The variance for partial match retrievals in k-dimensional bucket digital trees

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1 The variance for partial match retrievals in k-dimensional ucket digital trees Michael FUCHS Department of Applied Mathematics National Chiao Tung University January 12, 21 Astract The variance of partial match queries in k-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis eing long and complicated. In this paper, we introduce a new and much easier approach. Moreover, our approach works for k-dimensional PATRICIA tries, k-dimensional digital search trees and ucket versions as well. 1 Introduction and Results Data structures for storing and retrieving multidimensional data are of vital importance in several areas of computer science such as design of data ase systems, graphics algorithms, etc. One possile class of such data structures was introduced in [6] and is ased on digital trees. The common feature of this class is that the data is composed of infinite -1 strings. We assume throughout this work that these strings are generated y a symmetric Bernoulli source, i.e., every it is generated independently and with the same proaility. We will study stochastic properties of the resulting random trees. We will first descrie the data structures in the aove mentioned class. Therefore, we start with k- dimensional tries. Assume that we have n records, denoted y R 1,..., R n, of k-dimensional data, i.e., every record consists of k infinite -1 strings, e.g., for the i-th record we have ( ) R i,1 = R [1] i,1, R[2] i,1, R[3] i,1,...,. ( ) R i,k = R [1] i,k, R[2] i,k, R[3] i,k,.... We shuffle the aove k sequences to otain a one-dimensional record R i ( ) R i = R [1] i,1,..., R[1] i,k, R[2] i,1,..., R[2] i,k, R[3] i,1,..., R[3] i,k,.... Then, we recursively construct the trie from the data R 1,..., R n as follows: if n = 1, we place the only record into the root which is considered an external node; if n 2, then the root ecomes an (empty) internal node and the records are either directed to the left or to right sutree according to whether their 1

2 1 1 D D 2 1 D 2 D 3 D D 4 D 4 1 D 4 D 1 D 3 D 1 D 3 Figure 1: A 2-dimensional trie, PATRICIA trie and digital search tree uild from the data data D 1 D 2 D 3 D 4 R i, R i, first it is or 1; finally, the sutrees are uild recursively y the same procedure, where the first it is removed; see Figure 1 for an example. Since the input data was random this yields a random tree which is is called k-dimensional random trie of size n. A close variant of k-dimensional tries are k-dimensional PATRICIA tries. Here, a trie is constructed from the shuffled data R 1,..., R n as aove and then multiple one-way ranching is suppressed; again see Figure 1 for an example. This yields a more alanced tree improving the overall performance of tries. The corresponding random tree is called k-dimensional random PATRICIA trie of size n. A final type of multidimensional digital trees is given y the k-dimensional digital search tree. Here, the tree is constructed from R 1,..., R n as follows: the first record is placed in the root; all other records are directed to the left or right sutree according to whether their first it is or 1; finally, the sutrees are constructed y the same principle again with the first it of the records removed; see Figure 1 for an example. So, in contrast to tries and PATRICIA tries, no distinction etween internal and external nodes is necessary for digital search trees. The resulting random tree will e called k-dimensional random digital search tree of size n. Note that all the trees introduced aove have the common feature that nodes only hold at most one record. If we allow nodes to hold up to 1 records, then the resulting trees are called k-dimensional ucket digital trees. In this paper, we will study the cost of partial match retrievals in k-dimensional ucket digital trees. Therefore, consider a partial match query which is a k-dimensional vector R = (R 1,..., R k ) with some of its coordinates a string of -1 its and others unspecified. For instance, for k = 2, we might have R 1 = (,,,,...), R 2 = (, 1, 1,,...), Then, a shuffled record R is produced as efore. For the aove example this yields R = (,,, 1,, 1,,,...). The partial match query asks now for the retrieval of all records in the tree with R eing used as search query. Here, or 1 in R means either going to the left or right sutree of the current node, whereas means 2

3 that we have to proceed with our search in oth sutrees. The cost of such a partial match query will e measured y the numer of nodes visited (where we only consider internal nodes for tries and PATRICIA tries); so for the query R aove, we have a cost of 2 for the trie and Patricia trie of Figure 1 and a cost of 3 for the digital search tree. Before going on, we will fix some notation which we are going to use throughout the work. First, it should e clear that under our random model, the cost of a partial match query only depends on the partial match pattern q which is a k-tuple of symols from S, }, where the i-th coordinate is S if R i is specified and otherwise, We will fix such a q throughout this work and denote the numer of entries in q y u, where we assume that < u < k. Furthermore, we will consider cyclic shifts of the entries of q y one position to the left which will e denoted y q ; more generally, q (l) will denote the cyclic shift of the entries of q y l position to the left. Also, we will associate to a partial match pattern q an infinite sequence (δ 1, δ 2, δ 3,...) with δ i = 1 if q i mod k = S and δ i = 2 otherwise. Finally, we denote the random variale descriing the cost of the partial match query y X q,n for all three types of ucket digital trees of size n and ucket size 1 (for the sake of simplicity, we suppress the index ). In this paper, we will e concerned with stochastic properties of X q,n. Therefore, let us recall what is known aout this random variale. First, the mean value of X q,n for k-dimensional tries was investigated in [1] where the authors proved that E(X q,n ) n u/k P 1 (log 2 n 1/k ) with P 1 (z) a one-periodic function whose Fourier expansion was given in [1] as well. Similar results were susequently proved in [3] for k-dimensional ucket digital tries, k-dimensional PATRICIA tries, and k- dimensional digital search trees, too. As for the variance, it was conectured in [4] that for k-dimensional tries Var(X q,n ) n u/k P 2 (log 2 n 1/k ) (1) with P 2 (z) a one-periodic function. The authors proved this conecture for k = 2 in [4]. The general case was then settled in [7]. In this paper, we will give a new and more simpler approach to the proof of (1). Moreover, our approach works for k-dimensional ucket tries as well. In order to state our result, we need some notation. Set h q (z) = 2δ 1 e (z)e z Lq (z/2) + e (z)e z e (z) 2 e 2z, where e (z) = 1 + z + z 2 /2! + + z /! and L q (z) = exp z} n E(X q,n)z n /n!. Theorem 1. We have, with and Var(X q,n ) = n u/k P T 2 (log 2 n 1/k ) + O(n 2u/k 1 ) P T 2 (z) = where ω r = u/k + 2πir/(kL) and L = log 2. r= c T r e 2πirz c T r = 1 δ 1 δ l 2 ωrl z ωr 1 hq (l)(z)dz, kl Note that it is not immediately clear why the integral in the aove result should converge. This will, however, follow from the proof elow. With slightly more work, the Fourier coefficients can e further simplified. 3

4 Corollary 1. We have, ( ( c T r = Γ( ω ( ωr ) r) + δ(2 ωr ) 2 ωr kl ( )( l + ωr + l + where l +1 δ(z) = δ 1 δ z, σ(z 1, z 2 ) = ( )( ) ωr ) , 2 = 1 )( ωr ) ) 2 1 l σ(2 ωr, 2 l )), (2) l 1 2 lk+u 1, 2 = δ 1 δ z 1 1 z 2 2. For instance, for k = 2, s = 1, and q = (, S), the value of c T ecomes (1 + 2) ( π ( ) ) 1/2 (l 1)(l + 1/2)2 l , 2 ln 2 8 l 1 2 l+1/2 l 2 where the last approximated was computed with Maple. This value coincides with the one given in [4]. Note that the expression given in the latter paper is slightly different; we leave it as an exercise to the reader to show that they are the same. Next, our approach can also e straightforwardly applied to k-dimensional ucket PATRICIA tries. Here, we have the same result as aove only h q (z) replaced y h q (z) =2δ 1 (e (z)e z δ 1 e (z/2)e z + (δ 1 1)e z/2 ) L q (z/2) + e (z)e z δ 1 e (z/2)e z + δ 1 e z/2 (e (z)e z δ 1 e (z/2)e z + δ 1 e z/2 ) 2. Also, a similar explicit expression for the Fourier coefficients as in Corollary 1 can e given. Since, the resulting formula is more messy we do not give details. Finally, k-dimensional ucket digital search trees are slightly more involved. Here, a new approach which was introduced in [2] can e used to otain an asymptotic expansion of the variance. In order to state our result, we again need some notation. Therefore, set and Theorem 2. We have, with and c D r = Q(s) = 1 h q (z) = 1 klγ(1 + ω r ) ( (1 s 2 ), Q l = ( ) 2 L () q (z)) l ( ) 1 2 =1 ( ) ) () ( Lq (z) 2. Var(X q,n ) = n u/k P D 2 (log 2 n 1/k ) + O(n 2u/k 1 ) P D 2 (z) = δ 1 δ l 2 ωrl r= c D r e 2πirz ( s ωr ) e zs hq (l)(z)dz + p(s) ds, (3) Q( 2s) 4

5 where p(s) = (1 + s) 1 + ( 1). s + 2 Also here, it is not clear why the aove doule integral should e convergent. This will, however, again follow from the proof elow. Moreover, the Fourier coefficients can e further simplified here, too. We will state the result for = 1. Therefore, set π(x ω 1)/(sin(πω)(x 1)) if x 1; ϕ(ω; x) = πω/ sin(πω), if x = 1. Then, we have the following corollary. Corollary 2. Assume that = 1. Then, c D 1 r = δ 1 δ l 2 ωrl klq(1)γ(1 + ω r ) where 1, 2, 3 δ q, = l ( 1) 1 δq (l), 2 δq (l), 3 2 ( 1 2 )+(1 ω r) 1 ( 1) l 2 (l+1 2 ) Q l Q1 Q 2 Q 3 ϕ(ω r ; ), l+ δ h. We conclude the introduction y giving a short sketch of the paper. In the next section, we will treat k-dimensional ucket tries. Then, in Section 3, we will riefly discuss k-dimensional ucket PATRICIA tries. Finally, in Section 5, we will prove the results for k-dimensional ucket digital search trees. 2 k-dimensional Bucket Tries h=1 Note that from the definition of k-dimensional ucket tries, we have X d X q q,n =,I n + Xq,n I n + 1, if q = (,...); X q,i n + 1, if q = (S,...), (n + 1), where I n = Binom(n, 1/2), (X n) is an independent copy of (X n ) with X n d = X n, and X q, = X q,1 = = X q, =. Poissonization. Let P q (z, y) denote the Poisson generating function of E(expX q,n y}), i.e., P q (z, y) = e z n Then, we otain from the aove distriutional recurrence E(e Xny ) zn n!. P q (z, y) = e y Pq (z/2, y) δ 1 + e (z)e z (1 e y ). Next, y taking first and second derivatives with respect to y and setting y =, we otain the following functional equation for the Poisson generating function of the mean (denoted y L q (z)) L q (z) = δ 1 Lq (z/2) + 1 e (z)e z 5

6 and for the Poisson generating function of the second moment (denoted y M q (z)) 2 M q (z) = M q (z/2) + 4 L q (z/2) + 2 L q (z/2) e (z)e z, if q = (,...); M q (z/2) + 2 L q (z/2) + 1 e (z)e z, if q = (S,...). In order to go from the Poisson generating function ack to the original quantity, we have introduced in [2] the notation of J S -admissiility. Recall that f(z) is called J S -admissile if the following two conditions hold (where here and throughout this work, ɛ will denote a small constant whose value might change from one appearance to the next). (I) There exists an α R such that f(z) = O ( z α ) uniformly for arg(z) ɛ. (O) We have, uniformly for ɛ arg(z) π. f(z) := e z f(z) = O ( e (1 ɛ) z ) The importance of this notation is that if f(z) is the Poisson generating function of, say f n, then J S - admissiility of f(z) implies that the following asymptotic relation etween f n and its Poisson generating function holds f n f () (n) τ (n)! with τ (n) = n e z (see [2] for a more thorough discussion of this). In our context, J S -admissile is easily checked via the following result. Proposition 1. Assume that we have f q (l)(z) = δ l+1 fq (l+1)(z/2) + g q (l)(z), ( l < k), where all involved functions are entire. Moreover, assume that g q (l)(z) is J S -admissile for l < k. Then, f q (l)(z) is J S -admissile for l < k. Proof. We only show how to prove (I). Therefore, we start y iterating the recurrence. This yields f q (z) = 2 u fq (z/2 k ) + δ 1 δ l g q (l)(z/2 l ). Now set Then, y the assumptions, we otain B q (r) := max f q (z). z =r, arg(z) ɛ B q (r) 2 u Bz (r/2 k ) + O (r α ). Next, we define K q (r) y K q (r) = 2 u Kq (r/2 k ) + O (r α ). 6

7 Then, B q (r) K q (r). Moreover, we immediately otain y iteration r α, if α > u/k; K q (r) = r u/k log r, if α = u/k; r u/k, if α < u/k. This proves our claim. Using this result it is easily proved that oth L q (z) and M q (z) are J S -admissile. Also, note that we have L q (z) = O ( z u/k), Mq (z) = O ( z 2u/k) (4) uniformly as z and arg(z) ɛ. Next, we define the poissonized variance as Then, y a straightforward computation Ṽ q (z) = M q (z) L q (z) 2. Ṽ q (z) = δ 1 Ṽ q (z/2) + h q (z), where h q (z) was defined in the introduction. The importance of the poissonized variance is apparent from the following result which is proved from the J S -admissiility of L q (z) and M q (z). Proposition 2. As n, Var(X q,n ) = Ṽq(n) + O ( n 2u/k 1). Proof. First note that from the proof of Proposition 1 and (4), we have Ṽ q (z) = O ( z u/k) uniformly for z and arg(z) ɛ. Next, consider Var(X q,n ) = E(Xq,n) 2 (E(X q,n )) 2 ( M q () (n) τ (n)! L () q (n) τ (n)! ) 2 Ṽq(n) n L q(n) 2 n 2 Ṽ q (n) + smaller order terms. Using the aove ounds and Ritt s theorem (see [5]) demonstrates the claim. Asymptotic Expansion of L q (z). We will first look at the mean value which is needed in the proof of Corollary 1. Therefore, y using iteration as in the proof of Proposition 1, we otain L q (z) = 2 u Lq (z/2 k ) + δ 1 δ l g q (l)(z/2 l ), where g q (l)(z) = 1 e (z)e z. Next, we use Mellin transform. Therefore, note that y (4) and the trivial ound L q (z) = O ( z +1) as z, the Mellin transform of L q (z) exists in the strip 1, u/k. Applying Mellin transform to the aove functional equation then yields M [ L q (z); ω] = Γ(w) ( ) w + δ 1 2 ωk+u 1 δ l 2 ωl, R(ω) 1, u/k. (5) 7

8 Moreover, y inverse Mellin transform and shifting the line of integration to the right, we have where P 1 (z) = r= L q (z) z u/k P 1 (log 2 z 1/k ), (z ), (6) c r e 2πirz, c r = Γ( ω r) kl ( ωr + ) k 1 δ 1 δ l 2 ωrl. Note that due to the fast decay of (5) along vertical lines, (6) more generally holds uniformly for z and arg(z) π/2 ɛ. Asymptotic Expansion of Ṽq(n). functional equation, we have Using iteration as aove and applying Mellin transform to the resulting M [Ṽq(z); ω] = ωk+u δ 1 δ l 2 ωl M [ h q (l)(z); ω], R(ω) 1, u/k. Now, from (4), we have that as z h q (l)(z) = O(z β ) (l) for any β > and l < k. Oviously, h q (z) = O(z +1 ) as z for l < k. Hence, the Mellin transform of h q (l)(z) exists in the strip 1,. Our claimed result follows from this y inverse Mellin transform and shifting the line of integration to the right. Simplification of the Fourier Coefficients. Therefore, we concentrate on The main task is the evaluation of z ωr 1 ( 2δ l+1 e (z)e z Lq (l+1)(z/2) + e (z)e z e (z) 2 e 2z ) dz. z ωr 1 e (z)e z Lq (l+1)(z/2)dz, the remaining parts eing easy. Note that due to (5), we have ( ) M [ L q (l+1)(z/2); σ] = 2σ Γ(σ) σ + δ 1 2 σk+u q (l+1)(2 σ ), where Now, y inverse Mellin transform z ωr 1 e (z)e z Lq (l+1)(z/2)dz = δ q (l+1)(z) = δ l+2 δ l++1 z. = ( ) ( ) ( 2 σ Γ(σ) σ σk+u ( )( σ + ωr σ + 8 ) δ q (l+1)(2 σ ) z ωr σ 1 e (z)e z dzdσ ) 2 σ Γ(σ)Γ( ω r σ) 1 2 σk+u δ q (l+1)(2 σ )dσ,

9 where the outer integral is along the line R(σ) =. Finally, y shifting the line of the integration to the left, we otain the asolute convergent series z ωr 1 e (z)e z Lq (l+1)(z/2)dz = Γ( ω r ) ( )( )( ) l + ωr + l + ωr 2 l δ q (l+1)(2 l ) l 1 2. lk+u l +1 Collecting everything and standard computation yields the claimed result. 3 k-dimensional Bucket PATRICIA Tries Here, from the definition of Patricia tries, we have X q,i n + Xq,n I n, if I n, n}, X q X q,n =,I n + Xq,n I n + 1, otherwise, X q,i n, if I n = n, X q,i n + 1, otherwise, if q = (,...); if q = (S,...), (n + 1), where notation and initial conditions are as in the previous section. From this we then otain for the Poisson generating function of the mean (with notation as efore) where L q (z) = δ 1 Lq (z/2) g q (z) g q (z) = e (z)e z + δ 1 e (z/2)e z δ 1 e z/2, and the Poisson generating function of the second moment 2 M q (z) = M q (z/2) + 2 L q (z/2) 2 + 4(1 e z/2 ) L q (z/2) g q (z), if q = (,...); M q (z/2) + 2(1 e z/2 ) L q (z/2) g q (z), if q = (S,...). Moreover, we have for the poissonized variance Ṽ q (z) = δ q Ṽ q (z/2) + h q (z), where h q (z) was defined in the introduction. The remaining analysis now proceeds from these functional equational equations as in the previous section. 4 k-dimensional Bucket Digital Search Trees Here, we will use a slightly different approach compared with the approach aove. This approach was introduced in [2]. Since the current situation is similar to the examples discussed in [2], we will only give a sketch and refer the interested reader to the latter paper for more details. Again, we start from a distriutional recurrence for X q,n which for the current situation reads as follows X d X q q,n+ =,I n + Xq,n I n + 1, if q = (,...); (n ), X q,i n + 1, if q = (S,...), where the notation is as efore and initial conditions are given y X q, = and X q,1 = = X q, 1 = 1. 9

10 Poissonization. We again define Then, P q (z, y) = e z n E(e Xq,ny ) zn n!. ( ) P q (z, y) = e y Pq (z/2, y) δ 1. Taking derivatives yields for the Poisson generating function of mean and second moment (denoted as efore) ( ) L () q (z) = δ 1 Lq (z/2) + 1 (7) and ( ) M q () 2 (z) = M q (z/2) + 4 L q (z/2) + 2 L q (z/2) 2 + 1, if q = (,...); M q (z/2) + 2 L q (z/2) + 1, if q = (S,...). The first step is again to show that L q (z) and M q (z) are J S -admissile. Therefore, we need the following result which is proved y a reduction to the trie case (see [2] for similar results). Proposition 3. Assume that we have ( ) f () (z) = δ q (l) l+1 fq (l+1)(z/2) + g q (l)(z), ( l < k), where all involved functions are entire and at z =. Moreover, assume that g q (l)(z) is J S -admissile for l < k. Then, f q (l)(z) is J S -admissile for l < k. This together with the closure properties from [2] then shows that L q (z) and M q (z) are J S -admissile. Next, we consider the poissonized variance Ṽq(z) = M q (z) L q (z) 2. An easy computation proves that ( ) Ṽ () (z) = δ 1 Ṽ q (z/2) + h q (z), where h q (z) was defined in the introduction. Then, from the J S -admissiility of L q (z) and M q (z), we otain as for tries the following result. Proposition 4. As n, Var(X q,n ) = Ṽq(n) + O ( n 2u/k 1). Asymptotic Expansion of L q (z). Again, we first consider the mean value. Therefore, note that in comparison to tries, it is not possile to iterate (7). Hence, we first apply Laplace transform and otain (s + 1) L [ L q (z); s] = 2δ 1 L [ L q (z); 2s] + (s + 1) 1 /s. (8) Next, we normalize with Q(s) from the introduction. Therefore, set L q (s) = L [ L q (z); s]/q( s) and Ḡ(s) = (s + 1) 1 /(Q( 2s) s). Then, L q (s) = 2δ 1 Lq (2s) + Ḡ(s). 1

11 Now, we can iterate and otain L q (s) = 2 k+u Lq (2 k s) + 2 l δ 1 δ l Ḡ(2 l s). The next step is again Mellin transform. Therefore, note that the Mellin transform of L q (s) exists in a non-trivial strip. Moreover, due to the growth properties of Q(s) from [2], the Mellin transform of Ḡ(s) exists in the strip 1,. Applying Mellin transform yields M [ L q (s); ω] = M [Ḡ(s); ω] 1 2 k ωk+u δ 1 δ l 2 l ωl, R(ω) 1 + u/k,. Next, y inverse Mellin transform and shifting the line of integration to the left, we otain L q (z) r= c r s 1 u/k 2πir/(kL), (s ). Since, Q( s) = 1 + O( s ) as s, the same asymptotic expansion holds for L [ L q (z); s] as well. Finally, y inverse Laplace transform (see [2] for technical details), we have L q (z) z u/k P 1 (log 2 z 1/k ), (z ), where P 1 is a computale, 1-periodic function. A more careful analysis shows that the aove asymptotic expansion holds uniformly for z and arg(z) π/2 ɛ. Asymptotic Expansion of Ṽq(z). Here, we proceed as aove and otain V q (s) = 2 k+u Vq (2 k s) + δ 1 δ l 2 l Hq (l)(2 l s), were V q (s) = L [Ṽq(z); s]/q( s) and H q (l)(s) = (L [ h q (l)(z); s] + p(s))/q( 2s) with Now, oserve that h q (z) = p(s) = (1 + s) 1 + ( 1). s + 2 O(z 2u/k 2 ), if z ; O(1), if z +, where the first ound follows from the previous paragraph and Ritt s theorem (see [5]) and the second ound is trivial. This together with the growth properties of Q(s) then in turn yields O(1/s), if s ; H q (s) = O(s β ), if s +, where β > is an aritrary constant. Consequently, the Mellin transform of H q exists in the strip 1,. The remaining proof of Theorem 2 proceeds then as in the previous paragraph. 11

12 Simplification of the Fourier Coefficients for = 1. First, y iteration of (8), L [ L q (z); s] = 1 s where δ q, = l=1 δ l. Next, y partial fraction expansion, where L [ L q (z); s] = 1 s δ q,l = Consequently, y inverse Laplace transform δ q, (s + 1) (2 s + 1), ( 1) l 2 ( l+1 2 ) δ q, = 1 δ q,l, (2 l s + 1)Q l Q l s (2 l s + 1)Q l l ( 1) 2 (+1 2 ) Q δ q,+l. L q (z) = l δ q,l Q l (1 e z/2l ). This implies L q(z) = l δ q,l 2 l Q l e z/2l, L q (z) 2 = l,h δ q,l δq,h z/2l z/2h e. 2 l+h Q l Q h Plugging this into (3) (note that for = 1, we have h q (z) = L q(z) 2 ) and using 1 Q( 2s) = 1 ( 1) 2 ( 2) Q(1) Q (s + 2 ) together with some standard computation proves the claim. References [1] P. Flaolet and C. Puech (1986). Partial match retrival of multidimensional data, J. ACM, 33, [2] H.-K. Hwang, M. Fuchs, V. Zacharovas (21). Asymptotic variance of random digital search trees, sumitted. [3] P. Kirschenhofer and H. Prodinger (1994). Multidimensional digital searching - alternative data structures, Random Struc. Algorithms, 5:1, [4] P. Kirschenhofer, H. Prodinger, W. Szpankowski (1993). Multidimensional digital searching and some new parameters in tries Int. J. Found. Comput. Scie., 4, [5] F. W. J. Olver (1974). Asymptotics and Special Functions, Academic Press, Computer Science and Applied Mathematics. [6] R. L. Rivest (1976). Partial-match retrieval algorithms, SIAM J. Comput., 5, [7] W. Schachinger (1995). The variance of a partial match retrieval in a multidimensional symmetric trie, Random Struc. Algorithms, 7:1,