The Height of List-tries and TST

Transcription

1 Discrete Mathematics an Theoretical Computer Science (subm.), by the authors, 1 rev The Height of List-tries an TST N. Broutin 1 an L. Devroye 1 1 School of Computer Science, McGill University, 3480 University Street, H3A 2K6 Montreal, Canaa. nbrout,luc}@cs.mcgill.ca. receive 17th February 2007, revise 21 st May 2007, accepte tomorrow. We characterize the asymptotics of heights of the trees of e la Brianais (1959) an the ternary search trees (TST) of Bentley an Segewick (1997). Our proof is base on a new analysis of the structure of tries that istinguishes the bulk of the tree, calle the core, an the long trees hanging own the core, calle the spaghettis. Keywors: Tries, branching process, height, e la Brianais, TST. 1 Introuction Tries are ata structures use to manipulate an store strings by taking avantage of the igital character of wors. They were introuce by e la Brianais (1959). Apparently, the term of trie was coine by Frekin (1960) as a part of the wor retrieval. Their properties an uses are reviewe by Knuth (1973) an more recently by Szpankowski (2001). Consier n sequences of characters (or strings) from an alphabet A = 1,...,}. Each one of the sequences carves a path in an infinite roote -ary position tree T where the chilren of each noe are labele with the characters of A: starting from the root, the characters are use in orer to move own the tree. If all the sequences are istinct, the corresponing paths in T are istinct as well. The trie T n is efine to be the smallest subtree of T such that the paths corresponing to the sequences are istinct within T n. In this paper, we are intereste in tries built from n inepenent sequences. Each sequence is an infinite sequence of inepenent an ientically istribute (i.i..) characters istribute like A, where PA = i} = p i. We assume without loss of generality that 1 > p 1 p 2 p > 0. One of the quantities of interest uner this moel is the height H n of a of a trie built from n inepenent such sequences. It is well known that the height satisfies where H n logn n 2 log(1/q) Q = i=1 in probability, (1) p 2 i. (2) Research of the authors was supporte by NSERC Grant A3456 an a James McGill fellowship. subm. to DMTCS c by the authors Discrete Mathematics an Theoretical Computer Science (DMTCS), Nancy, France

2 2 N. Broutin an L. Devroye (See Régnier, 1981; Devroye, 1984; Pittel, 1985; Szpankowski, 1991, 2001). The trie is only an abstract ata structure, that is, it oes not specify the implementation (see Clément et al., 1998, 2001). The usual implementation of a trie uses an array for the branching structure of a noe (Frekin, 1960). Although this always ensures constant-time shunting of the strings, the space require may become an issue for large alphabets: many pointers woul be left unuse. To avoi this, one can replace the array by variable size structures. De la Brianais (1959) propose to use linke-lists, an we shall call the implementation a list-trie. More recently, builing on early ieas of Clampett (1964), Bentley an Segewick (1997) evelope an elegant structure base on binary search trees. It is known as the bst-trie, ternary search trie or TST for short. List-tries an TST aim at a trae-off between storage space an spee, an the access time to chilren is no longer constant. In particular, the height of the tree an the worst-case search time are ifferent in general. Both these structures may be seen as high-level tries whose eges are weighte to reflect the internal low-level structure of a noe (see Figure 2). This point of view has been taken by Clément, Flajolet, an Vallée (1998, 2001) who thoroughly analyze these hybri implementations of tries. In particular, they analyze the average size an average epth. The question of the worst-case search time in hybri-tries was left open. The worst-case search time is then given by the weighte height of the tree, we shall call it the height in the following. This paper aresses the question of the (weighte) height of hybri tries, an of list-tries an TST, in particular. 2 Weighte tries 2.1 An embeing to construct weighte tries In this section we propose an embeing to construct the weighte tries. Strictly speaking, since we are intereste in the weighte height, we only construct the sequence of weighte epths of the noes. Note that our embeing is only one way to buil tries with the esire istribution. We will see in Section 4 that hybri tries, like list-tries an TST, can be seen as weighte tries. Our construction emphasizes an unerlying structure consisting of inepenent ranom variables. However, in the couple tries built from the embeing, the ranom variables are epenent in general. Our construction goes in two steps: we first give a shape to the tree, an then assign the weights to the eges base on the shape. Consier the istribution p 1,..., p } over the alphabet A. THE SHAPE OF THE TRIE. The shape of the trie is simply an orinary trie: Each string efines an infinite path in T. The carinality N u of a noe u T is the number of strings whose paths in T intersect u. Then, the trie T n is constructe by pruning T own to every noe of carinality at most one. The sequences are istinct with probability one, an the strings efine istinct paths in T. Therefore, the trie T n is almost surely finite. The tree T n constitutes the shape of the weighte trie. We efine E i = log p i. For the ege e between u an its i-th chil in T, we let p e = p i an E e = log p e. DIFFERENT TYPES OF NODES. There are 2 types of noes, each type being characteristic of the branching structure of the noe. The branching structure of every noe u T n is escribe by a -vector τ u : if u 1,...,u are the chilren of u, then we efine τ u = ( 1[N u1 1],1[N u2 1],...,1[N u 1] ). The vector τ u inicates which one of the eges own u are part of some path in T n, an influences the cost of accessing the chilren of u.

3 The Height of List-tries an TST 3 THE WEIGHTS. Consier a sequence of ranom vectors Z τ : τ 0,1} }, where Z τ = (Z1 τ,...,zτ ). The vectors Z τ escribe the cost of accessing the chilren of the noes. We assume that for all τ 0,1}, maxz τ i : 1 i }, an for all types τ such that τ is a permutation of (1,0,...,0), Z τ = (1,...,1). Each noe of T is assigne an inepenent copy of the whole sequence. Consier a noe u T, an its sequence Z τ }. Weights are associate with the eges to u s chilren base on its type τ u. The ege e i between u an its i-th chil in T is given the weight Z ei = Z τ u i = τ 0,1} Z τ i 1[τ u = τ]. We use the notations Z τ i an Z e interchangeably. It shoul always be clear whether a subscript refers to an inex or an ege. Let π(u) be the set of eges on the path from u up to the root in T. The weighte epth of a noe u is efine by D u = e π(u) Z e. A tree that may be constructe by this proceure is calle a weighte trie. We are intereste in the weighte height of T n : H n = maxd u : u T n } = maxd u : N u 2} + 1. Surprisingly, the first asymptotic term of H n epens on two parameters only: the istribution p 1,..., p }, an Z = Z (1,...,1). In particular, the first orer asymptotics of H n stays the same if we moify Z τ,τ 0,1} } in such a way that Z remains unchange. This may be explaine using the structure of a trie. 2.2 The structure of a trie The profile of orinary tries can be explaine by istinguishing a so-calle core, that constitutes the bulk of the trie, an spaghetti-like trees hanging own the core (Broutin an Devroye, 2007a). This istinction is justifie by the following Lemma, which is at the heart of our proofs: Lemma 1. Let T n be a ranom trie. Let m = m(n) such that m = o(logn). There exists ω = ω(n), as n such that: (a) with probability 1 n ω, all the noes u with N u log 2 n have τ u = (1,...,1), (b) the maximum number of noes with N u m(n) an τ u (1,...,1) on a path own the root is o(logn) with probability 1 n ω, an (c) the maximum number of noes with N u m(n) an egree at least two on a path own the root is at most m(n) = o(log n). THE CORE OF A TRIE. What we call the core here shoul not be confuse with the graph-theoretic core, which happens to be empty for trees (see e.g., Janson et al., 2000). The core of the trie is efine to be the set of noes u T for which N u m(n), for m(n) an m(n) = o(logn). The core is enote by C. Since m, a noes in the core has type τ = (1,..., 1) with probability 1 o(1). As a consequence, in a weighte trie, the istribution of weights in the core shoul be closely approximate by Z = Z (1,...,1). The core can be escribe by its logarithmic profile φ(α,t) = lim n logep m (t logn,αlogn) logn t,α > 0, (3)

4 4 N. Broutin an L. Devroye where P m (k,h) enotes the number of noes u, k levels away from the root with N u m(n) an D u h. In other wors, assuming for now that the limit in (3) exists, we have EP m (t logn,αlogn) = n φ(α,t)+o(1), as n. The function φ(, ) is characterize in Theorem 2. HANGING SPAGHETTIS. The spaghettis are the trees remaining when pulling out the core from the trie. They lie in the part of the trie where the noes o not have chilren any more: the types of the noes may take all the values in 0,1}. However, the weighte height of a long spaghetti is close to its unweighte height, or number of levels. To see this, observe that the noes not in the core have carinality at most m(n) = o(logn), so each spaghetti stores at most m(n) sequences. Each time the type τ is not a permutation of (1,0,...,0), i.e., the noe is truly branching, at least one string is put asie from the longest path. This can happen at most o(logn) times, an hence the heights with an without the branching noes iffer by at most o(logn). If the number of levels is Θ(logn), as is the case for the highest ones, the ifference is negligible. Both the core an the spaghettis contribute significantly to the height of a weighte trie. By figuring out what the core looks like, we can etermine when the spaghettis take over. Roughly speaking, we then know if an ege s weight can be approximate by a component of Z or simply remain unweighte. The main result of this paper is the following theorem. Theorem 1. Consier a weighte trie built from n inepenent sequences consisting of i.i.. characters istribute as A, where PA = i} = p i, 1 i. Let H n be its weighte height. Let φ(α,t) be the logarithmic weighte profile of the core of T n. Let c = sup α + φ(α,t) } log(1/q) : φ(α,t) 0, where Q = i=1 p2 i. Then H n = clogn + o(logn) in probability, as n. Remark. (a) The respective contributions of the core an spaghettis are α an φ(α,t)/log(1/q). (b) The joint between the core an the spaghettis along the longest path, i.e., the level at which the longest path leaves the core, occurs far from the bottom of the core (see Figure 1). 3 The height of weighte tries 3.1 The shape of the core Consier a weighte trie efine as in section 2. We consier m = m(n) with m(n) = o(log n). Let L k be the set of noes k levels away from the root in T. Let P m (k,h) be the number of noes u L k with D u h an N u m. Since m, for n large enough, we have m b an P m (k,h) = u L k 1[N u m,d u h]. The asymptotic properties of the expecte profile are irectly tie to large eviation theory (Dembo an Zeitouni, 1998). The ranom vector of interest here is (Z,E) = (Z K, log p K ), where K is uniform in 1,...,} an Z = Z (1,...,1) = (Z 1,...,Z ). For λ,µ R, the associate generating function of the cumulants is [ Λ(λ,µ) = loge e λz+µe].

5 The Height of List-tries an TST 5 Then, for x,y R, we efine the convex ual Λ of Λ by, Λ (x,y) = supλx + µy Λ(λ,µ)}. λ,µ Theorem 2. Let m = m(n) with m = o(logn). Let k t logn an h αlogn for some positive constants t an α. Let φ(α,t) = t log t inf Λ (x,y) : x > α t,y < 1 }. (4) t If φ(α,t) >, then EP m (k,h) = n φ(α,t)+o(1), as n. Remarks. (a) Observe that Theorem 2 justifies the efinition of φ(, ) in (3). (b) The constraint that m(n) is o(logn) is only use to ensure that the spaghettis each store o(logn) only. Theorem 2 still hols with m(n) as large as n o(1). Unlike the profile of unweighte tries (Devroye, 2002, 2005; Park et al., 2006), that of weighte tries oes not seem concentrate. However, it is log-concentrate in the sense of the following theorem, an the true profile P m (k,h) is close enough to EP m (k,h): Theorem 3. Let m = m(n) as n such that m = o(logn). Let k t logn an h αlogn for some positive constants t an α. Then, for all ε > 0, as n, P P m (k,h) n φ(t,α) ε} 0, an P P m (k,h) n φ(t,α)+ε} n ε+o(1). n 3.2 How long is a spaghetti? The behavior of the spaghettis is raically ifferent from the one observe in the core. This is because the number of strings store by a single spaghetti is at most m(n). We first look at the profile, not of a single trie, but of a forest of inepenent tries. Let T 1,T 2,...,T n be n inepenent tries. We assume that T i is a weighte trie on m i = m i (n) sequences generate by a memoryless source with istribution p 1,..., p }. Also, we assume that for all i, m/ m i m. The roots of T i, 1 i n, all lie at level zero. Then, we let P s (k,h) count the number of noes u at level k with D u h lying in any T i. Since T i is a trie, we only count the noes for which N u 2. We are intereste in EP s (k,h) when k ρlogn an h γlogn. Recall that the ifference between the number of layers an the weighte height is at most m(n) = o(logn). Hence, EP m (k,h) = o(1) if h k + Ω(logn). Also, when h k + o(logn), only the number of levels is relevant. For n large enough that m/ 2, by linearity of expectation, we have nq k EP m (k,h) nm 2 Q k, since two strings are ientical up to the k-th character with probability Q k. In other wors, we have Theorem 4. Let T i, 1 i n, be a forest of n inepenent tries. Let T i store m i = m i (n) sequences. Assume that m/ m i m for all 1 i n. Let k ρlogn an h γlogn, as n, for positive constants ρ an γ. Then, EP s n 1+ρlogQ+o(1) if ρ γ (k,h) = o(1) otherwise. as n.

6 6 N. Broutin an L. Devroye So the logarithmic profile of our forest is 1 + ρlogq. Theorem 5 claims that the eepest noe occurs when this logarithmic profile vanishes. Let H 1,...,H n be the weighte heights of T 1,...,T n, respectively, an efine S n = maxh i : 1 i n}. Then: Theorem 5. Assume that p 1 < 1. Assume that m(n) an m(n) = o(logn). Let Then, S n log 1/Q n in probability, as n. Furthermore, for every ε > 0, there exists δ > 0 such that, as n, P Sn logn 1 log(1/q) + ε } = O ( n δ). (5) α t φ(α, t) t o c α o α o + φ(αo,to) log(1/q) t o + φ(αo,to) log(1/q) 1 Fig. 1: A geometric interpretation for the height: each point (α,t, φ(α,t)) of the logarithmic profile of the core throws a line whose irection is given by (1,1,logQ). The line intersects the plane φ = 0 at (α φ(α,t)/logq,t φ(α,t)/ logq, 0). The constant c is the largest coorinate of one of these point measure along the α-axis. So spaghettis correspon to parallel rays that are originate on the profile. In particular the ray leaing to the furthest projection (longest path) is born far from the eepest level of the core. 3.3 Projecting the profile of the core In this section, we give a sketch of the proof of Theorem 1. Consier a weighte trie T n. The spaghettis are roote at a noe u C, the external noe-bounary of the core C in T n (the noes u C are the chilren of some noe v in the core, but are not themselves in the core). Then, if we write W u for the height of the subtree roote at u, we have H n = maxd u +W u : u C} = max h,k h +W u : u C L k,d u h}, where the noes in C have been split into groups epening on their level k an weighte epth h. Then, we can rewrite H n = sup h,k h + maxw u : u C,D u h,u L k }}. We have thus separate the contributions of the core from that of the spaghettis, h an maxw u : u C,D u h,u L k },

7 The Height of List-tries an TST 7 respectively. Only the latter expression requires more thought. The set of noes u C L k with D u h roughly contains roughly n φ(α,t)+o(1) noes by Theorem 2. Hence maxw u : u C,D u h,u L k } is S n with n = n φ(α,t)+o(1), that is, φ(α,t)/log(1/q) by Theorem 5. This explains why H n clogn, where c = sup α + φ(α,t) }. log(1/q) Complete proofs of the theorems can be foun in Broutin (2007) or Broutin an Devroye (2007b). Theorem 1 can be interprete by projecting the logarithmic profile φ(α, t) on the horizontal plane going through the origin. See Figure 1. 4 Application: the height of hybri tries Let A = 1,...,} be the alphabet. Let A i,1 i n} be the n strings. In the following, we istinguish the noes that constitute the high-level trie structure from the slots which make the low-level structure of a noe, whether this latter be a linke-list or a binary search tree. The high-level tree between the noes of a hybri trie is just the orinary trie. Only the low-level structure of the noes iffer from the array-base implementation. Consier a noe u T. The subtree roote at u stores a subset of the strings A i, 1 i n. Let N u 1,...,n} be the set of their inices, an write N u = N u. As in orinary tries, for a noe u at level k in T, only the k-th characters of each sequence are use. Moreover, only set A u A matters, which consists of the characters appearing at the k-th position in the sequences A i, i N u. In an hybri trie, the orer in which the strings are use to buil the trie matters, an we let σ u be the permutation A u where the characters are orere by first appearance. The internal structure of the noe u is built by successive insertions of the elements of σ u into an originally empty linke list, or binary search tree. For list-tries we then have a ranom linke list on A u an for TST a binary search tree on A u. This is shown in Figure 2. The costs of branching to a subtree is then given by the number of eges to cross insie the noe to the subtree. The istribution clearly epens on the type τ u of noe u, an when A u = 1, the cost is simply 1. This explains why hybri tries are weighte tries in the sense of Section Fig. 2: The ifferent noe structures use for the stanar (top-left), list (bottomleft) an bst-trie (right) when the orer of appearance of the characters is 3, 5, 4, 1, 2, 6. The ashe arrows represent the pointers to further levels of the trie. 4.1 List-tries In the list-trie of e la Brianais (1959), the cost of branching to a character a is just the inex of a is the permutation σ u. For every noe u, for which A u = A, σ u is istribute as the sequence (in orer) of first appearance of characters in an infinite string generate by the source. This fully escribes the istribution

8 8 N. Broutin an L. Devroye of Z = Z (1,...,1) = (Z 1,...,Z ). Then Z i is the inex of i in σ, an (Z,E) = (Z K, log p K ), where K is uniform in 1,...,}. Theorem 6. Let H n be the weighte height of a list-trie on n sequences generate by the memoryless source with probabilities p 1,..., p }. Then, H n clogn in probability, as n, where c = sup α + φ(t,α) } log(1/q) : α,t > 0, an φ(, ) is the logarithmic profile of the trie weighte with (Z,E) is escribe above. It seems ifficult to explicitly obtain φ(α,t) for general p 1,..., p }. Example: symmetric list-tries. We assume here that p 1 = p 2 = = p. Then, the permutation σ is just a uniform ranom permutation. For any λ,µ R, we have [ Λ(λ,µ) = loge e λz] + µlog = log ( i=1e iλ ) + (µ 1)log. Then, Λ (x,y) = sup λ,µ λx + µy Λ(λ,µ)}. For x [1,], there exists λ = λ(x) such that x = Λ(λ,µ) λ = i=1 ieiλ. (6) i=1 eiλ Therefore, we have ( Λ λx log (x,y) = i=1 e iλ) + log if x [1,],y = log otherwise. For instance, for = 2, we can fin an explicit expression for Λ : for x [1,], ( ) ( ( ) ) 1 x Λ 1 x 1 x 2 (x,log2) = xlog log x 2 2 x + + log2. 2 x With more manipulation, we obtain for this specific example c = c() = log( i=1 i) log 2 log, for large. Numerical values for c = c() can be foun in Table 1. Observe that c() is not monotonic in, an that the minimum height is reache for an alphabet of size four. 4.2 Ternary search trees In the ternary search trees introuce by Bentley an Segewick (1997), the implementation of a noe uses a binary search tree. The cost of branching to a character i A at a noe u is one plus the epth of i in the binary search built from the (non-uniform) ranom permutation σ u. When the noe u is of type τ u = (1,...,1), the permutation σ u is istribute as σ, the orere list of first appearances of characters in an infinite string generate by the memoryless source with istribution p 1,..., p }. Let Z i be istribute as the epth of i in the binary search tree built from σ. Then, Z is istribute as (Z 1,...,Z ) an (Z,E) = (Z K, log p K ), where K is uniform in 1,...,}. By Theorem 1, we obtain:

9 The Height of List-tries an TST c() Tab. 1: Some numerical values of c = c() characterizing the height of symmetric list-tries. Theorem 7. Let H n be the weighte height of a TST on n sequences. Let c = sup α + φ(α,t) } log(1/q) : α,t > 0, where φ(α,t) is the logarithmic profile of the core of a trie weighte by (Z,E) escribe above. Then, H n clogn in probability, as n. The ranom vector (Z,E) seems complicate to escribe for general istributions p 1,..., p. Some parameters like the average value an the variance of Z i, 1 i, have been stuie by Clément et al. (1998, 2001) an Archibal an Clément (2006). Example: Symmetric TST. We assume here that p 1 = p 2 = = p. In this case, the permutation σ is just a uniform ranom permutation. Hence, Z i is the epth of the key i in a ranom binary search tree. Observe that unlike in the case of list-tries, the Z i, 1 i, o not have the same istribution. This is easily seen, since, for instance as, EZ 1 log whereas EZ /2 2log. However, we are only intereste in the istribution of Z, that is, the epth of a uniform ranom noe. This istribution is known exactly, an is ue to Lynch (1965) an Brown an Shubert (1984): PZ = k} = 2k 1! j=k [ ], (7) j [ ] where n k enotes the Stirling number of the first kin with parameter n an k (see Segewick an Flajolet, 1996; Mahmou, 1992). Using (7), it is possible to compute the cumulant generating function Λ, an φ(α,t). Observe that the when = 2, the TST is equivalent to the list-tries. In general, one obtains c = c() = = 1 log + 1 log 2 log( i=1 1 log + 1 log 2 log 2 i 1 [ ) ] i j=i! j ( (2) (2 + 1) (3 1)!!(2 1) ) log(27/4) log 2 (see, e.g., Mahmou, 1992, p. 79). Numerical values for the constant c = c() are given in Table c() Tab. 2: Some numerical values of c = c() characterizing the height of symmetric ternary search trees.

10 10 N. Broutin an L. Devroye 5 Concluing remarks Theorem 1 is not the most general one can obtain. Also, the weights for non-branching noes Z σ, with σ a permutation of (1,0,...,0), may also be ranom instea of unit values. Then, the height satisfies H n logn n supα + γφ(α,t) : α,t > 0} in probability, where γ is a constant epening on p 1,..., p } an the weights Z σ. Finally, similar results hol for b-tries, where the leaves are allowe to contain up to b strings. These extensions an complete proofs of the theorems can be foun in Broutin (2007) or Broutin an Devroye (2007b). 6 Aknowlegement We are very grateful to Julien Clément for bringing this problem to our attention an for his insightful comments. References M. Archibal an J. Clément. Average epth in binary search tree with repeate keys. In Fourth Colloquium on Mathematics an Computer Science, pages , J. L. Bentley an R. Segewick. Fast algorithm for sorting an searching strings. In Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages , N. Broutin. Sheing New Light on Ranom Trees. Ph thesis, McGill University, Montreal, N. Broutin an L. Devroye. The core of a trie. Manuscript, 2007a. N. Broutin an L. Devroye. Weighte height of ranom tries. Manuscript, 2007b. G.G. Brown an B.O. Shubert. On ranom binary trees. Mathematics of Operations Research, 9:43 65, H. A. Clampett. Ranomize binary searching with tree structures. Communications of the ACM, 7(3): , J. Clément, P. Flajolet, an B. Vallée. The analysis of hybri trie structures. In 9th annual ACM-SIAM Symposium on Discrete Algorithms, pages , Philaelphia, PA, SIAM Press. J. Clément, P. Flajolet, an B. Vallée. Dynamical source in information theory: a general analysis of trie structures. Algorithmica, 29: , R. e la Brianais. File searching using variable length keys. In Proceeings of the Western Joint Computer Conference, Montvale, NJ, USA. AFIPS Press, A. Dembo an O. Zeitouni. Large Deviation Techniques an Applications. Springer Verlag, secon eition, L. Devroye. Laws of large numbers an tail inequalities for ranom tries an PATRICIA trees. Journal of Computational an Applie Mathematics, 142:27 37, L. Devroye. Universal asymptotics for ranom tries an PATRICIA trees. Algorithmica, 42:11 29, 2005.

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