~ 0 V lambda[cb[l]] ~- 0 then go to L1 ; FINI:
lambda[cb[l]] := i; end
Algorithm 416 a number o f arguments equal to ord[i]. In this case dx[i] should
contain the difference between the argument o f highest index o f
f[i] and that o f f [ i - 1].
U p o n execution o f I N T P the coefficients of the desired poly-
Rapid Computation of nomial are stored in c in such a manner that the coefficient in
front of the power t ~-1 is contained in c[i]. Other parameters are
Coefficients of not changed. Caution: The given data must be such that it is
possible to construct N e w t o n ' s interpolation formula with
Interpolation Formulas tEl] divided differences from them. We must also have ord[1] = 1.
Observe that if derivatives o f f are given the corresponding
Sven-~kke Gustafson* [ R e c d . 21 A u g . 1969] divided differences with confluent arguments must be evaluated
Computer Science Department, Stanford University, and given as input data.
Examples o f use o f INTP:
Stanford, CA 94305 Example 1. Determine the polynomial o f degree less than n which
interpolates a function f at n distinct points x~, i = 1, 2, . . . , n.
Input d a t a : d x [ i ] = x l , f [ i ] = J~, ord[i] = 1, i = 1, 2 . . . . , n.
Example 2. Let x~, x2, x3, x4 be four given points. We know
Key Words and Phrases: divided differences, Newton's .A, ~ . ~, f2.a, and.A • Determine the polynomial of degree 3 which
interpolation formula reproduces these quantities. Input data: n = 4,
CR Category: 5.13 dx[1] =xx ord[1] = I f[l] =A
dx[2] =x2--xl ord[2] = 2 f[2] =./~.2
dx[3] =x~--x2 ord[3] = 2 f[3] =J~.3
Description dx[4] = x, ord[4] = 1 f[4] = A
This algorithm is a companion to [1] where the theoretical Example 3. The same problem when we are given f ( - 1 ) , f ' ( - - 1 ) ,
background is described
f " ( - - I ) , and f(1). Input data: n = 4,
References dx[1] = -1 ord[1] = 1 f[1] = f ( - 1 )
dx[2] = 0 ord[2] = 2 f[21 = f ' ( - - 1)
1. Gustafson, Sven-/~ke. Rapid computation of interpolation dx[3] = 0 ord[3] = 3 f[31 = 0 . 5 . / " ( - - 1 )
formulae and mechanical quadrature rules. Comm. A C M 14 dx[4] = 1 ord[4] = 1 f[4] = f(1)
(Dec. 1971), 797-801.
For further details see [I ];
integer i,j, k; real ai, h, d, xx;
Algorithm real array arg [1 : n];
procedure I N T P (dx, f, c, ord, n) ; comment Initiate phase DI;
value n; real array dx, f, c; for i : = 1 step 1 until n do
integer array ord; integer n; arg[i] : = if ord[i] = 1 then dx[i] else dx[i] q- arg[i-- 1];
begin comment Phase DI;
comment 1NTP determines the coefficients of the polynomial of de- for i : = 2 step 1 until n do
gree less than n which reproduces given function values and begin
divided differences. The parameters of I N T P are: j : = ord[i];
i f j = 1 then go to divde;
idenlifier type comment d : = f[i];
n integer for k : = i step -- 1 until i -- j + 2 dof[k] : = f [ k - - 1];
ord integer array Array bounds [1 :n] f [ i - - j + l ] : = d;
dx, f, c real array Array bounds [1 :n] h : = dx[i]; ai : = arg[i];
n is the number o f coefficients o f the interpolating polynomial.
ord gives the character o f the input data: if ord[i] = 1 then x[i]
should be an argument and f[i] the corresponding function value. * Present Address: Inst. F. lnformations Behandling (Numeisk an-
But if ord[i] > 1 thenf[i] should contain a divided difference with alys), K T H , 10044 Stockholm, Sweden.
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of Volume 14
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