Generation of Well-Formed Parenthesis Strings in Constant Worst-Case Time

Transcription

1 Ž. JOURNAL OF ALGORITHMS 29, ARTICLE NO. AL Generaton of Well-Formed Parenthess Strngs n Constant Worst-Case Tme Tmothy R. Walsh Department of Computer Scence, Unersty of Quebec at Montreal, Montreal, Quebec, Canada H3C 3P8 Receved June 3, 1997; revsed Aprl 24, 1998 Prosurows and Rusey ŽJ. Algorthms 11 Ž 1990., publshed a recursve algorthm for generatng well-formed parenthess strngs of length 2n and challenged the reader to fnd a loop-free verson of ther algorthm. We present two nonrecursve versons of ther algorthm, one of whch generates each strng n OŽ n. worst-case tme and requres space for only OŽ 1. extra nteger varables, and the other generates each strng n OŽ. 1 worst-case tme and uses OŽ n. extra space Academc Press 1. INTRODUCTION In PR, Prosurows and Rusey publshed a recursve algorthm for generatng well-formed parenthess strngs of length 2 n and challenged the reader to fnd a loop-free verson of ther algorthm. Loop-free generaton algorthms for other representatons of bnary trees have snce appeared ŽvB, LvBR. ; we present here what we beleve to be the frst loop-free generaton algorthm for well-formed parenthess strngs, although we have receved reports of unpublshed algorthms. In Secton 2 we descrbe the ProsurowsRusey Gray code. In Secton 3 we apply Chase s graylex order Ch to derve a nonrecursve verson of t that generates each strng n OŽ n. worst-case tme and uses OŽ 1. extra space Žthat s, OŽ 1. extra nteger varables rather than bts.. In Secton 4 we apply Ehrlch s auxlary array ŽBER, Eh. to derve another verson that generates each strng n OŽ. 1 worst-case tme and uses OŽ n. extra space. 2. THE PROSKUROWSKIRUSKEY GRAY CODE FOR WELL-FORMED PARENTHESIS STRINGS A well-formed parenthess strng, or Dyc word, of length 2 n s a strng of n 0 s and n 1 s such that no prefx contans more 0 s than 1 s. In PR $25.00 Copyrght 1998 by Academc Press All rghts of reproducton n any form reserved.

2 166 TIMOTHY R. WALSH Prosurows and Rusey present the followng recursve descrpton of a Gray code n whch each word of length 2n s changed to ts successor by transposng a sngle par of letters. For the Dyc word y 1 0 x, the Ž. 1 operatons flp and nsert are defned as flp y 1 01x and nsertž y x. These defntons are extended to lsts, and two other operatons on lsts are defned: A B means A followed by B and A R means A reversed. The lsts TŽ n,.,1n, whch conssts of all of the words of length 2n wth the prefx 1 0, are generated by means of the followng recurson: Ž. flp T Ž n,2. f 1 n, R flpž T Ž n, 1.. TŽ n,. Ž 2.1. nsertž T Ž n 1, 1.. f 1 n, 10 n n f n. Ž. The frst word n T n, turns out to be n Ž 10. f 1 and n 3, n1 frstž T Ž n, Ž 10. f 1 n or Ž 2.2. Ž 1 and n 2., 10 n n f n Žwe added the condtons and n 3 and or Ž 1 and n 2. to mae the formulae n PR wor for the trval case n whch 1 and. Ž. n 2. The last word n T n, s 1 0 Ž 10. n. In PR a recursve algorthm s gven that generates TŽ n,. n OŽ 1. average tme per word, and two Gray codes are gven for the lst of all of the Dyc words of length 2n: TŽ n 1, 1. wth the prefx 10 removed, and TŽ n, n. TŽ n, n 1. TŽ n,2. T R Ž n, A NONRECURSIVE GENERATION ALGORITHM THAT USES OŽ. 1 EXTRA SPACE To obtan a nonrecursve algorthm for generatng TŽ n,., we generated the lsts TŽ n,. for 1 n 5 by hand from the recurson Ž 2.1. Žsee Fg. 1. and observed the pattern followed by the moton of the th occurrence of 1 from the left, whch we abbrevate as 1, n an nterval of words n whch 11, 12,..., 11 stay n one place. By defnton, 11, 12,..., 1 are fxed n poston; we call the other occurrences of 1 free. The rghtmost poston that 1 can have s 2 1; we call a free 1 that s not n ts rghtmost poston a lberal 1.

3 LOOP-FREE PARENTHESIS GENERATION 167 Ž. FIG. 1. T n,, the lst of well-formed parenthess strngs of length 2n wth prefx 1 0, 1 n 5. THEOREM 1. The set of words of TŽ n,. n whch 1 1,1 2,...,11 stay n one place s an nteral of consecute words and s parttoned nto subnterals n whch 1 s statonary too. As we pass from subnteral to subnteral, 1 moes one poston at a tme, ether mong rghtward from ts leftmost poston adjacent to 1 Ž 1 except that 11 starts n poston 2 wth one zero between t and 1. to ts rghtmost poston Ž poston 2 1 n the word., or else mong leftward from ts rghtmost poston to ts leftmost poston. Fnally, 1 moes rghtward f and only f the number of lberal 1 s to ts left s een. Proof. For n the statement of the theorem s vacuously true, snce the one word n TŽ n, n. has no free 1 s. For n we assume t to be true for TŽ n, 1. and TŽ n 1, 1., and prove t for TŽ n,.. More precsely, we prove the last statement of the theoremabout the drecton of moton of 1snce the other statements of the theorem are easly verfed. Suppose that 1, so that, by the frst lne of Ž 2.1., TŽ n,1. flpžtž n,2... In TŽ n,1. flpžtž n, 2.., 1 s a free 1 n poston 3, ts 2

4 168 TIMOTHY R. WALSH leftmost and ts rghtmost poston; n TŽ n, 2., 12 s fxed. Each 1, 2, has the same number of lberal 1 s to ts left n the correspondng words n TŽ n, 1. and n TŽ n, 2., and t moves n the same drecton n both lsts. So the last statement of the theorem holds for TŽ n,1.. Now suppose 1, so that, by the second lne of Ž 2.1., TŽ n,. flpžt R Ž n, 1.. nsertžtž n 1, 1... We compare both the drecton of moton of each free 1 and the number of lberal 1 s to ts left n TŽ n,., TŽ n, 1., and TŽ n 1, 1.. The flp operator, actng on TŽ n, 1., converts 11 from a fxed 1 to a free 1 n poston 2, whch s not ts rghtmost poston 2 1. For each 1, 1, the number of lberal 1 s to the left of 1 s greater by 1 for a word n flpžtž n, 1.. than for the correspondng word n TŽn, 1.. Reversng the lst flpžtž n, 1.. reverses the drecton of moton of 1, so that t moves n flpžt R Ž n, 1.. n the drect opposte to that n whch t moves n TŽ n, 1., whch s consstent wth the dfference n the number of lberal 1 s to ts left. The nsert operator, actng on TŽ n 1, 1., creates 1 whch s fxed, so that each free 1 has the same number of lberal 1 s to ts left n the correspondng words n nsertžtž n 1, 1.. and n TŽ n 1, 1,. and t moves n the same drecton. In passng from the last word of flpžt R Ž n, 1.., whch s the frst word of flpžtž n, 1.., to the frst word of nsertžtž n 1, 1.., 1 1, whch has no free 1 s to ts left, moves rghtward from poston 2 to poston 3. So the last statement of the theorem holds for TŽ n,.,1n. The result follows by nducton frst on n and then, for each n, on n. COROLLARY. Let g be the poston of 1 n a word n TŽ n,.. Then the correspondng lst of words g1 gn has the property that the set of words n whch the prefx g1 g1 s fxed s an nteral of consecute words and s parttoned nto subnterals n whch g s fxed too, and as we pass from subnteral to subnteral, the sequence sž g g. 1 1 of alues assumed by g s gen by sž. Ž 2, 3,...,21. and sž g1 g2 g1. Ž g 1,...,21. f g 2 j 1 1 for an even number of j, 1 j 1, Ž 2 1,..., g 1 1. f g j 2 j 1 for an odd number of j, 1 j 1. j

5 LOOP-FREE PARENTHESIS GENERATION 169 Snce sž p. s monotone for any prefx p, the lst of words g1 gn s n graylex order Ch. The successor to a gven word s determned from the followng generc algorthm for fndng the next word n any lst that s n graylex order Žmore generally, n any lst of words n whch all of the words wth a gven prefx form an nterval of adjacent words; Wa1, Wa2. : Search for the potthe largest such that g s not at ts last value n sž g g. 1 1 ; f there s a pvot, then change g to ts next value n sž g g. 1 1 ; change each g, 1 j n, to ts frst value n sž g g. j 1 j1 ; otherwse g1 gn s the last word n the lst. Rather than store the auxlary array g1 g n, we modfy the generc algorthm so that t acts on each Dyc word drectly. For each from n down to 1, f there s an even number of lberal 1 s to the left of 1, then 1 s movng rght and ts last poston s 2 1, and otherwse 1 s movng left and ts last poston s adjacent to 1 Ž 1 never moves left.. If all of these 1 s are n ther last postons, then the word s the last one. Otherwse, the pvot s the ndex of the frst 1 we fnd that s not n ts last poston; 1 gets moved by one poston to ts rght or left, dependng on the drecton n whch t s movng, and f n, then all of the 1 s to ts rght must be moved to ther frst postons. By examnng Fg. 2, the reader can verfy that only 1 and 1 actually have to be moved: each 1, 1 j j 1, s n ts rghtmost poston, whch was ts last poston and becomes ts frst poston after 1 and 11 have moved, because the number of lberal 1 s to ts left always changes by 1. Whle scannng a Dyc word from rght to left to fnd the pvot, we could scan the word from left to rght each tme we encounter a 1 to count the lberal 1 s to ts left, mang the worst-case tme complexty On Ž 2.. To mae the algorthm run n On Ž. tme, we mantan a varable Odd that s true f the total number of lberal 1 s n a gven word s odd. Before scannng a word, we ntalze a Boolean varable Left to Odd, and we 1 1 FIG. 2. The suffx begnnng wth 1 before and after 1 moves. There are four cases to consder, dependng upon whether 1 moves rght or left, and whether or not t moves to or from ts rghtmost poston. The arrows ndcate the drecton of moton of each 1.

6 170 TIMOTHY R. WALSH change t whenever a lberal 1 s encountered, so that as each 1 s examned, the varable Left s true f that 1 s movng leftward. We also mantan a Boolean varable Last, whch becomes true f the current word turns out to be the last one n TŽ n,.. If we are generatng TŽ n,., then we generate frstžtž n,.. from Ž 2.2., we ntalze Last to false, and we ntalze Odd to true f n 2 and n, snce, by Ž 2.2., frstžtž n,.. has only one lberal 1 Ž 1. maxž3, 1., and otherwse we set Odd to false, because then frstžtž n,.. has no lberal 1 s. Then we execute the updatng algorthm Next gven n Fg. 3 untl Last becomes true. If we are generatng all of the Dyc words of length 2n by generatng TŽ n 1, 1. wthout the prefx 10, then the updatng algorthm can be executed as s: 12, whch s free but at ts rghtmost poston n TŽ n 1, 1., s smply renamed 1, whch s now fxed. 1 FIG. 3. Fndng the successor to a gven word n TŽ n,. n On Ž. worst-case tme and OŽ. 1 extra space.

7 LOOP-FREE PARENTHESIS GENERATION 171 To generate TŽ n, n. TŽ n, n 1. TŽ n,2. T R Ž n, 1., we begn as f we were generatng TŽ n, n., set n, and then execute the updatng algorthm wth followng changes: change lne 5 to whle 1 do ; n lne 7, change f j 2 1 to f j 2 1 and ; nsert the followng code after lne 22: f then We pass from TŽ n,. to TŽ n, 1. or to T R Ž n,1.. 4 pj 1 1; 1; Only 1 moves. 4 Odd not Odd; The drectons of moton are reversed for any ; f 1, then 11 s a new lberal 1; f 1, then the lst s reversed. 4 return end f; 4. FINDING THE NEXT WELL-FORMED PARENTHESIS STRING IN OŽ. 1 WORST-CASE TIME We store three auxlary arrays: the array g1g2 gn of postons of the occurrences of 1, the array dd 1 2 d n, where d 1 f 1 s movng rght Ž g s ncreasng. and 0 otherwse, and the array ee 1 2 en ntroduced n BER and Eh to fnd the pvot n OŽ. 1 worst-case tme. Once the pvot s found, the Dyc word and all of the auxlary arrays are also updated n OŽ. 1 tme. From Fg. 2 t would appear that d 2, d 3,...,dn must all be changed from 1 to 0; however, each of these dm changes when t s needed Ž when m becomes the pvot., so that the total wor done to generate each new Dyc word s OŽ. 1. For the frst word n TŽ n,., the values of g 1, g 2,..., gn can be obtaned from Ž 2.2., d 1, f maxž 1, 3., and 0 otherwse, and e for all. After generatng the frst word of TŽ n,., we execute the algorthm QucNext to Fg. 4 untl Last becomes true. To generate all of the length 2n Dyc words as the lst TŽ n, n. TŽ n, n 1. TŽ n,2. T R Ž n, 1., we begn as f we were generatng TŽ n, n., ntalze to n, and then execute QucNext wth the followng changes: change lne 3 to f 1 then ; change lne 22 to f then We pass from TŽ n,. to TŽ n, 1. or to T R Ž n,1.. 4 pj 1 1; 1; Only 1 moves. dm, m, wll be updated later. 4 else f n then

8 172 TIMOTHY R. WALSH FIG. 4. Fndng the successor to a gven word n TŽ n,. n OŽ 1. worst-case tme and On Ž. extra space. The algorthms of Fgs. 3 and 4, plus algorthms for fndng the poston of a Dyc word n TŽ n,. and the Dyc word n a gven poston n TŽ n,. n On Ž. arthmetc operatons, were programmed n Modula-2 and tested. The two generaton algorthms ran n roughly the same total tme, whch s often the case when average-case OŽ. 1 -tme algorthms are made loop-free. The elmnaton of auxlary arrays s not a sgnfcant advantage for a computer, but does mae the algorthm easer to execute by hand, so that both algorthms have ther advantages. The lstngs for these programs and other related results are contaned n a research report Wa1 that s avalable from the author on request. The observant reader may notce the

9 LOOP-FREE PARENTHESIS GENERATION 173 comments n Fg. 4 explanng the array ee e ntroduced n BER 1 2 n and Eh and used there and n many other places to fnd the pvot n OŽ. 1 worst-case tme, and may want to prove, or to obtan a proof ŽWa1, Wa2., that t wors for any lst of words n whch all of the words wth the same proper prefx g g g form an nterval of adjacent words Ž whether or not the lst s n graylex order., provded only that n each such proper subnterval of the lst, g assumes at least two dstnct values. REFERENCES BER J. R. Btner, G. Ehrlch, and E. M. Rengold, Effcent generaton of the bnary reflected Gray code and ts applcatons, Comm. ACM 19 Ž 1976., Ch P. J. Chase, Combnaton generaton and graylex orderng, Proceedngs of the 18th Mantoba Conference on Numercal Mathematcs and Computng, Wnnpeg, 1988, Congressus Numerantum 69 Ž 1989., Eh G. Ehrlch, Loopless algorthms for generatng permutatons, combnatons, and other combnatoral confguratons, J. ACM 20 Ž 1973., PR A. Prosurows and F. Rusey, Generatng bnary trees by transpostons, J. Algorthms 11 Ž 1990., LvBR J. M. Lucas, D. R. van Baronagen, and F. Rusey, On rotatons and the generaton of bnary trees, J. Algorthms 15 Ž 1993., vb D. R. van Baronagen, A loopless algorthm for generatng bnary tree sequences, Inform. Process. Lett. 39 Ž 1991., Wa1 T. R. Walsh, A Smple Sequencng and Ranng Method That Wors on Almost all Gray Codes, Research Report 243, Department of Mathematcs and Computer Scence, Unversty of Quebec at Montreal, Aprl Wa2 T. R. Walsh, Gray codes for nvolutons, submtted for publcaton.