Running Tite: Conict-Free Access of Paths Address for Correspondence: M.C. Pinotti IEI-CNR Via S. Maria, Pisa ITALY E-ai:

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1 Mappings for Conict-Free Access of Paths in Bidiensiona Arrays, Circuar Lists, and Copete Trees Aan A. Bertossi y and M. Cristina Pinotti Istituto di Eaborazione de' Inforazione Nationa Counci of Research Pisa, ITALY E-ai: fbertossi@science.unitn.it, pinotti@iei.pi.cnr.itg Abstract Since the divergence between the processor speed and the eory access rate is progressivey increasing, an ecient partition of the ain eory into utibans is usefu to iprove the overa syste perforance. The eectiveness of the utiban partition can be degraded by eory con- icts, that occur when there are any references to the sae eory ban whie accessing the sae eory pattern. Therefore, apping schees are needed to distribute data in such a way that data can be retrieved via reguar patterns without conicts. In this paper, the probe of conict-free access of arbitrary paths in bidiensiona arrays, circuar ists and copete trees is considered for the rst tie and reduced to variants of graph-cooring probes. Baanced and fast appings are proposed which require an optia nuber of coors (i.e., eory bans). The soution for bidiensiona arrays is based on a cobinatoria obect siiar to a Latin Square. The functions that ap an array node or a circuar ist node to a eory ban can be cacuated in constant tie. As for copete trees, the apping of a tree node to a eory ban taes tie that grows ogarithicay with the nuber of nodes of the tree. Key Words: Bidiensiona array, circuar ist, copete tree, conict-free access, apping schee, utiban eory syste, path tepate. This wor has been supported by the "Provincia Autonoa di Trento" under a research grant. y On eave fro University of Trento, Departent of Matheatics, Trento, Itay i

2 Running Tite: Conict-Free Access of Paths Address for Correspondence: M.C. Pinotti IEI-CNR Via S. Maria, Pisa ITALY E-ai: List of Footnotes: This wor has been supported by the "Provincia Autonoa di Trento" under a research grant. ii

3 1 Introduction In recent years, the traditiona divergence between the processor speed and the eory access rate is progressivey increasing. Thus, an ecient organization of the ain eory is iportant to achieve high-speed coputations. For this purpose, the ain eory can be equipped with cache eories { which have about the sae cyce tie as the processors { or can be partitioned into utibans. Since the cost of the cache eory is high and its size is iited, the utiban partition has osty been adopted, especiay in shared-eory utiprocessors []. However, the eectiveness of such a eory partition can be iited by eory conicts, that occur when there are any references to the sae eory ban whie accessing the sae eory pattern. To expoit to the fuest extent the perforance of the utiban partition, apping schees can be epoyed that avoid or iniize the eory conicts [15]. Since it is hard to nd universa appings { appings that iniize conicts for arbitrary eory access patterns { severa speciaized appings, designed for accessing reguar patterns in specic data structures, have been proposed in the iterature (see [1, ] for a copete ist of references). In particuar, for bidiensiona arrays, Budni and Kuc [7], Baarishnan et a. [4], Ki and Prasanna [1], and Das and Sarar [8] studied appings that provide conict-free access to rows, couns, positive and negative diagonas, subarrays, and distributed subarrays. The techniques used range fro Latin squares to Perfect Latin squares, fro inear appings to quasi-groups [11]. Subsequenty, appings for other data structures ie copete trees and binoia trees have been devised. In particuar, appings that provide conict-free access to copete subtrees, root-to-eaves paths, subeves, and coposite patterns obtained by their cobination, have been investigated in [8, 9, 1, 1, 14]. The apping schees proposed in those papers are optia, i.e., they use as few eory odues as possibe; baanced, i.e., the nodes of data structures are distributed as eveny as possibe aong the bans; fast, i.e., the ban address to which a node is assigned is coputed quicy with no nowedge of the entire structure apping; and exibe, i.e., they can be used for tepates of dierent size. In the present paper, optia, baanced and fast appings are designed for conict-free access of paths in bidiensiona arrays, circuar ists, and copete trees. With respect to the above entioned papers, paths in bidiensiona arrays and circuar ists are deat with for the rst tie. Moreover, access to any (not ony to root-to-eaves) paths in copete trees is provided. The reainder of this paper is organized as foows. In Section, the conict-free access probe is foray stated. In Section, the probe of accessing paths in bidiensiona arrays is soved. The proposed soution is a variant of a graph-cooring, which requires an optia nuber of coors and is achieved using a cobinatoria obect siiar to a Latin Square. As a byproduct, the eory ban to which an array node is assigned is coputed in constant tie. In Section 4, the probe of accessing paths in circuar ists is optiay soved and the function that aps a circuar ist node to a eory ban can be cacuated in constant tie. In Section 5, the sae probe on copete trees is aso optiay soved via a variant of a graphcooring probe. The tie needed to assign a tree node to a eory ban grows ogarithicay with the nuber of nodes of the tree. Concusions are oered in Section 6. 1

4 Conict-Free Access When storing a data structure D, represented in genera by a graph, on a eory syste consisting of N eory bans, a desirabe issue is to ap any subset of N arbitrary nodes of D to a the N dierent bans. This probe can be viewed as a cooring probe where the distribution of nodes of D aong the bans is done by cooring the nodes with a coor fro the set f; 1; ; : : : ; N? 1g. Since it is hard to sove the probe in genera, access of reguar patterns, caed tepates, in specia data structures { ie bidiensiona arrays, circuar ists, and copete trees { are considered hereafter. A tepate T is a connected subgraph of D. The occurrences ft 1 ; T ; : : : ; T g of T in D are the tepate instances. For exape, if D is a copete binary tree, then a path of ength can be a tepate, and a the paths of ength in D are the tepate instances. After cooring D, a conict occurs if two nodes of a tepate instance are assigned to the sae eory ban, i.e., they get the sae coor. An access to a tepate instance T i resuts in c conicts if c + 1 nodes of T i beong to the sae eory ban. Given a eory syste with N bans and a tepate T, the goa is to nd a eory apping U : D! N that coors the nodes of D in such a way that the nuber of conicts for accessing any instance of T is inia. In fact, the cost for T i coored according to U, Cost U (D; T i ; N), is dened as the nuber of conicts for accessing T i. The tepate instance of T with the highest cost deterines the overa cost of the apping U. That is, Cost U (D; T; N) def = ax Cost U (D; T i ; N): T i T A apping U is conict-free for T if Cost U (D; T; N) =. Aong desirabe properties for a conict-free apping, a apping shoud be baanced, fast, and optia. A apping U is tered baanced if it eveny distributes the nodes of the data structure aong the N eory bans. For a baanced apping, the eory oad is aost the sae in a the bans. A apping U wi be caed fast if the coor of each node can be coputed quicy (possiby in constant tie) without nowedge of the cooring of the entire data structure. Aong a possibe conict-free appings for a given tepate of a data structure, the ore interesting ones are those that use the iniu possibe nuber of eory bans. These appings are caed optia. It is worth to note that not ony the tepate size but aso the overapping of tepate instances in the data structure deterine a ower bound on the nuber of eory bans necessary to guarantee a conict-free access schee. This fact wi be ore convincing by the arguent beow for accessing paths in D. Let G D = (V; E) be the graph representing the data structure D. The tepate P is a path of ength in D. The tepate instance P [x; y] is the path of ength between two vertices x and y in V, that is, the sequence x = v 1 ; v ; : : : ; v +1 = y of vertices such that (v h ; v h+1 ) E for h = 1; ; : : :. The conicts can be eiinated on P [x; y] if v 1 ; v ; : : : ; v +1 are assigned to a dierent eory bans. The conict-free access to P can be reduced to a cassica cooring probe on the associated graph G DP obtained as foows. The vertex set of G DP is the sae as the vertex set of G D, whie the edge (r; s) beongs to the edge set of G DP i the distance d rs between the vertices r and s in G D satises d rs, where the distance is the ength of the shortest path between r and s. Now, coors ust be assigned to the vertices of G DP so that every pair of vertices connected by an edge is assigned

5 a coupe of dierent coors and the iniu nuber of coors is used. Hence, the roe of axiu cique in G DP is apparent for deriving ower bounds on the conict-free access on paths. A cique K for G DP is a subset of the vertices of G DP such that for each pair of vertices in K there is an edge. By we-nown graph theoretica resuts, a cique of size n in the associated graph G DP ipies that at east n dierent coors are needed to coor G DP. In other words, the size of the argest cique in G DP is a ower bound for the nuber of eory bans required to access paths of ength in D without conicts. On the other hand, the conict-free access to P on G D is equivaent to coor the nodes of G D in such a way that any two nodes which are at distance or ess apart have assigned dierent coors. Unfortunatey, this atter cooring probe is NP-copete [1] for genera graphs. This usties the investigation either for good heuristics for genera graphs or optia agoriths for specia casses of graphs. In the next three sections, optia appings for bidiensiona arrays, circuar ists and copete binary trees wi be derived for conict-free accessing P. Accessing Paths in Bidiensiona Arrays Let a bidiensiona array A be the data structure D to be apped into the utiban eory syste. An array r c has r rows and c couns, indexed respectivey fro to r? 1 (fro top to botto) and fro to c? 1 (fro eft to right), with r and c both greater than 1. The graph G A = (V; E) representing A is a esh, whose vertices correspond to the eeents of A and whose arcs correspond to any pair of adacent eeents of A on the sae row or on the sae coun. For the sae of sipicity, A wi be used instead of G A since there is no abiguity. Thus, a generic node x of A wi be denoted by x = (i; ), where i is its row index and is its coun index. Lea 1 At east M = (+1) eory bans are required for conict-free accessing P in A. Proof Consider a generic node x = (i; ) of A, and its opposite node at distance on the sae coun, i.e., y = (i? ; ). A the nodes of A at distance or ess fro both x and y are utuay at distance or ess, as shown in Figure 1. Therefore, in the associated graph G AP, they for a cique, and they ust be assigned to dierent coors. In detais, such a cique, denoted as K A (x; ), is dened as foows: K A (x; ) = n n (i? + t;? t) ; : : : ; (i? + t; + t) : t i? + t;? + t ; : : : ; i? Suing up over t, the size of the cique resuts to be bx c K A (x; ) = (t + 1) + Hence, at east M = K A (x; ) = t= (+1) d e X t=1 + t; + o? t? t + 1 = S : 1 t & ( + 1) ' o coors are required. :

6 y x (a) y x (b) Figure 1: A subset K A (x; ) of nodes of A that fors a cique in G AP : (a) =, (b) = 4. Beow, a conict-free apping is given to coor a the nodes of an array A using as few coors as in Lea 1. Therefore, the apping is optia. Fro now on, the coor assigned to node x is denoted by (x). Agorith Array-Cooring (A; ); Set M = (+1) ( + 1 if is even and = if is odd Assign to each node x = (i; ) A the coor (x) = (i + ) od M: Intuitivey, the above agorith rst covers A with a tesseation of basic sub-arrays of size M M. Each basic sub-array S is coored in a Latin Square fashion as foows: the coors in the rst row of S appear fro eft-to-right in the sequence ; 1; ; : : : ; M? 1; the coor sequence for a generic row is obtained fro the sequence at the previous row by a eft-cycic shift. For =, the cooring of A, decoposed into 6 basic sub-arrays of size M M, is iustrated in Figure. Theore 1 The Array-Cooring apping is optia, fast, and baanced. Proof To prove optiaity, it ust be shown that the apping is conict-free and that the iniu nuber of coors is used. 4

7 Figure : An array A of size 16 4 with a tesseation of 6 sub-arrays of size 8 8 coored by the Array-Cooring agorith to conict-free access P. Consider a generic node x = (g; f) of A and the associated cique K A (x; ), dened in Lea 1. In order to prove that the apping is conict-free, one ony needs to show that a the nodes of K A (x; ), which are utuay at distance no ore than, are assigned by the Array-Cooring agorith to dierent coors. Foray, consider an arbitrary pair of nodes w = (i; ) and z = (h; `) beonging to K A (x; ), such that i? h (if i? h <, the roes of w and z coud be swapped). Then the apping is conict-free if the Array-Cooring agorith guarantees that the coors (w) and (z) are dierent. Moreover, et (w; z) ((w)? (z)) od M (i + )? (h + `) od M be the dierence between the two coors assigned to w and z. Then, the apping is conict-free if the foowing two conditions siutaneousy hod: 8 >< >: (w; z) 6= od M; i? h +? ` : In order to show that the conditions in (1) hod for any pair of nodes of K A (x; ), the two cases even and odd ust be distinguished. (+1) When is even, one has that M = = ++ = ( + 1) and = + 1. Observe that (w; z) < (i? h) +? ` < + = ( + 1) + < M and (w; z) >?? ` >?M: Then, the congruence (w; z) 6 od M is equivaent to (w; z) 6= and (w; z) 6= M. Ceary, (w; z) = i (i? h)( + 1) = `?, which is veried ony if either z = w or `? is a utipe of + 1. But, since `? ipies (w; z) 6=, no two distinct nodes of K A (x; ) can have the sae coor. Thus, it reains to prove that (w; z) 6= M. Assue by contradiction that (w; z) = (i? h)( + 1) +? ` = M. Therefore, three cases ay occur: (1) 5

8 h (i) i? h ; M i? 1, +1 (ii) i? h = (iii) i? h = M +1 M +1,. In case (i), (w; z) = M ipies? ` = (w; z)? (i? h)( + 1) M? which contradicts the fact that? `. In case (ii), (w; z) can be equa to M if and ony if? ` = M? is, (w; z) = M if and ony if? ` = M +1? 1 ( + 1) > ( + 1) since b M +1 c =, that. + 1 Thus, in case (ii), for any pair of nodes z and w of K A (x; ) which do not satisfy the rst condition in (1), it resuts that? ` is equa to a positive integer and precisey,? ` = + 1: But this vioates the second condition in (1) because (i? h) + (? `) = = + 1. Finay, in case (iii), (w; z) = M if and ony if? ` = M? + 1 ( + 1). That is, for any pair of nodes z and w of K A (x; ) not satisfying the rst condition in (1), it yieds? ` <, and, precisey,? ` =? : But again this vioates the second condition in (1) because the distance between w and z is (i? h) +? ` = = + 1. In concusion, for even, any two nodes whose coors dier exacty by M are + 1 apart, and their reative positions are depicted in Figure (a). (+1) When is odd, it foows that M = = ++1 = Moreover, =. Observe that (w; z) < + = ( + 1) < M and (w; z) >?M. Then, (w; z) 6 od M is again equivaent to (w; z) 6= and (w; z) 6= M. Ceary, (w; z) = i (i? h) = `?, which is veried ony if either w = z (i.e., i? h = `? = ) or i? h 1 and `? is a utipe of. Hence, two distinct nodes of K A (x; ) which have the sae coor are at distance (i? h) + `? >. It reains to prove that (w; z) 6= M. As before, three cases ay occur: h (i) i? h ; M i? 1, (ii) i? h = M M, (iii) i? h =. Note that M = +1 and M = Repeating the sae reasoning done for even, one can show again that any two nodes whose coors dier by M are + 1 apart. Their reative positions are iustrated in Figure (b). So, the Array-Cooring Agorith is conict-free. Moreover, since it uses the iniu nuber of coors, the proposed apping is optia. 6

9 / (+1)/ / + 1 (+1)/+1 / + 1 / (+1)/ (+1)/-1 (a) (b) Figure : Reative positions in A of two nodes which are assigned to the sae coor: (a) even, (b) odd. It is easy to see that the tie required to coor a the n = rc nodes of an array is O(n). Moreover, to coor ony a singe node x = (i; ) of the tree requires ony O(1) tie, since (x) = (i + ) od M, and hence the apping is fast. In order to prove that the apping is baanced, observe that each coor appears once in each sub-row of size M. Hence, the nuber of nodes with the sae coor veries rb c M c rd c M e: Observe that the Array-Cooring Agorith guarantees conict-free access to soe paths onger than. Specicay, it is possibe to access without conicts any horizonta path of ength M and any vertica M path of ength L = because L is the iniu integer such that L od M. Finay, g.c.d.(m;) since the distance between two consecutive nodes on the sae diagona of A is, any b c consecutive eeents on a diagona can be accessed with no conicts. 4 Accessing Paths in Circuar Lists Let a circuar ist C be the data structure D to be apped into the utiban eory syste. A circuar ist of n nodes, indexed consecutivey fro to n? 1, is a sequence of n nodes such that node i is connected to both nodes (i? 1) od n and (i + 1) od n. The graph G C = (V; E) representing C is a ring, whose vertices correspond to the eeents of C and whose arcs correspond to any pair of adacent eeents of C. For the sae of sipicity, C wi be used instead of G C since there is no abiguity. 8 >< Lea Let M = n od (+1) >: ( + 1) + n if n < + 1; n (+1) if n + 1: At east M eory bans are required for conict-free accessing P in C. Proof For conict-free accessing P in C two nodes with the sae coor ust be at distance at east + 1. When n < + 1, a the nodes are utuay at distance ess than andust a be coored with dierent coors. When n + 1, each coor ay appear at ost t = ties. Therefore, n (+1) 7

10 Figure 4: Conict-free access to P 4 in a circuar ist C of 1 nodes coored by the Circuar-List-Cooring agorith with M = 7. at east n t east M = n t coors are needed. Observed that n = n = ( + 1) + n od (+1) n (+1) (+1) ( + 1) + (n od ( + 1)), it foows that at eory bans are required. Beow, an optia conict-free apping is provided to coor a the nodes of a circuar ist C using as few coors as in Lea. As before, the coor assigned to node x is denoted by (x). Agorith Circuar-List-Cooring (C; ); Set M = 8 >< >: n if n < + 1 ( + 1) + n od (+1) n (+1) if n + 1 ( n od (M? 1); if n od M 6= Set = sm where s = n ; if n od M = M ( x od M if x < Assign to node x C, the coor (x) = (x? ) od (M? 1) if x Note that a inear (that is, non circuar) ist L can be optiay coored to conict-free access P with M = + 1 coors, which atches the trivia ower bound given by the nuber of nodes in P. In fact, L can be optiay coored by a naive agorith which assigns to node x the coor (x) = x od M. Such a naive agorith does not wor for circuar ists. For exape, consider the circuar ist C of 1 nodes, shown in Figure 4, to be coored to access P 4. Appying the naive agorith with M = 5, ony the rst 1 nodes can be feasiby coored with 5 coors, but additiona coors are then required for feasiby cooring the ast nodes, for a tota of 8 coors. In contrast, the optia Circuar-List- 8

11 (a) (b) Figure 5: A circuar ist C of 17 nodes coored to conict-free access P according to: (a) the Circuar- List-Cooring agorith (where M = 5), (b) the naive agorith with M = 5. Cooring agorith requires 7 coors ony. Moreover, it is worth to point out that the naive agorith does not aways wor for circuar ists even when appied with M = M = ( + 1) +. For n od (+1) n (+1) instance, for n = 17 and =, Lea gives M = 5. Appying the naive agorith with M = 5 to this instance, 15 nodes can be coored using 5 coors, but additiona coors are needed for feasiby cooring the ast nodes for a tota of 7 coors (as shown in Figure 5(b)). Instead, the optia cooring provided by the Circuar-List-Cooring agorith uses ony 5 coors, as shown in Figure 5(a). Indeed, the naive agorith aways produces a feasibe (athough not necessariy optia) cooring if appied using M = ( + 1) + n od ( + 1). Theore The Circuar-List-Cooring apping is optia, fast, and baanced. Proof To prove optiaity, two cases ay be distinguished. If n od M, Lea gives M = + 1 and the Circuar-List-Cooring agorith reuses the sae coor at distance M. Hence, no conict arises. If n 6 od M, Lea gives M +. Two nodes get the sae coor ony if they are at distances M or M?1, which are both greater than or equa to +1. Hence, as before, no conict arises. Since the agorith uses as few coors as possibe, the apping is optia. It is aso fast since each node is coored in constant tie. Finay, each coor is assigned to exacty n nodes when n is a M utipe of M, and no ore than in(n;) + ax(n?;) nodes are coored with the sae coor in M M?1 a the other cases. It is interesting to note at this point that, given a circuar ist of n nodes, the iniu nuber M = M(n; ) of coors required to conict-free access P satises the foowing properties (see Figure 6): Up to n = ( + 1)? 1, M(n; ) = n resuts, i.e. a the nodes ust have dierent coors. Indeed, a of the are utuay at distance no ore than and, therefore, they for a cique on the graph G CP. When n > ( + 1)? 1, M(n; ) depends on both n and, and, for a xed, is not a onotone 9

12 14 M(n, 6) Figure 6: The nuber of coors M(n; 6) required to conict-free access P 6 when n ranges between 1 and 58. function of n. In contrast, for arrays and trees (as wi be proved in the next section), M depends ony on and is onotone. 5 Accessing Paths in Copete Trees Let a rooted copete binary tree B be the data structure to be apped into the utiban eory syste. The eve of node x B is dened as the nuber of edges on the path fro x to the root, which is at eve. The axiu eve of the nodes of B is the height of B. Let Lev B (i) be the set of a nodes of B at eve i. A copete binary tree of height H is a rooted tree B in which a the eaves are at the sae eve and each interna node has exacty chidren. Thus, Lev B (i) contains i nodes. The h-th ancestor of the node (i; ) is the node (i? h; b h c), whie its chidren are the nodes (i + 1; ) and (i + 1; + 1), in the eft-to-right order. Fro now on, the generic node x, which is the -th node of Lev B (i), with counting fro eft to right, wi be denoted by x = (i; ). Therefore, the generic path instance P [x; y] wi be denoted by P [(i; ); (r; s)], where x = (i; ) and y = (r; s). Lea At east M = b c+1 + d e? eory bans are required to conict-free access P in B. Proof Consider a generic node x = (i; ). A the b c+1? 1 nodes in the subtree S of height b c rooted at the b c-th ancestor of x are utuay at distance not greater than. In addition, consider the d e nodes, 1; ; : : : d e, ancestors of x, on the path I of ength d e fro the b c-th ancestor of x up to the -th ancestor of x. A these nodes are at distance not greater than fro node x, and together with the nodes of S they are at utua distance not greater than. 1

13 µ µ 1 x S (a) µ µ 1 S x (b) Figure 7: A subset K B () of nodes of B that fors a cique in G BP : (a) =, (b) = 4. Moreover, for 1 d e? 1, consider the +1? 1 nodes in the copete subtree of height =? b c?? 1, rooted at the 's chid which does not beong to I. Such nodes are at distance not greater than fro x. Furtherore, these nodes, aong with the nodes of S and I, are a together at utua distance not greater than. Hence, in the associated graph G DP b c+1? d X e?1 =1 there is at east a cique of size +1? 1 = b c+1? d X e? h= h+1? 1 : Fro that, the cai easiy foows. Figure 7 shows a subset K B () of nodes of B which are at pairwise distance not greater than, for = and 4, and hence fors a cique in the associated graph G BP. An optia conict-free apping to coor a copete binary tree B acts as foows. A basic subtree K B () dened as in the proof of Lea is identied and coored. Such a tree is then overaid to B in such a way that the upperost eves of B coincide with the owerost eves of K B (). Then, the copete cooring of B is produced eve by eve by assigning to each node the sae coor as an aready coored node. Foray, for a given, dene the binary tree K B () as foows: K B () has a eftost path of + 1 nodes. the root of K B () has ony the eft chid; a copete subtree of height i? 1 is rooted at the right chid of the node at eve i on the eftost path of K B (). 11

14 (a) (b) Figure 8: Cooring of B for conict-free accessing: (a) P, (b) P 4. (Both K B () and K B (4) are depicted by dash spines.) The b c+1 + d e? nodes of K B () ust be coored with b c+1 + d e? dierent coors. Thus, the upperost eves of B are aready coored. For the sae of sipicity, to coor the reaining part of B, the eves are counted starting fro the root of K B (). That is, the eve of the root of B wi be renubered as eve x = (; ), the agorith to coor B acts as foows Now, xed Agorith Binary-Tree-Cooring (B; ); Set M = b c+1 + d e? ; Coor K B () with M coors; Visit the tree B in breadth rst search, and for each node x = (i; ) of B, with + 1, do: { Set = od b c, = dog(+1)e, =?+1 and =? 1?1 { Assign to x the sae coor as that of the node y = (r; s), where and r = i? + s = 8 >< >: if = +?1 +? od?1 if 6= od ; Exapes of coorings to conict-free access P and P 4 are iustrated in Figure 8. 1

15 τ = α = δ = 5 S 6 S 5 y S x π = Figure 9: For = 6, node x = (i; ) inherits the sae coor as node y = (r; s). Theore The Binary-Tree-Cooring apping is optia, fast and baanced. Proof. To prove that the apping is optia, it ust be shown that it is conict-free and it uses as few coors as those given by Lea. First, observe that the b c eaves of a subtree of height are at utua distance not greater than, and therefore they ust be coored with a dierent coors. Thus, et each eve of B be partitioned (starting fro the eftost node) into consecutive bocs of size b c. The boc b(i; w), with w, at eve i of B consists of the b c consecutive nodes (i; w b c ); (i; w b c + 1); : : : ; (i; (w + 1) b c? 1), which ust a be assigned to a dierent coor. Consider the node x = (i; ) to be coored. The node x = (i; ) beongs to the boc b x = b i; b, c and it appears in the ( + 1)-th position inside the boc. Consider the eftost node z of b x, where z = i; b c b c. Then, a generaization K B (z; ) of K B () can be dened depending on z. K B (z; ) incudes the foowing nodes of B: the nodes on the path? of ength fro the father of z up to the ( + 1)-th ancestor of z; + q, the nodes of the copete binary tree S q of height? q rooted at the chid, for which does not beong to?, of the q-th ancestor of z; the nodes of the copete binary tree S of height rooted at the + 1 -th ancestor of z. It is crucia to note that a the foowing nodes are at distance + 1 fro a the nodes in b x : (i) the root of K B (z; ), (ii) the eaves of S q, with + q, 1

16 (iii) the eaves of S, which are not parents of any node in b x. The nodes of b x = b i; are coored fro eft to right copying the sae coors used in the b c nodes of K B (z; ) specied in (i), (ii), and (iii) above, and considered by increasing eve and fro eft to right, as iustrated in Figures 1 and 11 for even and odd, respectivey. In particuar, z = i; b is assigned to the sae coor as the root of K c B (z; ), which is the ( + 1)-th ancestor of x; for q +, the?q nodes of b x, (i; b c b c +?q? 1 + 1); : : : (i; b c b c +?q? 1 +?q ), are assigned to the sae coors as the eaves of the tree S q. Observe that the nuber of nodes coored with the two steps above is 1 + P q=b c+?q = d e?1 : When is odd, this is enough to coor the entire boc since d e?1 = b c+1?1 = b c. In fact, the set of nodes of K B (z; ) specied in (iii) above is epty for odd. In contrast, when is even, ony the rst haf of the boc has been coored since d e?1 = b c?1 : Thus, to coor the second haf of the boc, one further step is required, which uses the coors of the nodes of K B (z; ) specied in (iii) above: The rightost b c?1 nodes of b x are assigned to the sae coors as the rightost (resp., eftost) b c?1 eaves of the copete binary tree rooted at ( fact that the -th ancestor of z is a eft (resp., right) chid of its father. + 1)-th ancestor of z, depending on the In order to prove that the apping is conict-free, an inductive reasoning on the eve i of the tree is foowed. The basis for the induction is i =, when the tree coincides with K B () and it is coored, by denition, with a dierent coors. For i >, consider a generic node x = (i; ), its boc b x and its eftost node z. By inductive hypothesis, a the nodes in the tree up to eve i? 1 are coored in a conict-free anner, but with coor repetitions. In particuar, the subtree K B (z; ) is conict-free and since its nodes are utuay at distance at ost they ust have been assigned to a dierent coors. The agorith coors b x copying the coors of soe nodes in K B (z; ), specied in (i), (ii), and (iii), which are exacty at distance + 1 fro the nodes of b x. Therefore, there are no coor repetitions in b x and no conict can arise. Note that nodes in dierent bocs at eve i ay inherit the sae coor, but since any two nodes in dierent bocs are at distance at east + 1 no conict can arise. Therefore, a the nodes in the tree up to eve i are coored in a conict-free anner. Finay, since the tree is coored with the coors of K B (), whose nuber equas the ower bound of Lea, the tree-cooring apping is optia. It is easy to see that the tie required to coor a the n nodes of a tree is O(n). However, to coor ony a singe node x of the tree requires ony O(og n) tie since, in the worst case, a the nodes aong a path fro x up to the root ust have been coored. One can readiy see that, if the height H of the tree B is a utipe of, then the nodes of B can be partitioned into = d H+1?1 e subsets, each of which induces a copy of M K B(). Therefore, each coor is used ties, and the apping is baanced. 14

17 Γ S 6 S 5 S z Figure 1: The generaization K B (z; 6) of K B (6) for the node z. The root of K B (z; 6), the eaves of the subtrees S 6 ; S 5, and the rightost eaves of S are used to coor the nodes in the boc b z. Γ S 5 S 4 S z Figure 11: The generaization K B (z; 5) of K B (5) for the node z. The root of K B (z; 5) and the eaves of the subtrees S 4 and S 5 are used to coor the nodes in the boc b z. 15

18 The resuts shown for binary trees can be extended to a q-ary tree Q, with q. Coroary 1 At east M = qb c+1 d? (q? 1) q? 1 X e? q h+1? 1 q? 1 h= = 1 + qb c+1? 1 + q d e? q q? 1 eory odues are required to conict-free access P in a q-ary tree Q. Siiary to the binary case, for a given, dene a q-ary tree K q Q () as foows: K q Q () has a eftost path of + 1 nodes; the root of K q Q () has ony the eftost chid; a copete subtree of height i? 1 is rooted at the q? 1 rightost chidren of the node at eve i on the eftost path of K q Q (). Such a K q Q () is then overaid to Q in such a way that the upperost the owerost eves of Q coincide with eves of K q Q (). Then, the copete cooring of Q is produced eve by eve by assigning to each node the sae coor as an aready coored node. For the sae of sipicity, to coor the reaining part of Q, the eves are again counted starting fro the root of K q (). That is, the eve of the root of Q Q wi be renubered as eve the agorith to coor Q is the foowing: + 1. Now, Agorith q-ary-tree-cooring (Q; ); Set M = 1 + qb c+1?1+q d e?q q?1 ; Coor K q Q () with M coors; Visit the tree Q in breadth rst search, and for each node x = (i; ) of Q, with + 1, do: { Set = od q b c, = dogq ( + 1)e, =? + 1 and = { Assign to x the sae coor as that of the node y = (r; s), where and r = i? + s = 8 >< >: q if = q q + q?1 +? od q?1 if 6= + 1 od q; q?1 By a reasoning siiar to that epoyed for copete binary trees, the optiaity of the q-ary-tree- Cooring Agorith easiy foows. 16

19 6 Concusions In this paper, the probe of conict-free accessing arbitrary paths P in particuar data structures, such as bidiensiona arrays, circuar ists and copete trees, has been considered for the rst tie and reduced to variants of graph-cooring probes. Optia, fast and baanced appings have been proposed. Indeed, the eory ban to which a node is assigned is coputed in constant tie for arrays and circuar ists, whie it is coputed in ogarithic tie for copete trees. However, it reains as an open question whether a tree node can be assigned to a eory ban in constant tie. On the other hand, the conict-free access to P on an arbitrary data structure D is NP-copete [1], and this usties the investigation of good heuristics. This probe is equivaent to the cassica node cooring probe in the associated graph G DP. Therefore, it can be soved by the ost eective cooring heuristic nown so far, that is, the saturation-degree heuristic [6], which wors as foows. Let N(x) be the neighborhood of node x in the associated graph G DP. At each iteration, the saturationdegree heuristic seects the node x to be coored as one with the argest nuber of dierent coors aready assigned in N(x). Ties between nodes are broen by preferring the node x with the argest nuber of coored nodes in N(x). Once seected, node x is assigned the owest coor not yet assigned in N(x). As experientay proved in [5], the saturation-degree heuristic is especiay eective when the iniu nuber of coors is given by the size of the argest cique K of G DP. Therefore, it shoud wor ecienty aso for the conict-free access probe, and, in particuar, for d-diensiona arrays as we as for generic, i.e. not necessariy copete, trees. Indeed, it is expected in such cases that the iniu nuber of required eory bans be equa to the ower bound given by the size of the argest cique K of G DP, as happened for bidiensiona arrays and copete trees. Unfortunatey, the resuting cooring is not guaranteed to be optia, fast or baanced. Moreover, it is sti an open question to deterine whether the probe of conict-free accessing paths on d-diensiona arrays and generic trees is NP-copete. Finay, in a ore practica perspective, the nuber of eory bans avaiabe coud be xed to a constant, depending on the eory conguration. Then, if the nuber of eory odues M() required for a given P is arger than, no conict-free access is possibe. However, assue that P the ongest path that can be accessed without conicts using eory bans, i.e. M( ). Then, accessing P, no ore than d e conicts ay arise. Hence, the proposed appings are scaabe. is Acnowedgeent The authors are gratefu to Richard Tan for his hepfu coents, and to Thoas McCoric for having provided the reference [1]. 17

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