A Global Approach to the n-dimensional Traveling Salesman Problem: Application to the Optimization of Crystallographic Data Collection

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1 Appl. Cryst. (1987). 20, A Glbal Apprach t the n-dimensinal Traveling Salesman Prblem: Applicatin t the Optimizatin f Crystallgraphic Data Cllectin BY JEFFREY n. WEINRACH AND DENNIS W. BENNETT* Department f Chemistry, University f Wiscnsin-Milwaukee, Milwaukee, Wiscnsin 53201, USA (Received 13 April 1987; accepted 24 July 1987) Abstract An algrithm fr the ptimizatin f data cllectin time has been written and a subsequent cmputer prgram tested fr diffractmeter systems. The prgram, which utilizes a glbal statistical apprach t the traveling salesman prblem, yields reasnable slutins in a relatively shrt time. The algrithm has been successful in treating very large data sets (up t 4000 pints) in three dimensins with subsequent time savings f ca 30%. Intrductin The prgramed mvement f fur-circle diffractmeters is an inherently inefficient prcess as perfrmed n mst diffractmeter systems. Imprvement in the prcedure invlves srting reflectins in such a manner that the time required t cllect data is minimized. This is essentially a prblem f finding the shrtest path in {20, X, ~P} space, a tangible representative f a class cllectively knwn as 'the traveling salesman prblem' (Fld, 1956). The traveling salesman prblem (TSP) is a classic example f a branch f mathematics dealing with cmbinatrial ptimizatin; that is, minimizatin f a functin ver nn-cntinuum cnditins fr which differential calculus cannt be applied. The basic premise f the TSP is t span an array f data pints in n-space with the shrtest path pssible. Fr mst applicatins the system under study has been twdimensinal and the determined path has been a clsed ne (the end pint alng the path is als the starting pint). Many appraches have been develped t slve such cnstrained prblems, with mderate success. Recently a review f the TSP and the appraches taken t slve specific cases have been published in a cmprehensive bk by Lawler, Lenstra, Rinny Kan & Shmys (1985). Fr the mst part the prblems addressed in this bk cnsist f a few hundred pints arranged in a tw-dimensinal * Authr t whm crrespndence shuld be addressed. array. Appraches t the slutins f these prblems fall int tw general categries, ne invlving decisins based n the immediate envirnment f pints alng the path, a 'lcal' apprach, and ne in which decisins are made n the set f pints as a whle, a 'glbal' apprach. In the limit the lcal strategy becmes ne f cnnecting each pint t its clsest neighbr, prviding an bviusly pr slutin. Extensins beynd this limit apprach a functin f n! in difficulty, where n is the number f independent neighbrs included in a given lcal search, and thus expand quickly t frmidable prprtins. In additin, the c.p.u, time required fr lcal appraches increases apprximately as a functin f n(n-1), making the algrithms themselves unreasnably inefficient fr large data sets. On the ther hand, glbal appraches exhibit their wn difficulties, principally in develping an algrithm which will treat a large number f randmly distributed pints simultaneusly. In general, mst such strategies nly begin glbally, and are 'lcalized' as sn as pssible in rder t render the prblem tractable (Grtschel & Padberg, 1978). Karp's 'divide and cnquer' algrithm (Karp, 1986) prvides an elegant example. It begins by partitining the space int rectangular regins, then generates ptimal lcal turs within each regin which are subsequently cllected int a final path. Unfrtunately, the efficiency f such algrithms depends critically n the way in which the space is divided, since a large number f pints in a subdivisin simply creates a lcal traveling salesman prblem, whereas t few pints in each regin yield an apprach t the 'clsest neighbr' slutin. Thus, neither lcal strategies nr thse which begin glbally have prven t be bth practical and general. Furthermre, nne f the appraches t date extraplates readily t three r mre dimensins, a pint f bvius cncern t the practicing crystallgrapher. In this paper we describe a new algrithm which btains a shrt-path slutin t the uncnstrained traveling salesman prblem, characterized by a randm array f pints in n-dimensinal space with n /87/ Internatinal Unin f Crystallgraphy

2 508 THE n-dimensional TRAVELING SALESMAN PROBLEM restrictins n the beginning and end pints f the path. The algrithm readily handles thusands f pints, is cmpletely general, and demands nly very mdest cmputatinal time and mass strage. Thery Of the tw appraches t a general slutin f the TSP, the glbal apprach is in principle superir t the lcal apprach, since the slutin must depend n the cmplete distributin f pints. Cmputers, hwever, make binary decisins, and lend themselves naturally t the lcal apprach, which cnsiders nly ne neighbr at a time. It fllws frm this that a wrkable algrithm shuld cnsider the ensemble f pints cllectively, but instead f building a path pint by pint, the algrithm shuld cnsider the ensemble as it makes individual binary decisins. This can be accmplished by sequential divisin and srting f the ensemble int smaller and smaller subgrups, until the final set is reached which cnsists f an rdered set f grups each cntaining ne pint, a path. Since the prcedure invlves the sequential divisin and srting f an array f pints the algrithm can be designed t be recursive, that is, it can perate n itself. In this manner the prcedure becmes increasingly mre efficient as it prgresses twards a slutin. The descriptin which fllws is restricted t three dimensins, 20, g and ~, with the added cnstraint that nly ne f the three independent variables will determine a given decisin. Thus the algrithm described herein is specifically designed fr the ptimizatin f crystallgraphic data cllectin in which all angles are traversed simultaneusly and independently until the final angle is reached. Nevertheless, mdificatins t the prcedure fr ther prblems shuld be cmpletely straightfrward. The array t be srted cnsists f a cmplete set f angular crdinates, scaled if necessary t take int accunt any difference in relative mtr speeds. The array is then partitined int tw subsets by srting it int lwer and higher grups with respect t a selected variable; fr example, grup I might cntain the smaller 20's, grup II the larger 20's. A simple insertin srt is used fr the first half f the riginal array with all these pints becming members f the lwer grup. At this pint the largest value in this grup is the critical value. Any pint with a value larger than the critical value becmes a member f the higher grup. If an insertin int the lwer grup needs t be made, the element f the smaller grup with the critical value becmes an element f the higher grup and the next highest value then becmes the critical value. This saves cnsiderable time, since many elements can be placed directly int the higher grup, thus circumventing the insertin srt fr these pints. Each grup cntains exactly half f the data pints unless the ttal number f elements is dd, in which case ne grup will cntain an extra element. It is imprtant t mentin this since the algrithm invlves the switching f array elements which d nt match in the case f even/dd parity. This makes it necessary t carry the starting and ending indices f all subgrups in an auxiliary array thrughut the srting prcess. Each divisin defines tw subsets, ne f which will be spanned cmpletely in the srt sequence befre entrance int the ther subset alng the path. After the initial divisin (which requires nly an arbitrary decisin regarding the rdering f the tw initial subsets), the entire decisin-making prcess simply invlves the chice f which f tw subsets t span first. Fllwing the first divisin anther variable is selected and grup I is divided int tw new grups (e.g. Ia and Ib) with respect t this variable. At this juncture the algrithm must determine the rder f these tw subgrups. It des this by cmputing the mean psitin f the pints in grups Ia, Ib, and II, then determining whether the mean psitin in grup Ia r Ib is 'clser' t the mean psitin in grup II. In the specific applicatin cnsidered here 'clser' and 'distance' take n special meanings. Because nly a single variable (20, ;~ r ~) is rate limiting, the maximum abslute value f the difference between the mean 20's, the mean X's and the mean q~'s is the actual criterin fr deciding which grup is 'clser' t anther. The grup clser t grup II becmes the secnd f the tw t be spanned, since grup II fllws grup I in sequence. Grup II is then divided with respect t the same variable, with mean psitins fr subgrups IIa and lib calculated and distances t the 'clser' subgrup frm grup I used t determine the rder f grups IIa and lib. This prcess is cntinued, selecting a new variable each time the divisin prcess has cvered the entire ensemble, until each subgrup cntains ne and nly ne pint. The prcedure becmes a bit mre cmplex when cnsidering nnterminal subgrups, i.e. all thse nt at the beginning r end f the array. In these cases the mean psitins f the grups bth preceding and fllwing the divided grup must be evaluated in the decisin. Cnsider, fr example, grups IV, Va, Vb, and VI. The IV, Va, Vb, VI and IV, Vb, Va, VI distances are cmputed with the shrter f the tw paths determining the srt rder. The algrithm thus cntains sectins t handle the first grup, the middle grups, and the final grup in the array fr each cycle thrugh the entire ensemble. The algrithm In rder t prvide as much generality as pssible the algrithm utlined belw is written in symblic lgic using symbls which are easily transfrmed int apprpriate cmputer cde.

3 List f symbls NV V(l) N ~(N) Symbl G(N).-+G(N') S(N) --V(I)--. G( N) + G(N') M(N) #(N) NI N~ Ni MAB TCP{ V(I), V(2), V(3) } Traveling salesman algrithm Initial partitining JEFFREY B. WEINRACH AND DENNIS W. BENNETT 509 Definitin Number f independent variables (NV = 3 fr 20, X, ~0) Variable I (I = 1, 2, 3 fr 20, Z, tp) Grup index (in the limit N is the reflectin sequence number). Grup with sequence number N, in array 1 Renumber grups: (G(1), 6(2)..., G(N), G(N'), 6(N + 1),...) (G(1), 6(2),..., G(N), G(N + 1), G(N + 2),...)~ Switch cntents f grup N and grup N' (effectively switches rder f grups) Srt grup N int tw new grups with respect t variable V(I), N < N'. Mean psitin f pints in grup. N 2 Number f pints in grup N? Ttal number f pints in n space ( {20, Z, ~} space) Current grup index Ttal number f current grups with different indices (number f indexed grups) Maximum abslute value Array cntaining diffractmeter angles t be srted Cmment Set up array t be srted (TCP) NV= 3 Ni= 1 I=1 S(1) --V(I)--. G(1) + 6(1') Ni= 2 1=2 S(1) --V(l)+ 6(2) + 6(2') IF MAB[M(2') - M(I') ] > MAB[M(2) - M(1') ] THEN 6(2),-~ 6(2') G(1) = G(2) t#(n) = i: - i,, where i, is the starting index fr grup N and i/is the final index fr grup N. i: and i, must be carried as auxiliary variables since they are necessary fr mapping variables during the switching prcess and determining the mean lcatin M(N). 3 6(2) = 6(2') Ni= 3 S(1') --V(I)--, G(3) + G(Y) IF MAB[M(3)- M(2)] > MAB[M(Y)- M(2)] THEN G(3) ~ G(3') TCP= {G(I), G(2), G(3), G(4)} First grup algrithm I=I+1 IFI>NVTHEN I=1 N~= 1 IF#(1)=I THEN GO TO2 S(1) --V(I)~ G(1) + G(1') IF MAB[M(1') - M(2) ] > MAB[M(1) - M(2)] THEN G(l) ~-, O(l') N~ = N~ + 1 Middle grup algrithm Nc = N~ + 1 IF N~ = N i THEN GO TO 3 IF #(N )= 1 THEN GO TO 2 S(Nc) --V(I)~ G(Nc) + G(N/) IF MAB[M(N~- 1) - M(Nc)] + MAB[M(N~ + 1) - M(N/)] > MAB[M(Nc- 1) - M(N/)] + MAB[M(Nc + 1)- M(N~)] THEN G(Nc) ~ G(N/) N = N~ + 1 Value f variables in Fig. 1 determined here GO TO 2 Last grup algrithm IFNc=NtTHEN END Each grup cntains ne pint IF #(N)= 1 THEN GO TO 1 S(N~) --V(I)--+ G(N~) + G(N/) IF MAB[M(N~- 1)- M(Nc)] > MAB[M(N~ - 1)- M(N/)] THEN G( N~) ~--,, G( N/) N~ = N~ + 1 GO TO 1 Results and discussin Fig. 1 illustrates a simple wrked example in tw dimensins t assist the prgrammer in develping this algrithm fr his wn particular system. In rder t test the efficiency f the algrithm a number f similar tw-dimensinal paths were generated and cmpared with the actual shrtest rutes, determined

4 5 l0 THE n-dimensional TRAVELING SALESMAN PROBLEM by calculating all pssible paths. In general nly very small sets f pints can be treated in this manner, since the time required t btain 'brute-frce' slutins fr n pints increases as a functin f n! In every case the path prduced by the algrithm was ne f the shrtest 0"03% f the paths, and always differed negligibly frm the ptimal slutin, in spite f requiring nly a minuscule fractin f the time necessary fr the crrespnding brute-frce slutins. Fr example, a ten-pint slutin such as that shwn in Fig. 1 typically requires less than 1 c.p.u, secnd cmpared with 1800 c.p.u, secnds needed t determine the ptimal path by brute frce. Fig. 2 shws the results btained frm a test prgram designed t find a shrt-path slutin fr a general tw-dimensinal array f randmly distributed pints. A study f this path reveals sme f the characteristics f the slutin btained frm the algrithm. In general, the algrithm tends t migrate frm regins f lw pint density twards regins f high pint density, thus creating a path in which minimal time is spent in regins where there are few pints. The circled areas are places where crssver has ccurred alng the path, resulting frm the arbitrary nature f the space-partitining prcess in the algrithm. These areas are always small, bth in size and number, but they limit the efficiency f the algrithm. This particular prblem arises when a grup divisin separates pints that are clse t ne anther. In rder t keep the prcedure simple and general we have made n attempts at ptimizatin, but the crssver prblem can cnceivably be reslved by treating the pints as 'vibrating' rigid bdies in n-space, thus allwing them t verlap. The space can then be partitined such that verlapping pints are placed in the same grup. It is als prbable that the prcedure can be imprved by taking advantage f the specific nature f the variables invlved in frmulating cnstraints fr particular cases. Fr crystallgraphy such infrmatin might include Laue-grup/space- "" ::!: :.. ~ O ~ 6) ~ 0 6) OO O L (a) -t ~ J Fig. 1. A wrked example f a simple tw-dimensinal applicatin f the algrithm. The initial set is characterized by values f 1, 1 and 1 fr I (the reference number f the variable cnsidered), Ni (ttal number f current grups with different indices) and Nc (current grup index). Successive frames in the sequence have the fllwing values f I, Ni and N: 2, 3, 1; 2, 3, 1; 2, 4, 1; 1, 5, 1; 1,6,3; 1,7,5; 1,8,7; 2,9,4 and 2, 10,7. (b) Fig. 2. (a) A randm 100-pint array in 2-space. (b) A shrt-path slutin thrugh these pints generated by the algrithm.

5 JEFFREY B. WEINRACH AND DENNIS W. BENNETT 511 Table 1. Efficiency f algrithm The time is measured in arbitrary units; time saved rati--(time befre- time after)/time befre. Number f Time befre Time a~er Time saved reflectins srt srt rati " "17 0" "75 0" " "41 0" " "88 0"302 grup infrmatin and the cyclic nature f q~ and Z (bth variables can be reached by traversing thrugh bth psitive and negative angles). As illustrated in Table 1, the applicatin f this algrithm t crystallgraphic data sets results in substantial savings in data cllectin time. The rati f 'time saved' t ttal time befre srting indicates savings ranging frm 15 t 37%. Since the unsrted data sets apprximate 'clsest-neighbr' pathways, the algrithm clearly prvides significantly shrter turs. It shuld be nted that these cmparisns are relative since they als depend n the path calculated prir t applicatin f the algrithm. The results in the table were generated n an IBM-PC with a prgram cmpiled in Micrsft Quickbasic ~. The selectin f the srting rutine is critical t the cmputatinal efficiency f the prcedure. The prgram described here makes use f an insertin srt rutine, since the algrithm nly requires that a minimum f half an array be srted during any ne peratin. A bubble srt wuld be extremely inefficient in this applicatin, while there may be ther mre sphisticated rutines which are mre efficient (Amsterdam, 1985). As with the path determinatin, we have made n attempt t ptimize the prgram itself. On the IBM-PC, a pint pathway typically requires abut 5 c.p.u, h, while the savings in data cllectin can amunt t numbers f days. This serves t establish smething f an upper limit which can readily be decreased with faster cmputers and imprvements in cde. DWB wuld like t acknwledge Dr Ward T. Rbinsn at the University f Canterbury fr initial discussins in the area f data cllectin time ptimizatin. References AMSTERDAM, J. (1985). Byte Magazine, September, pp FLOOD, M. M. (1956). Oper. Res. 4, GROTSCHEL, M. & PADBE, M. W. (1978). Optimizatin and Operatins Research, edited by R. HENN, B. KORTE & W. OETTLI, pp Berlin: Springer. KARP, R. M. (1986). Cmmun. ACM, 29, LAWLER, E. L., LENSTRA, J. K., RINNOOY KAN, A. H. G. & SHMOYS, D. B. (1985). (Editrs.) The Traveling Salesman Prblem: A Guided Tur f Cmbinatrial Optimizatin. Chichester: Jhn Wiley.