c-perfect Hashing Schemes for Binary Trees, with Applications to Parallel Memories

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1 -Perfet Hashing Shemes for Binary Trees, with Appliations to Parallel Memories (Extended Abstrat Gennaro Cordaso 1, Alberto Negro 1, Vittorio Sarano 1, and Arnold L.Rosenberg 2 1 Dipartimento di Informatia ed Appliazioni R.M. Capoelli, Università di Salerno, 84081, Baronissi (SA Italy {alberto,vitsa,ordaso}@dia.unisa.it 2 Dept. of Computer Siene, University of Massahusetts Amherst Amherst, MA 01003, USA rsnbrg@s.umass.edu Abstrat. We study the problem of mapping tree-strutured data to an ensemble of parallel memory modules. We are given a onflit tolerane, and we seek the smallest ensemble that will allow us to store any n- vertex rooted binary tree with no more than tree-verties stored on the same module. Our attak on this problem abstrats it to a searh for the smallest -perfet universal graph for omplete binary trees. We onstrut suh a graph whih witnesses that only O ( (1 1/ 2 (n+1/(+1 memory modules are needed to obtain the required bound on onflits, and we prove that Ω ( 2 (n+1/(+1 memory modules are neessary. These bounds are tight to within onstant fators when is fixed as it is with the motivating appliation. 1 Introdution Motivation. This paper studies the effiient mapping of data strutures onto a parallel memory system (PMS, for short whih is omposed of several modules that an be aessed simultaneously (by the proessors of, say, a multiproessor system. Mapping a data struture onto a PMS poses a hallenge to an algorithm designer, beause suh systems typially are single-ported: they queue up simultaneous aesses to the same memory module, thereby inurring delay. The effetive use of a PMS therefore demands effiient mapping strategies for the data strutures that one wishes to aess in parallel strategies that minimize, for eah memory aess, the delay inurred by this queuing. Obviously, different mapping strategies are needed for different data strutures, as well as for different ways of aessing the same data struture. As a simple example of the seond point, one would map the verties of a omplete binary tree quite differently when optimizing aess to levels of the tree than when optimizing aess to root-to-leaf paths. The preeding onsiderations give rise to the problem studied here. Given a data struture represented by a graph, and given the kinds of subgraphs one

2 2 wants easy aess to (alled templates, our goal is to design a memory-mapping strategy for the items of a data struture that minimizes the number of simultaneous requests to the same memory module, over all instanes of the onsidered templates. Our Results. This paper presents the first strategy for mapping a binary-tree data struture onto a parallel memory in suh a way that any rooted subtree an be aessed with a bounded number of onflits. Our results are ahieved by proving a (more general result, of independent interest, about the sizes of graphs that are almost perfet universal for binary trees, in the sense of [5]. Related Work. Researh in this field originated with strategies for mapping two-dimensional arrays into parallel memories. Several shemes have been proposed [3, 4, 8, 10] in order to offer onflit-free aess to several templates, inluding rows, olumns, diagonals and submatries. While the strategies in these soures provide onflit-free mappings, suh was not their primary goal. The strategies were atually designed to aommodate as many templates as possible. Strategies for mapping tree strutures onsidered onflit-free aess for one elementary template either omplete subtrees or root-to-leaf paths or levels [6, 7] or ombinations thereof [2], but the only study that even approahes the universality of our result i.e., aess to any subtree is the C-template ( C for omposite of [1], whose instanes are ombinations of different numbers of distint elementary templates. The mapping strategy presented for the C- templates instanes of size K, with M memory modules, ahieves O(K/M + onflits. Bakground. For any binary tree T : Size (T is its number of verties; Size (T, i is its number of level-i verties; Hgt (T is its height (= number of levels. For any positive integer, a -ontration of a binary tree T is a graph G that is obtained from T via the following steps. 1. Rename T as G (0. Set k = Pik a set S of verties of G (k that were verties of T. Remove these verties from G (k, and replae them by a single vertex that represents the set S. Replae all of the edges of G (k that were inident to the removed verties by edges that are inident to S. The graph so obtained is G (k Iterate step 2 some number of times. A graph G n = (V n, E n is -perfet-universal for the family T n of n-vertex binary trees if every -ontration of an n-vertex binary tree is a labeled-subgraph of G n. By this we mean the following. Given any -ontration G (a = (V, E of an n-vertex binary tree T, the fat that G (a is a subgraph of G n is observable via a mapping f : V V n for whih eah v V is a subset of f(v. The simplest 1-perfet-universal graph for the family T n is the height-n omplete binary tree, T n, whih is defined by the property of having all root-to-leaf paths of ommon length Hgt (T n = n. The perfet-universality of T n is witnessed by the identity map f of the verties of any n-vertex binary tree to the verties of T n. Of ourse, T n is a rather ineffiient perfet-universal graph for

3 3 T n, sine it has 2 n 1 verties, whereas eah tree in T n has only n verties. It is natural to ponder how muh smaller a 1-perfet-universal graph for T n an be. The size in number of verties of the smallest suh graph is alled the perfetion number of T n and is denoted Perf(T n. In [5], Perf(T n is determined exatly, via oinident lower and upper bounds. Theorem 1 ([5] Perf(T n = (3 (n mod 2 2 (n 1/2 1. In this paper, we generalize the study of storage mappings for trees in [5] by allowing boundedly many ollisions in storage mappings. We thus relax the one to one demands of perfet hashing to boundedly many to one. 2 Close Bounds on Perf (T n Our study generalizes Theorem 1 to -perfet-universality, by defining Perf (T n, for any positive integer, to be the size of the smallest -perfet-universal graph for T n ; in partiular, Perf 1 (T n = Perf(T n. We first derive our upper bound on Perf (T n by expliitly onstruting a graph G that is -perfet-universal for T n. Theorem 2 For all integers n and > 1, ( Perf (T n < / 2 (n+1/(+1 + O(n. (1 We onstrut our -perfet-universal graph G via an algorithm A that olors the verties of T n in suh a way that the natural vertex-label-preserving embedding of any n-vertex tree T into T n (whih witnesses T s being a subgraph of T n never uses more than verties of any given olor as homes for T s verties. When we identify eah like-olored set of verties of T n i.e., ontrat eah set to a single vertex in the obvious way we obtain the -perfet-universal graph G, whose size is learly an upper bound on Perf (T n. Algorithm A proeeds in a left-to-right pass along eah level of T n in turn, assigning a unique set of olors, C i, to the verties of eah level i, in a roundrobin fashion. A thereby distributes the 2 i level-i verties of T n equally among the C i level-i verties of G. Clearly, thus, Size (G = i C i. The remainder of the setion is devoted to estimating how big the sets C i must be in order for G to be -perfet-universal for T n. Auxiliary results. We begin by identifying some speial subtrees of T n. Let m and i be integers suh that 3 log m i n, and let x be a binary string of length i log m. Consider the subtree T (i,m (x of T n onstruted as follows. 1. Generate a length-(i log m path from the root of T n to vertex x. 2. Generate the smallest omplete subtree rooted at x that has at least m leaves; easily, this subtree all it T (x has height log m. 3 All logarithms are to the base 2.

4 4 3. Finally, prune the tree so onstruted, removing all verties and edges other than those needed to inorporate the leftmost m leaves. Easily, every tree T (i,m has exatly m leaves, all of length i. Lemma 1. The trees T (i,m are the smallest rooted subtrees of T n that have m leaves at level i. Proof. (Sketh For any level j of any binary tree T, we have Size (T, j Size (T, j. One verifies easily from our onstrution that the trees T (i,m ahieve this bound with equality. Lemma 2. For all i and m, 2m + i log m 1 Size ( T (i,m 2m + i 1. (2 Proof. (Sketh We merely sum the sizes of the omplete subtree on T (x, plus the size of the various paths needed to apture them. Thus we have Size ( T (i,m = (i hpm m p m (3 where p m is the number of omplete subtree on T (x and h pm 1 is the height of the smallest omplete subtree on T (x. We obtain the bound of the lemma from the exat, but nonperspiuous, expression (3, from the fats that there is at least one 1 in the binary representation of m and that p m 1. Let us now number the verties at eah level of T n from left to right and say that the distane between any two suh verties is the magnitude of the differene of their numbers. Lemma 3. For any integers 0 i n and 0 δ i < i, the size of the smallest rooted subtree of T n that has m leaves at level i of T n, eah at distane 2 δi from the others, is no smaller than Size ( T (i,m + (m 1δi. Proof. If two verties on level i are distane 2 δi apart, then their least ommon anestor in T n must be at some level i (δ i + 1. It follows that the smallest rooted subtree T of T n that satisfies the premises of the lemma must onsist of a opy of some T (i δi,m, with m leaves at level i δ i, plus m vertex-disjoint paths (like tentales from that level down through eah of the δ i levels i δ i + 1, i δ i + 2,..., i of T n. Equation (3 therefore yields: Size (T mδ i + Size ( T (i δi,m = (m 1δi + Size ( T (i,m. The upper bound proof. We propose, in the next two lemmas, two oloring shemes for A, eah ineffiient on its own (in the sense that neither yields an upper bound on Perf (T n that omes lose to mathing our lower bound, but whih ombine to yield an effiient oloring sheme. We leave to the reader the simple proof of the following Lemma.

5 5 Lemma 4. Let T n be olored by algorithm A using 2 δi olors at eah level i. If eah 2 δi def κ i = 2 i /, then any rooted n-vertex subtree of T n engenders at most ollisions. Lemma 5. Let T n be olored by algorithm A using 2 δi olors at eah level i. If eah 4 2 δi def λ i = 1 ( n + log( + 1 i 4 exp2, (4 then any rooted n-vertex subtree of T n engenders at most ollisions. Proof. We onsider a shallowest tree T that engenders +1 ollisions at level i of T n. Easily, the offending tree T has +1 leaves from level i of T n. By the design of Algorithm A, these leaves must be at distane 2 δi from one another. We an, therefore, ombine Lemma 3, the lower bound of (2 (both with m = + 1, and (4 to bound from below the size of the offending tree T. Size (T Size ( T (i,+1 + δi n 2 + log( + 1 i 2( i log( n + 1. Sine we are only about ( n-vertex subtrees of T n, the lemma follows. A bit of analysis verifies that the respetive strengths and weaknesses of the oloring shemes of Lemmas 4 and 5 are mutually omplementary. This omplementarity suggests the ploy of using the κ i -sheme to olor the top of T n and the λ i -sheme to olor the bottom. A natural plae to divide the top of T n from the bottom would be at a level i where κ i λ i. Using this intuition, we hoose level i def n + 1 = + log to be the first level of + 1 the bottom of T n. (Sine n, trivial alulations show that n > i. Using our hybrid oloring sheme, then, we end up with a -perfet-universal graph 1 n 1 G suh that Size (G = κ j + λ k. Evaluating the two summations j=0 k=i in turn, we find the following, under the assumption that > 1 (sine the ase = 1 is dealt with definitively in [5]; see Theorem 1. 1 j=0 1 κ j = 2 j / j=0 1 exp2 ( n (2 j / + 1 = 1 2i + i 1 j=0 + log + O(n < 2 2 (n+1/(+1 + O(n(5 4 To enhane the legibility of powers of 2 with ompliated exponents, we often write exp2(x for 2 X.

6 6 n 1 n 1 ( n + log( + 1 k λ k = exp2 2 k=i k=i i + 1 < 2 2 (n k/ 2 exp2 k=i < ( + 11/ 2 ( ( n n + 1 ( log n 1 k=i 2 (n k/ < 2 1 1/ 2 (n+1/(+1. (6 The bounds (5, 6 yield the laimed upper bound (1 on Perf (T n. Beause of spae limitations, we defer the proof of the following lower bound to the omplete version of this paper. Theorem 3 For all > 1 and all n, Perf (T n > exp2 ( n Aknowledgment. The researh of A.L. Rosenberg was supported in part by US NSF Grant CCR Referenes 1. V. Auletta, S. Das, A. De Vivo, M.C. Pinotti, V. Sarano, Optimal tree aess by elementary and omposite templates in parallel memory systems. IEEE Trans. Parallel and Distr. Systs., 13, V. Auletta, A. De Vivo, V. Sarano, Multiple Template Aess of Trees in Parallel Memory Systems. J. Parallel and Distributed Computing 49, 1998, P.Budnik, D.J. Kuk. The organization and use of parallel memories. IEEE Trans Comput., C-20, 1971, C.J.Colbourn, K.Heinrih. Conflit-free aess to parallel memories. J. Parallel and Distributed Computing, 14, 1992, F.R.K. Chung, A.L. Rosenberg, L. Snyder. Perfet storage representations for families of data strutures. SIAM J. Algebr. Disr. Meth., 4, 1983, R. Creutzburg, L. Andrews, Reent results on the parallel aess to tree-like data strutures the isotropi approah, Pro. Intl. Conf. on Parallel Proessing, 1, 1991, pp S.K. Das, F. Sarkar, Conflit-free data aess of arrays and trees in parallel memory systems, Pro. 6th IEEE Symp. on Parallel and Distributed Proessing, 1994, pp D.H.Lawrie. Aess and alignment of data in an array proessor. IEEE Trans. on Computers, C-24, 1975, R.J. Lipton, A.L. Rosenberg, A.C. Yao, External hashing shemes for olletions of data strutures. J. ACM, 27, 1980, K.Kim, V.K.Prasanna. Latin Squares for parallel array aess. IEEE Trans. Parallel and Distributed Systems, 4, 1993, A.L. Rosenberg and L.J. Stokmeyer, Hashing shemes for extendible arrays. J. ACM, 24, 1977, A.L. Rosenberg, On storing ragged arrays by hashing. Math. Syst. Th., 10, 1976/77,