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
of Volume 14
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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