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346 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS September
An Algorithm for Path Connectionsand Its Applications*
C. Y. LEEt, MEMBER, IRE
Summary-The algorithm described in this paper is the outcome Based on this model, we will consider a class of well-of an endeavor to answer the following question: Is it possible to find defined optimal path problems. A general procedure forprocedures which would enable a computer to solve efficiently path- s t c o p... ~solving this class of problems will then be given in Sec-connection problems inherent in logical drawing, wiring diagram-
t
ming, and optimal route finding? The results are highly encouraging. tion IIIWithin our framework, we are able to solve the following types of Within this class of problems is the shortest-route prob-problems: lem on which there has been earlier definitive work.
1) To find a path between two points so that it crosses the least Algorithms for finding shortest paths have been givennumber of existing paths. by Dantzig,l Ford and Fulkerson,2 Moore3 and Prim.4
2) To find a path between two points so that it avoids as much as The minimal-distance illustrationis (Section V) makepossible preset obstacles such as edges.
3) To find a path between two points so that the path is optimal use of one of Moore's algorithms, which is a specializawith respect to several properties; for example, a path which is tioIn of algorithm A given in Section III. These experi-not only one of those which cross the fewest number of exist- iments were tried out before the abstract model wasing paths, but, among these, is also one of the shortest. completed. Once we have at our disposal the abstract
The minimal-distance solution has been programmed on an IBM model, it came as a pleasant surprise that problems704 computer, and a number of illustrations are presented. The class such as the minimal-crossing and minimal edge-effectof problems solvable by our algorithm is given in a theorem in Section problems, which had appeared difficult to us previously,III. A byproduct of this algorithm is a somewhat remote, but unex- all yielded immediately to algorithm A. The possibilitypected, relation to physical optics. This is discussed in Section VI. I
Of J'oint mlinimization is also a direct consequence of1. INTRODUCTION algorithmn A. A further outcome was the "diffraction"
N processing information conisisting of patterns, patterns. These experiments would not have been at-rather than numbers or symbols, on a digital com- tempted if we had not noticed the patterns obtained inputer, we may wish to know how a computer, with- the minimal-distance experiments.
out sight and hearing, can be made to deal compete ntly II. AN ABSTRACT MODEL AND THE PATHwith situations which appear to require coordination, PROBLEMinsight, and perhaps intuition. It is not our intention toconsider the general problem of pattern detection and A. e-Space, an Abstract Modelrecognition by machines. We will consider rather the Let Cbe a set of elements called cells: C= {cl, C2,following simpler, and therefore perhaps more basic, For each cell ci in C, there is defined a subset of C calledproblem in pattern processing by machines. a 1-neighborhood N(ci) of ci: N(ci) = I cli, C2, * * * X C, -
Let a pattern of some sort be presented to a machine. The following rules hold for 1 neighborhoods:We then want the machine to construct some optimal N1) Every 1-neighborhood has in it exactly n cells,path subject to various constraints imposed by the pat- where n, n > 1, is some predetermined number depend-tern. The problem is to find efficient procedures, which, ing on the specific model involved.if followed by the machine, would lead to an optimal N2) If c'CN(ci), then ciCN(cj). We will call thesolution. function N with domain C and range subsets of C the
Ideally, many situations would fall within this de- 1-neighborhood function.scription. We might present to the machine a map of Together with the 1-neighborhood functions, thereManhattan and ask it to find the shortest-time route are n functions di, d2, - , d", on C to C defined as fol-between, say, the United Nations and Yankee Stadium, lows: If ci is any cell in C and N(ct) = { cli, c2i, *, c,i,using only public transportation. With sufficient care, thenit is possible to make a problem such as this unambigu-ous. In most cases, however, it wvould be too great a dk(ei) =CkiJ k = 1, 2, , n.struggle just to present the problem in a way that iscompletely and consistently stated. For this reason, we l'G. B. Dantzig, "Maximization of a Linear Function of Variableshave decided to present an abstract model in Section II. Subjelct to Flinear Inequalities, " Cowles Commission; 1951.
work," Can. J. Malth., vol. 8, pp. 399-404; 1956.F. F. VMoore, "vShortest path through a maze," in "Annals of the
* Received by the PGEC, December 2, 1960. This material was Computation Laboratory of Harvard University," Harvard Univer-presented as part of the University of Michigan Engineering Summer sity Press, Cambridge, Mass., vol. 30, pp. 285-292; 1959.Session, Ann Arbor, June 19-30, 1961. 4R. C. Prim, "Shortest connection networks and some general-
t Bell Telephone Labs., Inc., Whippany, N. J1. jzations," Belt Sys. Tech. J., vol. 36, pp. 1389-1401; November, 1957.
1961 Lee: An Algorithm for Path Connections and Its Applications 347
That is, dk(ci) is the kth coordinate cell in the 1-neigh- cell c**, find an admissible path pl2 ..r(C*, c**) which isborhood of ci. minimal with respect to (fl, f2, * fr).
Let S be a finite set of symbols: S= s1I s52 * stm}. In the following section we will show an algorithmS is called the alphabet of space e. which solves the path problem P for a certain set of
Let F be a map on C to CXS. That is, for every vectors F.cicC, P(ci) =(c, s), siCS. In general we will writeF(ci) = (ci, s(ci)). Therefore, to every cell ciCC is asso- III. A SEARCH AND TRACE PROCEDUREciated some symbol s(ci) CS. The map F gives then the A. MIIontone Functions and M11onotone Vectorscell-symbol configuration for a particular path problem. Let a C-space (C, S, N, F, M) be given. A function fGenerally speaking, we may keep in mind the analogy on r*(ci, ci) to the set of non-negative integers I is saidthat each function F corresponds to some sort of a street to be monotone if for every path p(ci, ci), we have themap for a particular city or town, and that the path inequalitvproblem is to finid an optimal path fromn one point toanother in that city or town. f(p(ci, ck)) < f(p(c ci)),
Let ci, ci be two distinct cells in C. By a path p(ci, c') where p(ci, Ck) is any subpath of p(ci, ci). A vector F ofis meant a set of cells called a chain: p(ci, ci) monotone functions (fi, f2, f,f is said to be a
0=icc cl, c2, . , =cj asuch that ci+'lCN(ci) for onotone vector.s=0, 1, nm-i. By ir(ci, ci) is meant the set of allpaths p(ci, ci) between cell ci and cell ci. B. An Algorithm
Let 211 be a map called an admission map with domainir(ci, ci) and range the two-element set 0O, 11. Anly path Let c* and c** be the initial and final cells in a C-p(ci, ci) such that M(p(ci, ci)) = 1 is said to be an ad- space (C, S, N, F, M). The main procedure will be given
mIssible path. Otherwise, p(ci, ci) is said to be inadmis- in terms of a number of subprocedures.sible. The set of all admissible paths will be denoted by D) Cell list: A cell list L IS an ordered list containings*(ci, ci) names of cells.The quintuple (C, S, N, F, A) is called a C-space. D2) Cell mass: With each cell in list L will be associ-
ated an r-tuple called a cell mass (mi1, Mi2, - * , mi). Thecell masses are ordered lexicographically. Thus, ifB. The Path Problemm = (mi1, Mi2, , mr), and m' = (M1', m2', me') are
Let F be a vector of r functions (fi, f2, ' fr) where two cell masses, (M1, M2, , mr) < (mi',IM21, mr/)each function ft, Ti= 1, 2, r, is on 7r*(ci, ci), the set if mi = mi', i =0, 1, k, but mk+l <m+l; O < k <r.of admissible paths, to I, the set of noni-negative inte- D3) Chain coordinate: Associated with each cell ingers. A path p'(ci, ci) of wr*(c, ci) is said to be minimal the list L is also a chain coordinate which is one of thewith respect to fi if 1-nieighborhood coordinate functions dk.
f1(p'(c6, Ci)) < f1(p(ci, C)) D4) Auxiliary list Li: An auxiliary cell list L1 is pro-vided for momentary storage of names of cells.
for all p(ci, ci) Cr*(ci, ci). A path pl2(ci, ci) is said to be RI) Procedurefor constructing auxiliary list L1: Let cminimal with respect to (fi, f2) if be a cell in list L. By a path p(c*, ci, c**) we mean a path
(i) pl2(Ci7 Cj) E_ pI(Ci7fC) p(c*, c**) of which p(c*, ci) is a subpath. A cell ci is said(i)p12(c , ci) CP'(c~, Ci) to be admissible if p(c*, ci, c**) is an admissible path
where P1(ci, ci) is the set of all paths in r*(ci, ci), which and if the cell mass for c' has not yet been determined.are minimal with respect tofi; that is, Let {ci} CN(c) be the set of all admissible cells in N(c).
Append to list L1 the set ci1. L1 is constructed by re-P(ci=,ci) PI ci) Mf(p1(ci, ci)) < fi(p(ci, ci)) peating this process for every entry c in L, under the
for all p(ci, ci) C w*(ci, ci) }; condition that a cell should not be listed more than once.L1 is therefore the set of all distinct cells ci such that ci
and is an admissible cell in N(c) for some cell c in the list L.
(ii) f2(p'2(Ci, Ci)) < f2(p(Ci, Ci)) R2) Procedure for assigning cell masses and chain co-ordinates:L,et ac cell in L1. A possible cell mass for c
for all p(c v ci) C P'(c , ci). is determined as follows: For each ciCNV(ci) whose cell
Therefore, p12(ct, ci) iS minimal with respect to (f', f2) mashsbeXeemnd osrc nrtpeif among all paths minimal with respect tofi, p12(c1, cl) (fi(p(c*, ci, ci)), fr(p(c*, ci, c1))).is also minimal with respect to f2. In a similar way, wemay define a path p12 .r(ci ci) which is minimal with Now apply rule R3 below to find a cioCN(ci), for whichrespect to (f1, f2, * ,fr). this r-tuple is a minimum. A possible cell mass for ci isThe path problem we are considering is the following: then the r-tuple (fi(p(c*, cia, Ci)), * * * , fr(P(6*, cia, c1))),P) Path problem: Given a C-space (C, S, N, F, M), and a possible chain coordinate for c1 is dkc where dk;(c1)
a vector F= fi, f2, * * * ,f4,} an initial cell c* and a final = cio.
348 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS September
Next, find possible cell mnasses for all ciGL1. Amnonig cluding the initial cell, be called the path length of thatthe cells in L1, let {c' } be the set of cells whose possible path. For the path p(c*, c**), therefore, the path lengthcell masses are a miniiiinmumi. We then assign to each is q+1.CIE c } the samiie cell imiass aind the samiie chaini co- For admissible paths of lenigth 1,' it follows fromii ruleordinate as its possible cell milass aid its possible chaini R2 that m(c**) < F(c*, c**). Let us nlow assume valid thecoordinate. iniduction hypothesis that m(c**) < F(c*, c**) for allThe cells whose possible cell imiasses are niot miinimial adinissible paths of path length n-ot greater than q.
are no longer conisidered, and the list L1 is cleared. Consider the subpath p(c*, cq) of p(c*, c**). By theR3) Selection rule: In R2, a cell ci and a selected sub- induction hypothesis,
set I ci } of N(ci) were given. The rule which was invoked m(c) _ F(c*, c2).to select one cio from the set {ci} is called a selection (). F(c*, ca).rule. Let us now suppose m(c**) > F(c*, c**). Since F is mono-
R4) Procedure for updating list L: Let {c} be the set tone, we then have the inequalitiesof cells whose cell nmasses had been determined by R2. m_(o)_ F(c*, Cq) < F(C*, C**) < m(c**)Append first to list L the set Ic'} . Next examine each M(Cq). M(C*, iF c**) < m(c**)
a
cell c in L to see if the cell mnasses of all its admissible Since now m(c) <m(c**), it follows from Lemma 1 that1 neighbors had been determnined. If so, erase c from f(c*, Cq) < (C*, c**). Therefore the cell mass m(cq) isthe list L. determined before the cell mass m(c**) is determined.
R5) Initialization: Cell c* is given the cell mass From rule R2 it follows that as soon as m(cq) is de-(0, O, , 0). The list L containis initially the single termined, the cell Cq becomes a member of cell list L.entry c*. The list L, is initially cleared. Since by rule R2,With the aid of rules R1-R5, we may now state the fmin {F(p(c*, ci, c**)) ci C N(c**)}
main procedure. Mc* =main procedure. m(c**) = l~~~~~~~~over all c2 for which m(ct) iS defined,
A: The Search and Trace Algorithms:Al: Search algorithm: Apply rule R5 for iniitializa- and since Cq is one of such cells ci, it follows that this
tion. Apply rules Rt, R2, R4 to entries of L until either minimum cannot be greater than F(p(c*, c**)). Thisc** appears in L or the list L has been exhausted. In contradicts our earlier supposition, and the lemma fol-the former case, we will proceed to the trace algorithm lows.A2. In the latter case, there is no admissible path The basic result is embodied in the following:p(c*, c**) in the e, space. Theorem: Let a C-space (C, S, N, r, AM) be given.A2: Trace algorithm: Begin at c**, follow the chain Let P be a path problem with respect to a vector F.
coordinates until c* is reached. This determines a unique If F is monotone, then algorithm A yields a pathpath p(c*, c**). p(c*, c**) satisfying P.
Proof: Let p(c*, c**) be any admissible path fromC. The Procedulre Applied to the Path Problem c* to c**. Since F is monotone, we may apply Lemma 2LetaeCspace(C,S,N, IF,M), avectorF =(fif2, ,f,) to get m(c**). F(p(c*, c**)). It follows from rule R2
and an initial cell c* be given. A cell ci which is reached thatfrom c* after exactly t application of rules Rt to R4 is m(C**) = (fi(p(c*, C**)), * * ,fr(P(c* C**))).said to have a chain index I (c*, ci) of t from c*. We will . .begin with two lemmas.Lemma 1: Let F be a monotone vector. Let ci and ci F(p(c*, c**)) = (fl(p(c*, c**)), f* *,fr(p(c*, c**))).
be two cells with cell masses m(ci) and m(ci) and chair The theorem therefore follows by the lexicographicindexes (c*, ci) and f(c*, ci), respectively. If m(ci) <m(c'), ordering of these r-tuplesthen f(c*, ci) <((c*, ci).
Proof: Assume 1(c*, ci) >f(c*, ci). This means that IV. APPLICATIONSafter f(c*, ci) applications of rules RI to R4, the cellmass m(ci) has been determined, but the cell mass Id the setiC to bea t squares her planewithm(ci) has not. By the nature of rule R2, the cell masses the sual 1niboroos as shown in Fig 1.aWe willare obtained by evaluating the vector F for the paths in the alpb setgSoconsis of th Fig:let, thflphabe set S'~ [onr1Qt of the4c fnlo1"lT;g:CYquestion. Since F is monotone by hypothesis, m(ct) 1 iisfo o9>m(ci) and the lemma follows.Lemma 2: Let F be a monotonle vector. Let p(c*, 2) All letters of the English alphabet.
c**) be any admissible path from c* to c**. TIhen m(c**) 3) The symbols: +, -, ., J, /, ~,-, K, 7, 1, r
Proof: Let p(c*, c**) consist of the chainl of cells The i-neighborhood function N and the coordinate* ** _ {*_ o1 2 q ** _ +1} functions d1, d,, d8, and d4 are defined as follows: Given
p(c*, c**) = c* -c, c1, c2, * , c**c -c . a cell ci, d1(ci) is the cell to the right of ci, d2(ci) is the
W;e will let the number of cells contained in a path, ex- cell above ci, d3(ci) is the cell to the left of ci and d4(ct)
1961 Lee: An Algorithm for Path Connections and Its Applications 349
16 15 14 13
5 4 3 12
6 2
7 8 9 10
Fig. 1-The set C of squares in the plane.
is the cell below ci. N(ci) is theni the set 'di(ci), d2(ci), We assert first that f is moniotone. This is so by thed3(ci), d4(ci) } iterative nature of the definition of f; the value of f
Let a funiction r7 be given, so that to each cell ciE C is never decreases as a path increases in length. Hence, weassociated a symbol s(ci) CS. The function 111 is given may apply the basic theorem to get:as follows: Corollary 1: Algorithm A solves the minimnal-crossinig
Let p(c*, c**) = {c° = c*, cl, , c-1, Cn = c**} be a problem.path fromn c* to c**. p(c*, c**) is admissible, i.e., Example 1: Consider the cell configuration given inMAlr(p(c*, c**)) = 1, if
Fig. 2. The path AA consisting of the chain of cells1) s(ci)= blank,-, or for i1, 2, n-1. {6, 5, 16, 15, 14, 3, 2, 11, 28} is already present. We2) s(ci+l) £- whenever ci+A=d1(ci) or ci+1 =d3(ci), wish to find a minimal-crossing path from c* (cell 18),
i=O, 1, * n-i. to c** (cell 13).3) s(ci+1)5# whenever ci+1=d2(ct) or c(+ =d4(cC), Applying algorithm A, we find that list L consists of
i=O , n, n-i. the single entry { 181 to begin with. List L1 therefore
Thus, except for the function F which depends on the has in it entries {5, 17, 19}. Note that cell 39 is not an
applicationi in question, the specialization of the e admissible 1-nieighbor of cO. The possible cell massesspace to our applications has been fully described. for cells {5, 17, 19} are 1, 0 and 0, respectively. Hence,
by rule R2, cells 17 and 19 are assigned cell mass 0 and
A. A Minimal-Grossing Problem are appended to the list L. Moreover, the chain co-
ordinates of cells 17 and 19 are respectively d4 and d2.Given a set of squares in the plane, the problem Of The cell masses and chain coordinates for these cells are
fin-ding a path p(c*, c**) from c* to c** such that properly denoted in Fig. 2. Thus, in cell 17, we have thep(c*, c**) crosses over the fewest number of existing pair (t 0) meaning that the chain coordinate is d4paths is called the miniimal-crossing problem. We will (i.e., downward) and the cell mass is 0.formulate the minimi-al-crossing problemi as a path prob- Applying rules Rl to R4 again, we find that list L,lem in som-le appropriate C space and then solve it with has in it now the entries {5, 201. This is so since cellsthe aid of algorithm A. 16, 36, 38, 39, 6 and 40 are all not admissible. The pos-
For this problem the vector F would consist of a sible cell masses for cells 5 and 20 are respectively 1single function f given as follows: and 0. Thus, the cell mass for cell 20 is 0, the chain co-
F1) f(p(c*, c*))=0. ordinate for cell 20 is d2, and cell 20 is appenided to
F2) If s(ci) =blank, then list L. Moreover, cells 17 and 19 are erased from list L.Fig. 3 shows the cell mass and chain coordinates for
(min {f(p(c*, ci)) ci AT(ci) } over all ci all the cells reached by the application of Algorithm A.
f(p(c*,ci))=6 for whichf(p(c*, c')) has been defined; The trace algorithm then traces out a solution pathundefIned otherwise. p(c*, c**) as shown. The boundary cells have been(undefined otherwise. oitdi i.3
F3)Ifs(c =-, he E<xample 2: Consider again the cell configuration in((min{f(p(c*, d2(ci))),f(p(c*, d4(cD))) })+1 Fig. 2. In this case, however, we stipulate that it costs
f(p(c*, ci)) = if either one of the values off is defined; 3 units to cross a -, but 1 unlit to cross a |. That is,I ~~~~~~~~~~thedefinition F3 iS changed to read:
oundefined otherwise.
F4) If s(c) = , then F3'. If s(c'-)=- then1
f(p(c*, c2)) = |if either one of the values of f is defined; f(p(c*, c&)) = j if either one of the values off is defined;
lundefined otherwise. tundefined otherwise.
350 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS September
371 36 35 34 33 32 31
x X X X X X X 17 16 IS 14 13 30
38 17 16 15 14
13 30 Blx -B K 4 ____ _ _ _IX39IBIS I 4 I Zl 29l 1 I181 5 4 3 12 29
IS ~ ~ ~ 15 4 3 1229BIJ
(0) ___I_
(0) (* 1) (,1) (0,) (-,I) (-) ___ (,) __
x A x L A x j AL 2A2
41 20 7 a 9 10 27,0) __ _ _ _ _ _ _ _ __ _ __ ( , ) _ _ _ _
42 __2 7 8 10 274221 22 23 24 25 26
Fig. 3-Cell configuration after algorithm A has beenFig. 2-A minimal-crossing example. applied to Example 1.
17 16 15 14 13 30
Sn - g(4,0) __________ (4,2)
18 5 4 3 12 29
(0) (*- I I) (*- ,I) (-2) (42) _19 6 I2 II 28
A x L - A
20 7 8 9 10 27
(4,0) I(*--0 (4 04,0) (4- 0) (, 0)
Fig. 4-Cell configuration after algorithm A has beenapplied to Example 2.
Since F is still monotone, algorithmn A miay be ap- g(p(c*, c))plied to solve this modified path problem.
In this case, after applying algorithm A, we have the '(min { g(p(c*, ci)) |C C N(c')}) + R(ci)cell configuration and the solution path shown in Fig. 4. = over all Ci for which g(p(c*, ci)) has been defined;
In Example 1, the function f is not only monotone,but grows a single unit at a timne. For such functions andfor vectors made up of such functions, it is possible tosimplify algorithm A to solve the path problem. In where R(ci) =the number of cells ci in N(ci) in each ofExample 2 the functioni f nio longer grows one unit at a which the symbol is neither blank, nor -, nortime. The machinery of C5-space is needed to cope withthis more general class of monotone functions. G3) If s(ci)= , then
B. A AMinimal-Edge-Effect Problem g(p(c*, ci))
Let us begin with the C,-space consisting of squares {(min{g(p(c*, di())), g(p(c*, d4)Ci)))*)+ R(ci)in the plane described earlier. Let an initial cell c* anda final cell c** be given. We wish now to consider the = if either value of g is defined;problem of finding a path from c'* to c* * which avoids, undfedotherwise:as much as possible, any symbol other than -, |andthe blank symbol. In other words, we want a path G4) If s(ct) = thenwhich does not tend to "cling to edges."'
For this application, the vector F would again con-*sist of a single function g given as follows: f(min {g(p(c, d1(ct))), g(p(c*, d3(ci)))})
Gi) g(p(c*, c*)) =0. g(p(c*, ci)) = + R(ci) if either value of g is defined;G2) If s(c9 =blank, then ~undefined otherwise.
1961 Lee: An Algorithm for Path Connections and Its Applications 351
64 63 62 61 60 59 58 57
X X A
37 36 35 34 33 32 31 56
xI( I,0) 0) (01
38 17 16 15 14 13 30 55
x x
1-0 (, ) -( ,o) (-, ) ('-.0139 148 5 4 3 12 29 54
x(I ,0) 1 (0) ('-. 0) ) 9,01
40 89 6 2 II 28 53x x
(9,0)f 1,0) ___
41 20 7 8 9 10 27 52A X(0) (4,0) 10) 01,0)
42 21 22 23 24 25 26 51
(4,0) ('-0) (9,10) (---0) ('-0) (-0) (-0O)I" 44 45 4~f6 47 48 49 50
(9,'0) ('-0) (-0O) ('-0) )-0) (- 01 (--0) (0Fig. 5-A miinimal edge-effect example.
From this it follows that F is again mnonotone. erty was the niumber of crossings, anid in the second, theTherefore, we have: property was edge effect.
Corollary 2: Algorithmii A solves the minimal edge- On the other hand, the C-space mnodel was intendedeffect problemn. to solve joint miiimizationi problems; i.e., the vector FExample 3: Consider the cell configuratioin giveli ini may have several compoinent funictions. In order to
Fig. 5. We wish to construct a path AA from cell 41 to illustrate the possibility of joint miinimization, let uscell 58 satisfying the path problem with respect to the consider a minimal distance then edge-effect problem.vector F given by Gt to G4. What we wish to find here is a path which is, first of all,To begin with, the list L has in it the single enitry one of the shortest, and secondly, among all shortest41 1. The list L1, therefore, has in it enitries t 20, 40, 42 }. paths, this path is also mninimal with respect to edge
The possible cell masses for these cells are, respectively, effect. The vector F for this problem has, therefore, two1, 0, 0. Therefore, by rule R2, m(40) =m(42) ==0. The component functions F= (h, g), where h is the distancechaiin coordinates for cells 40 and 42 are also determinied function and g is the edge-effect function.and are respectively I and T . The edge-effect funictioni g will be taken to be exactly
List L is now updated to contain cells 41, 40, 421. the same as that defined previously in Gt to G4. TheFrom this, we get for list L1 the cells 120, 19, 39, 21, 43 }. distance function h is given as follows:Following rule R2, we get therefore m(39) =m(43) HI) The function h satisfies Fl, F3, F4.=m(21) =0, and also their chain coordinates. By rule H2) If s(ci) =blank, thenR4, cell 42 is erased from list L.The new list L IIow has in it cells 141, 40, 39, 43, 211. ((min{ h(p(c*, c)) c1 ( N(c1) }) ± lover all
Therefore, cells 120, 19, 18, 38, 22, 44} all belong to list h(p(c*, ci)) = ci for which h(p(c*, c) has been defined;L1. Consider now cell 44. It is clear that m(44) =0. We undefined otherwise.must, however, invoke the selection rule R3 to get itschain coordinate. We may assume, in this case, that From these definitions it again follows that both h andthe left neighbor is always preferred over the other 1- g are monotone functions. F= (h, g) is therefore a mono-nieighbors. The chain coordinate for cell 44 then be- tone vector. Therefore, we have:comes <-. Corollary 3: Algorithm A solves the joint minimiza-
Continuing in this way, we arrive at the path shown tion problem with respect to first distance then edgein Fig. 5. We see that the path so constructed avoided effect.all the X 's which may be considered as edges. Example 4: Let us consider again the original con-
joint **~~~~~~~~* ~figuration shown in Fig. 5. We wish to find a path AAC. MitMnimization which solves the path problem with respect to the vec-
In the last two applications, the vector F in each tor F= (h, g) for this configuration.case consisted of a single function. This was So because Following rule R5, we begin with cell 41 in list L.we were looking for paths which were minimal with By rule R1, cells 20, 40 and 42 therefore belong to listrespect to a single property. In the first case, the prop- L1. The possible cell masses for cells 20, 40 and 42 are
352 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS September
64 63 62 61 60 59 58 57
K X A(4, 5,2) -,6,3) (4-7,4) ((12,2) (4,13,1)
37 36 35 34 33 32 31 56
K(4,4,21 (4,8,51 (,9,7) (4,10,3) ( I,I1,11 (4-,12,1)
38 17 16 15 14 13 30 55
K IC
14,3,1) 98_(,98) (4,9,2) (4,10,) (-, 11,1) -,12,1)39 18 2 4 3 12 29 54
(4,2,0) (4,3,2) (4,7,5) +8,1) (4-9,1) (4,10,2) (4-11,2)40 19 6 2 II 28 53
4+,1,0) (4,2,11 (4,6,3) (4,7,1) ( 8,2) (4,10,2)
41 20 7 8 9 10 27 52
A IC
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(f,2,0) (,3,0 ) (4-,4,0 ) (4-,5,0) (-,6,0) ( * ,7,0) (4,8,0) (*-,9,0)
Fig. 6-A joint minimization example.
respectively (1, 1), (1, 0) and (1, 0). Therefore, m(40) 1-neighborhood funciton N* satisfies the finiteness con-= (1, 0) and m(42) = (1, 0). The chain coordinates for dition NO rather than conditions Ni and N2 of Sectioncells 40 and 42 are respectively I and 1. II. Accordingly, we must also make minlor changes toFrom rule R4, the list L now contains cells 41, 40 and algorithm 4. For instance, we must redefine the coor-
42. Therefore, by Ri, list L1 has now in it cells { 20, 19, dinate functions di, and modify our process of assigniing39, 43, 21} with possible cell masses respectively: chain coordinates. Let us agree to call the modified al-(1, 1), (2, 1), (2, 0), (2, 0), (2, 0). Thus m(20) = (1, 1) gorithm A*. Then our basic theorem would read:and cell 20 has chain coordinate <-. Let a e*-space (C, S, N*, F, Al) be given. Let P be a
Continuing in this mannier, and using again the selec- path problem with respect to a vector F. If F is mono-tion rule R3 that the left direction is to be preferred tone, then algorithm A4* yields a path p(c*, c**) satisfy-over all other directions, we arrive at the path shown in ing P.Fig. 6. We see that in this case, since we are interested In regardi to the second question, our knowledge isin the shortest path, the path AA makes contact once very meager. It is quite possible that an essentiallywith an edge. We had seen before that there are paths different algorithm is needed to deal with non-monotonewhich make no contact with any edge. vectors.
Coming back to our basic theorem, we ought to makeD. Generalizations it clear that even when a vector F is nmonotone, it may
\Ve should, perhaps, take a moment at this time to be so pathological that the process of applying algo-reflect on the following two questions. In Section II, we rithm A could become extremely tedious. To be specific,have set up an abstract model, our C5-space, in such a let p(cl, cn) be an admissible path consisting of the chainway that algorithm A can be used to solve path prob- of cells:lems formiulated withini this model. The first questionwe will ask is whether algorithm A4 can be applied to p(c', cn) = Ic1, c2, , c}.still more general situations that is, whether our ab-stract model of C-space may be further generalized. In A monotone function f is said to be 1-hereditary ifa similar way, we may wish to relax the monotone con- f(p(c1, ca)) depends only on f(p(cl, cn-l)) and on cellsdition on vectors F. The second question is, therefore, Cn-' and cr1. A monotone function f is said to be 2-whether our basic theorem may be generalized to in- hereditary if f(p(cl, Cn)) depends only on f(p(cl, Cn-2))dlude also perhaps a subclass of non-monlotone vec- and f(p(c', c4-')) and cells c-2, cn- and c8 In a similartors. manner, we may define p-hereditary functions for p_.1.The answer to the first question can be given in the All of the exarmples of monotone vectors given in this
affirmlative, and may appear a bit surprising. Specifi- section happen to be 1-hereditary. In such cases thecally, we may redefine the 1-neighborhood funlction N process of applying algorithmn A is much simplified.such that N needs to satisfy (neither rule Ni nor rule N2 \V\e may also apply algorithml A to solve the minimalof Section II. The only condition that N mnust satisfyr is corner problem; that is, to find a path with the leasta finiteness condition: number of corners. This problem presents an interestingNO. Every i-neighborhood is finite. twist, since the corner function is monotone but 2-Let us call a space (C, S, N*, F, Al) a C5-*space if the hereditary. In the same way, one may construct p-
1961 Lee: An Algorithm for Path Connections and Its Applications 353
hereditary functionis for arbitrary p. Indeed, one may minimal-distance problem, it occurred to us that theconstruct m-ionoton-e functions which are not finitely search algorithmn is, in a remote sense, a computernodelhereditary. of waves expanding from a source unlder a form of
straight-line geometry. Those cells havinig the same cellV. MINIMAL-DISTANCE SOLUTIONS-A MA/AZE AND miiass mllay be thought of as the locations of the wave-
OTHER ILLUSTRATIONS fronit at the mth unit of timie. Fig. 8, for example, may
As the reader can see, the statement of algorithm A be taken to represenit the wavefront originating from
lends itself quite directly to computer programming. source A as it expanded.Such a step for the minimal distance problem has been Following this line of thought, we proceeded to run
carrieout. A natural cell configuration Is dtr Ined a few simple experimenits suggested by optics. In Fig.byrthedprinter ntassociated compiurater. Tetpriner 15, the obstacle consisted of a vertical barrier with a slitcantheprinte120 characters winhoe line,tand usu printsr in the middle. The source is at the left-hanid end of the60 lines toa pa cell configuran is thereforena diagram. The blank spaces may serve as an indication60retag a arra o12 cell byg60 cs thesets o of the propagation of the wavefraont. The reader may see
symbols are, in this case, the set of characters onthe that the wave pattern to the right of the obstacle is theprinter, same as that which would have been produced by a
In Fig. 7 is shown an input into the computer. The source located at the slit.In~~~~~~Fig71is shw aniptit h optr hIn the same way, Fig. 16 shows the effect of havingbounidary cells and the obstacle cells are all marked X. I the staeway Figs.16shw th efe oaving
In this illustration, we wish to find a miniimal-distance a 2-slit obstacle and Figs. 17 and 18 are patternspath~~~~~~betee celAadelB created by having a number of multiple-slit obstacles.pathbetween cell A and cell B.
There are several differences between the modelFig. 8 shows the result of applying algorithm Al (thesearch algorithm) to the cell configuratioin given in Fig.7. Since it is not possible to print mnore thani one char- The geometry assumed in this model is, in the first place,acter in each cell, we have chosen to print out the least not Euclidean. Since distances in this geometry are
significant octal digit of the cell masses. The reader can measured roughly as distances are understood by taxi
see that the immediate neighbors of cell A has cell .ass drivers on the island of Manhattan, this geometry is
1. The neighbors of these have cell mass 2, etc. This sometimes called Manhattan geometry.search-expansion process continues until cell B is In addition to the difference in geometry, and the fact
reached. that we are dealing with a discrete space, we have also
Fig. 9 shows the result of applyXig algorithm A2 (the not taken into account the phenomenon of interference.trace algorithm) to the cell configuration. The selection The comiputer instead has a built-in first-come-first-rule used here is to order the admissible neighbors ac- served rule. That is, where there are several sources pres-
cording to the following list of preference: right, up, ent, the amplitude of the wave at any point is deter-
left, down. The path shown is the machinie's solution to mined b the source closest to it.the original path problem. Although there is only a remote resemblanice between
Fig. 10 depicts a three-stage adder circuit; each box this model and optics, these experiments seem to sug-
designates one of the circuit stages. We wish to apply gest the possibility of microsimulation of physicalalgorithm A to establish all appropriate coinections. phenomena on a computer and the possibility of look-
Fig. 11 shows the result of applying algorithm A. One ing for effects in this way if the laws of nature were
miay note that in these paths, there are many more nodified. In our case, it would be possible to inicludecorners than necessa'ry. The appearance of these extra also interference. The model would then be a reasoni-* ... 1 ~~~~~~~ablv faithful representation of elenienltary wave phe-coriners is due to our simple selection rule. To a large I
extent, the appearance of paths can be conitrolled by nomena under discrete M\anhattai geometry.incorporatinig appropriate selection rules inito algo-rithm-i A. VI I. AcKNOWLEDGMENT
Shortly after the program was written, we realized In the course of this work, the writer has had thethat this program, without change, can be used to also benefit of stimulating conversation-s with a number ofsolve maze problemls. In this sense then, and with people. Among these are E. F. Moore, T. H. Crowley,proper modifications, the program may serve in particu- D. H. Evans, S. H. Washburn, R. W. Hamming andlar as a 704 version of Shannon's mnaze-solving ma- C. A. Lovell. The "diffraction" experimlents were anchine.5 An illustration is the Hampton court maze outgrowth of a conversation with D. H. Evans. Theshown in Filg. 12. Filg. 13 shows the result of applying wrriter also wishes to acknowledge the unltirinlg assist-the search algorithmn to the mtaze conlfiguration. The ance for Sections V and VI from 5. 0. Four of IBM4.machine's solution is given by Fig. 14.
5C. E. Shannon, "vPresentation of the maze-solving machine,"Trans of the 8th Gybernetics Conf., Josiah MVacy Jr. Foundationl, New
VI. THE SEARCH ALGORITHM AND HUYGENS York, N. Y., pp. 173-180; 1952.PRINCIPLE
In the course of programming algorithm A for the (See Figs. 7-18, on follZowing pp. 354-365.)
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Fig.
15-A
single-slit
experlimien-t.
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16-A
double-slit
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17-Begirininig
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Fig.
18Ne
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t
