On Parity based Divide and Conquer Recursive Functions

Transcription

1 O Parity based Divide ad Coquer Recursive Fuctios Sug-Hyu Cha Abstract The parity based divide ad coquer recursio trees are itroduced where the sizes of the tree do ot grow mootoically as grows. These o-mootoic recursive fuctios called fog () ad fog () are strictly less tha liear, o() but greater tha logarithm, Ω(log ). Properties of fog () such as o-mootoicity, upper ad lower bouds, etc. are examied ad prove. These fuctios are useful to aalyze computatioal complexities of certai algorithms, especially problems of fidig various properties of -ary divide ad coquer trees or size balaced -ary trees. Several iteger sequeces based o the divide ad coquer recursive relatios are ewly discovered as well. Keywords: divide ad coquer, Nomootoic growth fuctio, Aalysis of Algorithms 1 Itroductio I the aalysis of algorithms, the computatioal time complexity fuctios, T (), are ofte limited to mootoically icreasig [1] or evetually o-decreasig [2] growth fuctios, i.e., T ( 1 ) T ( 2 ) if there exist such that < 1 < 2. Liear ad logarithm fuctios are examples of mootoically icreasig fuctios where their asymptotic relatioships ca be discussed. Here some o-mootoic fuctios are itroduced to aalyze computatioal complexities of certai algorithms. The divide ad coquer techique, which solves problems by breaig them ito two or more smaller subproblems, is oe of most popular algorithm desig techiques [1]. This techique produces the implicit size balaced -ary tree [3] whose sizes of its childre trees are the same or differ by oly oe. This divide ad coquer -ary tree gives itriguig iteger sequeces with regard to its iput size [3]. Neil Sloae s Olie Ecyclopedia of Iteger Sequeces [4] cotais a zoo of divide ad coquer iteger sequeces. Yet, several ew divide ad coquer iteger sequeces geerated from the o-mootoic recursive fuctios are discovered i this article. Most classical recursive divide ad coquer algorithms have their computatioal time complexities i a stadard Mauscript received Jue 25, 212; revised Auguest 1, 212. S.-H. Cha is with the Computer Sciece Departmet, Pace Uiversity, New Yor, NY, 138 USA Tel/Fax: / scha@pace.edu (see scha). recursive form give i (1) where a is the umber of subproblems ad /b is the size of the sub-problems. T () = at ( ) + f() (1) b These typical recursive divide ad coquer algorithms form a full a-ary recursio tree ad the asymptotically equivalet growth fuctio ca be determied by the Master theorem [1]. For example, cosider the problem of fidig the value of the th power of c, c where is a positive iteger. Two differet divide ad coquer recursive algorithms give i (2) ad (3) ca solve the problem. { c, if = 1 pow(c, ) = pow(c, 2 ) pow(c, (2) 2 ), if > 1 c, if = 1 pow(c, ) = pow(c, 2 )2, if is eve (3) pow(c, 2 )2 c, if is odd The former algorithm i (2) always call two half sized subproblems which result i the full biary divide ad coquer recursio tree as show i Figure 1 (a). The latter algorithm i (3) always call oly oe subproblem of the half size which results i the uary divide ad coquer recursio tree as show i Figure 1 (b). The algorithm i (3) is called the biary method which appeared before 2 B.C. [5, 6]. Assumig that multiplicatio operatio taes a costat time, the computatioal time complexities for algorithms i (2) ad (3) have the stadard recursive forms, T () = 2T (/2) + 1 ad T () = T (/2) + 1, respectively. They are asymptotically liear Θ() ad logarithm Θ(log ), respectively, which ca be trivially show by the Master Theorem [1]. Cosider a algorithm i (4) which calls oe subproblem if is eve but calls twice if is odd. c, if = 1 pow(c, ) = pow(c, /2) 2, if eve (4) pow(c, 2 ) pow(c, 2 ), if odd The computatioal time complexity of the algorithm i (4) is strictly less tha liear, o() but greater tha logarithm, Ω(log ) ituitively as show i Fig. 1 (c). Ufortuately, the Master theorem does ot help to fid the

2 (a) full biary (b) Uary (c) Biary Θ() Θ(log()) Θ(fog()) Figure 1: Three ids of divide ad coquer recursio tree. exact asymptotic closed formula for this recursive relatio i (4). Eve more geeral divide ad coquer recurrece solvig theorem, such as the Ara-Bazzi method [7], caot hadle this simple ad rather elemetary divide ad coquer relatio, either. Albeit the algorithms i (2) ad (4) should ot be used for the problem of evaluatig iteger powers, it is a good example to realize that there exists a recursive fuctio that is strictly less tha liear ad greater tha or equals to the logarithm as figuratively explaied i Fig 1. This mysterious parity based divide ad coquer recursive fuctio shall be called fog (). I [3], the problem of fidig the sum of heights of a size balaced -ary tree or a divide ad coquer recursio tree was cosidered. This article shall prove that the computatioal complexity of solvig this problem i [3] is Θ(fog()) ad ivestigate other problems whose computatio complexities are Θ(fog()). The subsequet sectios are costructed as follows. The sectio 2 formally defies the fog () ad its properties are examied ad prove. Aother similar leave-oe-out divide ad coquer recursive fuctio called fog () is itroduced as well. I sectio 3, various problem examples whose computatioal ruig time complexities are either Θ(fog ()) or Θ( fog ()) are give. Fially, the sectio 4 cocludes this wor. 2 Divide ad coquer recursive fuctio 2.1 Defiitio fog () Computatioal ruig time complexity of the algorithm (4) ca be represeted as a recursio tree where the ode has oe or two childre depedig o its parity Fig. 2 (a) eumerates the first 16 recursio trees. Let the size of this biary recursio tree be fog 2 () as defied i (5). 1, if = 1 fog 2 () = fog 2 ( 2 ) + 1, if %2 = fog 2 ( 2 ) + fog 2( 2 ) + 1, if %2 (5) fog 2 () is ot mootoically growig fuctio but fluctuates as show i Fig 2 (b) (a) fog 2 () recursio trees fog 2 () log 2 () (b) fog 2 () graph i compariso to ad log. Figure 2: fog 2 () recursio trees ad graph. The cocept i (5) ca be geeralized for the -ary divide ad coquer cases where each ode has up to childre. The sizes of -sub trees follow the iteger partitio ito balaced parts defied i (6). ( {}}{ ) BIP(, ) =,...,,,..., } {{ } =% (6) For examples, BIP(17, 3) = (6, 6, 5) ad BIP(22, 4) = (6, 6, 5, 5). If is divisible by, all childre have the uique size. If is ot divisible by, there are exactly two ids of childre, i.e., ad as defied i (6). The biary divide ad coquer algorithms i (2) ad (3) ca be geeralized to -divide ad coquer algorithms which have Θ(log ) uary recursio tree ad Θ() - ary recursio tree, respectively. The algorithm i (4) ca be also geeralized to -divide ad coquer where it oly calls oe sub-problem if the size is divisible by or calls two sub-problems if ot. This algorithm has a biary recursio tree regardless of. The computatioal time

3 I the best case, the iput size is divisible by ad its sub-problem s size is also divisible by all the way to the base case. This case is Θ(log ). This best case sceario occurs at = m where m is a positive iteger as depicted i Fig. 4. fog2() 15 log2() (a) f og2 () (a) first 21 f og3 () 6 fog3() 5 log3() (b) best - uary (c) typical - biary (d) worst case - full biary (b) f og3 () 2 fog () 4 log4() 15 Figure 3: f og3 () recursio trees. 1 5 complexity of this algorithm ca be defied as i (7). 1, if = 1 2, if f og () = ) + 1, if % = f og ( f og ( ) + f og ( ) + 1, if % = (7) For the example of terary ( = 3) case, Fig 3 (a) shows the first 21 recursio trees. I the worst case, it has a full biary tree as show i Fig 3 (d) while it has a uary tree i the base case as give i Fig 3 (b). The Table 1 lists first 1 iteger sequeces for f og2 (), f og3 (), ad f og4 (). 2.2 Properties of f og () Property 1 No-mootoicity: f og (1 ) f og (2 ) if 1 < 2. The first property of f og () is straightforward as show i Figs 2 4. Aother obvious property of f og () is its lower boud. Property 2 The lower boud: f og () = Ω(log ()) (c) f og4 () 1 fog4() 8 log () (d) f og5 () The fuctio i (7) is amed as f og for two reasos. The first oe is to be cosistat with logarithm fuctio itroduced by Napier ad the secod reaso is depicted i Fig 4 where f og () iteger values are ploted as dots istead of lies. This o-mootoic fuctio loos lie fogs (e) f og1 () Figure 4: various f og () plots. The worst case or the upper boud of f og () ca be derived usig the Master theorem. Theorem 1 The upper boud: f og () = O(N log 2 ). Proof: I the worst case, the iput size ad its all sub-childre s sizes are ot divisible by. As depicted i Fig 3 (d), it forms a full biary tree ad thus T () = 2T (/) + 1. Usig the Master theorem case 2, T () = Θ(log 2 )

4 Table 1: Θ(fog ()) Iteger Sequeces. Iteger sequece for = 1,, 1 2 1, 2, 4, 3, 7, 5, 8, 4, 11, 8, 13, 6, 14, 9, 13, 5, 16, 12, 2, 9, 22, 14, 2, 7, 21, 15, 24, 1, 23, 14, 19, 6, 22, 17, 29, 13, 33, 21, 3, 1, 32, 23, 37, 15, 35, 21, 28, 8, 29, 22, 37, 16, 4, 25, 35, 11, 34, 24, 38, 15, 34, 2, 26, 7, 29, 23, 4, 18, 47, 3, 43, 14, 47, 34, 55, 22, 52, 31, 41, 11, 43, 33, 56, 24, 61, 38, 53, 16, 51, 36, 57, 22, 5, 29, 37, 9, 38, 3, 52, 23, 3 1, 2, 2, 4, 4, 3, 5, 5, 3, 7, 7, 5, 9, 9, 5, 8, 8, 4, 9, 9, 6, 11, 11, 6, 9, 9, 4, 11, 11, 8, 15, 15, 8, 13, 13, 6, 15, 15, 1, 19, 19, 1, 15, 15, 6, 14, 14, 9, 17, 17, 9, 13, 13, 5, 14, 14, 1, 19, 19, 1, 16, 16, 7, 18, 18, 12, 23, 23, 12, 18, 18, 7, 16, 16, 1, 19, 19, 1, 14, 14, 5, 16, 16, 12, 23, 23, 12, 2, 2, 9, 24, 24, 16, 31, 31, 16, 24, 24, 9, 22, 4 1,2,2,2,4,4,4,3,5,5,5,3,5,5,5,3,7,7,7,5,9,9,9,5,9,9,9,5,8,8,8,4,9,9,9,6,11,11,11,6,11,11,11,6,9,9,9,4,9,9,9,6,11,11,11,6,11,11,11,6,9,9,9,4,11,11,11,8,15,15,15,8,15,15,15,8,13,13,13,6,15,15,15,1,19,19,19,1,19,19,19,1,15,15,15,6,15,15,15,1, For = 3 ad = 4, log 3 2 =.639 ad log 4 2 =.5, respectively. The tigher upper boud for the biary case is give as follows. Theorem 2 The upper boud for = 2: fog 2 () = O(N log 2 (φ) ). (a) fog 2 () recursio trees. Proof: A odd umber is always divided ito odd ad eve parts. A eve umber is either a sum of two smaller eve umbers i the best case or two smaller odd umbers i the worst case. I the worst case, we have a Fiboacci tree as show i Fig 5. I the stadard divide ad coquer form, a = 1f h+2f h+1 f h +f h+1 = 1+ f h+1 f h+2 = 1+ 1 φ = φ. Sice T () = φt (/2) + 1 belogs to the case 1 i the Master Theorem, T () = Θ(N log 2 (φ) ) Θ(N.6942 ) (b) fog 3 () recursio trees. Figure 6: fog () recursio trees. Corollary 1 fog p() fog () for a positive iteger, p. Proof omitted. Figure 5: Worst case Fiboacci tree. The followig obvious iequality for the logarithm fuctio, i.e., log 1 () log 2 () for 1 > 2, does ot apply to the fog fuctios. Fallacy 1 fog 1 () fog 2 () for 1 > 2. Proof: While there are cases that the claim is true, e.g., (fog 3 (64) = 18) < (fog 2 (64) = 7), there are also umerous couter examples lie (fog 3 (96) = 16) (fog 2 (96) = 9), (fog 4 (63) = 9) (fog 3 (63) = 7), etc. Hece, fog 1 () fog 2 () for 1 > 2. Despite the Fallacy 1, fog is gettig lower ad fadig as icreases, i.e., the lower ad upper bouds of fog 1 () are lower tha those of fog 2 () for 1 > 2 as show i Fig 4. The followig corollary 1 is a exceptioal case of Fallacy Leave-oe-out divide ad coquer, fog () There are two ids of divide ad coquer recursio trees. Oe is the stadard oe where is the umber of leaf odes. The other is the leave-oe-out divide ad coquer where is the total umber of both iteral ad leaf odes. I the leave-oe-out divide ad coquer tree, umber of subtrees have their sizes of either ( 1)/ or ( 1)/. The media split tree [8] is a example of the biary leave-oe-out divide ad coquer tree. I [3], -ary leave-oe-out divide ad coquer are categorized as simply -ary size-balaced tree. The parity based divide ad coquer fuctio defied i (7) ca be altered to aalyze the leave-oe-out divide ad coquer algorithms. Let s deote this altered fuc-

5 Table 2: fog () Iteger Sequeces. Iteger sequece for = 1,, 1 2 1, 2, 2, 4, 3, 5, 3, 7, 5, 8, 4, 9, 6, 9, 4, 11, 8, 13, 6, 14, 9, 13, 5, 14, 1, 16, 7, 16, 1, 14, 5, 16, 12, 2, 9, 22, 14, 2, 7, 21, 15, 24, 1, 23, 14, 19, 6, 2, 15, 25, 11, 27, 17, 24, 8, 24, 17, 27, 11, 25, 15, 2, 6, 22, 17, 29, 13, 33, 21, 3, 1, 32, 23, 37, 15, 35, 21, 28, 8, 29, 22, 37, 16, 4, 25, 35, 11, 34, 24, 38, 15, 34, 2, 26, 7, 27, 21, 36, 16, 41, 3 1, 2, 2, 2, 4, 4, 3, 5, 5, 3, 5, 5, 3, 7, 7, 5, 9, 9, 5, 8, 8, 4, 9, 9, 6, 11, 11, 6, 9, 9, 4, 9, 9, 6, 11, 11, 6, 9, 9, 4, 11, 11, 8, 15, 15, 8, 13, 13, 6, 15, 15, 1, 19, 19, 1, 15, 15, 6, 14, 14, 9, 17, 17, 9, 13, 13, 5, 14, 14, 1, 19, 19, 1, 16, 16, 7, 18, 18, 12, 23, 23, 12, 18, 18, 7, 16, 16, 1, 19, 19, 1, 14, 14, 5, 14, 14, 1, 19, 19, 1, 4 1, 2, 2, 2, 2, 4, 4, 4, 3, 5, 5, 5, 3, 5, 5, 5, 3, 5, 5, 5, 3, 7, 7, 7, 5, 9, 9, 9, 5, 9, 9, 9, 5, 8, 8, 8, 4, 9, 9, 9, 6, 11, 11, 11, 6, 11, 11, 11, 6, 9, 9, 9, 4, 9, 9, 9, 6, 11, 11, 11, 6, 11, 11, 11, 6, 9, 9, 9, 4, 9, 9, 9, 6, 11, 11, 11, 6, 11, 11, 11, 6, 9, 9, 9, 4, 11, 11, 11, 8, 15, 15, 15, 8, 15, 15, 15, 8, 13, 13, 13, tio as fog (). fog () = 1, if = 1 2, if + 1 fog ( 1 ) + 1, if ( 1)% = fog ( 1 ) + fog ( 1 ) + 1, if ( 1)% (8) Figure 6 shows the first 14 fog 2 () ad 25 fog 3 () recursio trees ad the the table 2 lists first 1 iteger sequeces for fog 2 (), fog 3 (), ad fog 4 (). (a) biary (b) terary Figure 7: iclusive heights of stadard divide ad coquer tree. 3 Applicatios I [3], the problem of fidig the sum of heights of a size balaced -ary tree, Z () or a leave-oe-out divide ad coquer recursio tree was cosidered. Although the value ca be computed i liear time by traversig the tree i the depth first order maer, it ca be computed faster i Θ( fog ()) because we eed to traverse oly oe sub-tree if all sub-trees sizes are the same or oly two sub-trees otherwise. Here some other problems whose computatioal complexities are either Θ(fog ()) or Θ( fog ()) are examied. Let Z () be a stadard divide ad coquer recursio tree. Let H(Z ()) be the sum of each ode s height i Z () ad it is defied recursively as i (9)., if 1 H(Z ()) = log + H ( Z ( ) ) +( ) H ( Z ( ) otherwise ) (9) Note that = % as defied i (6). Table 3 shows the first 1 iteger sequeces of H(Z 2 ()) ad H(Z 3 ()). The direct defiitio based recursive algorthm i (9) would tae Θ( log 2 ). However, if the followig coditio i (1) is added to (9), the value ca be computed i Θ(fog ()). H(Z ()) = log + H (Z ( ) ) if % = (1) plicit tree data structures are more popular tha the sum of exclusive heights or simply heights. Hece, Fig 7 eumerates the first eight stadard biary ad terary divide ad coquer trees with each ode cotaiig the iclusive height. With a little modificatio to (9) ad (1), the sum of the iclusive heights ca be computed i Θ(fog ()) as well. Table 4 shows the first oe hudred iteger sequeces of the sum of iclusive heights of biary ad terary trees. Similarly, computig the several other properties of stadard or leave-oe-out divide ad coquer trees would have their computatioal complexities of Θ(fog ()) or Θ( fog ()) aturally. For example, the path legth of a rooted tree [9], P (T ) is aother importat property ad P (Z ()) ad P (Z ()) ca be computed i Θ(fog 2()) ad Θ( fog 2 ()), respectively. Strahler umberig of a biary tree, T, S(T ) is aother importat property of a biary tree. S(T ) i (11) ca be computed i liear time if oe uses the postorder traversal [1]., if T is empty S(T ) = max(s(t L ), S(T R )), if S(T L ) S(T R ) S(T L ) + 1, if S(T L ) = S(T R ) (11) I Neil Sloae s Olie Ecyclopedia of Iteger Sequeces [4], the sum of iclusive heights [3] of various ex- Yet, S(Z 2 ()) ad S(Z 2()) ca be computed much faster

6 Table 3: Sum of heights of stadard divide ad coquer tree iteger sequeces. Iteger sequece for = 1,, 1 2, 1, 3, 4, 7, 9, 1, 11, 15, 18, 2, 22, 23, 24, 25, 26, 31, 35, 38, 41, 43, 45, 47, 49, 5, 51, 52, 53, 54, 55, 56, 57, 63, 68, 72, 76, 79, 82, 85, 88, 9, 92, 94, 96, 98, 1, 12, 14, 15, 16, 17, 18, 19, 11, 111, 112, 113, 114, 115, 116, 117, 118, 119, 12, 127, 133, 138, 143, 147, 151, 155, 159, 162, 165, 168, 171, 174, 177, 18, 183, 185, 187, 189, 191, 193, 195, 197, 199, 21, 23, 25, 27, 29, 211, 213, 215, 216, 217, 218, 219, 3, 1, 1, 3, 4, 5, 5, 5, 5, 8, 1, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 22, 25, 28, 3, 32, 34, 36, 38, 4, 41, 42, 43, 44, 45, 46, 47, 48, 49, 5, 51, 52, 53, 54, 55, 56, 57, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 63, 67, 71, 74, 77, 8, 83, 86, 89, 91, 93, 95, 97, 99, 11, 13, 15, 17, 19, Table 4: Sum of iclusive heights of stadard divide ad coquer tree iteger sequeces. Iteger sequece for = 1,, 1 2 1, 4, 8, 11, 16, 2, 23, 26, 32, 37, 41, 45, 48, 51, 54, 57, 64, 7, 75, 8, 84, 88, 92, 96, 99, 12, 15, 18, 111, 114, 117, 12, 128, 135, 141, 147, 152, 157, 162, 167, 171, 175, 179, 183, 187, 191, 195, 199, 22, 25, 28, 211, 214, 217, 22, 223, 226, 229, 232, 235, 238, 241, 244, 247, 256, 264, 271, 278, 284, 29, 296, 32, 37, 312, 317, 322, 327, 332, 337, 342, 346, 35, 354, 358, 362, 366, 37, 374, 378, 382, 386, 39, 394, 398, 42, 46, 49, 412, 415, 418, 3 1, 4, 5, 9, 12, 15, 16, 17, 18, 23, 27, 31, 34, 37, 4, 43, 46, 49, 5, 51, 52, 53, 54, 55, 56, 57, 58, 64, 69, 74, 78, 82, 86, 9, 94, 98, 11, 14, 17, 11, 113, 116, 119, 122, 125, 128, 131, 134, 137, 14, 143, 146, 149, 152, 153, 154, 155, 156, 157, 158, 159, 16, 161, 162, 163, 164, 165, 166, 167, 168, 169, 17, 171, 172, 173, 174, 175, 176, 177, 178, 179, 186, 192, 198, 23, 28, 213, 218, 223, 228, 232, 236, 24, 244, 248, 252, 256, 26, 264, 268, usig (12) i Θ(fog 2 ()) ad Θ( fog 2 ()), respectively. Refereces [1] T. H. Corme, C. E. Leiserso, ad R. L. Rivest,, if Z() is empty Algorithm, MIT Press, Cambridge, Massachusetts, S(Z()) = max(s(z( 2 )), S(Z( ))), if %2 S(Z( 2 )) + 1, if %2 = [2] D. S. Mali ad M. K. Se, Discrete Mathematic (12) Structures: Theory ad Applicatios, Thomso Course Techology, 24 4 Coclusios Both stadard ad leave-oe-out divide-ad-coquer recursio trees are pervasive i computer sciece. This article itroduced the parity based divide ad coquer recursio trees. The fuctios of sizes of stadard ad leaveoe-out divide ad coquer trees were deoted as fog () ad fog (), respectively. Properties ad some applicatios of these fuctios were discussed. We strogly believe that there are a plethora of applicatios of these fuctios i various problems which use the divide ad coquer algorithms. Aother cotributio of this article is discoverig ew iteger sequeces. All iteger sequeces i Tables 1 4 are surprisigly ot curretly i Neil Sloae s Olie Ecyclopedia of Iteger Sequeces [4]. These are importat ad pervasive iteger sequeces which ivolve divide ad coquer algorithms. The divide ad coquer recursio is oe of the widely studied areas ad Master theorem ad Ara-Bazzi method attempt to geeralize these recursive formulae. However, they caot hadle fog () ad fog () fuctios. Studyig the more geeralized parity based recursive forms is oe of the future wors. [3] S.-H. Cha, O Iteger Sequeces Derived from Balaced -ary trees, i Proceedigs of America Coferece o Applied Mathematics, Cambridge, MA, pp , Ja 212. [4] N. J. A. Sloae. The O-Lie Ecyclopedia of Iteger Sequeces. [5] D. E. Kuth, The Art of Computer Programmig, vol 2: Semiumerical Algorithms, 2d ed. Addiso- Wesley, 1981 [6] B. Datta ad A. N. Sigh, History of Hidu Mathematics, vol 1, Bombay, 1935 [7] M. Ara ad L. Bazzi, O the solutio of liear recurrece equatios, Computatioal Optimizatio ad Applicatios, vol. 1(2), pp , [8] B. A. Sheil, Media Split Trees: A Fast Looup Techique for Frequetly Occurrig Keys, Comm. ACM, vol. 21,. 11, pp , [9] T. C. Hu ad K. C. Ta, Path Legth of Biary Search Trees, SIAM Joural o Applied Mathematics, vol. 22,. 2, pp , [1] P. Kruszewsi, A ote o the Horto-Strahler umber for radom biary search trees, Iformatio Processig Letters, vol. 69,. 1, pp. 4751, 1999.