Flexible Word Design and Graph Labeling

Transcription

1 Flexible Word Design and Graph Labeling Ming-Yang Kao Manan Sanghi Robert Shweller Abstrat Motivated by emerging appliations for DNA ode word design, we onsider a generalization of the ode word design problem in whih an input graph is given whih must be labeled with equal length binary strings of minimal length suh that the Hamming distane is small between words of adjaent nodes and large between words of non-adjaent nodes. For general graphs we provide algorithms that bound the word length with respet to either the maximum degree of any vertex or the number of edges in either the input graph or its omplement. We further provide multiple types of reursive, deterministi algorithms for trees and forests, and provide an improvement for forests that makes use of randomization. We also onsider generalizations of this problem to weighted graphs and graphs with optional edges. Finally, we explore the extension from simple adjaeny queries to more general distane queries and show how to obtain distane labelings for rings and paths by applying properties of hyperube traversal. Supported in part by NSF Grant EIA Department of Eletrial Engineering and Computer Siene, Northwestern University, Evanston, IL 6008, USA. kao@s.northwestern.edu. Department of Eletrial Engineering and Computer Siene, Northwestern University, Evanston, IL 6008, USA. manan@s.northwestern.edu. Department of Eletrial Engineering and Computer Siene, Northwestern University, Evanston, IL 6008, USA. shwellerr@s.northwestern.edu. 1

2 1 Introdution This work an be viewed either as a generalization of odeword design or a speial restrited ase of the more general graph labeling problem. The problem of graph labeling takes as input a graph and assigns a binary string to eah vertex suh that either adjaeny or distane between two verties an be quikly determined by simply omparing the two labels. The goal is then to make the labels as short as possible (see [14] for a survey). Early work in the field [8, 9] onsidered the graph labeling problem with the restrition that the adjaeny between two nodes must be determined solely by Hamming distane. Speifially, the labels for any two adjaent nodes must be below a given threshold, while the nodes between non-adjaent nodes must be above it. We return to this restrited type of graph labeling motivated by growing appliations in DNA omputing and DNA self-assembly. A basi requirement for building useful DNA self-assembly systems and DNA omputing systems is the design of sets of appropriate DNA strings (ode words). Early appliations have simply required building a set of n equal length ode words suh that there is no possibility of hybridization between the words or Watson Crik omplement of words [1, 4, 6, 7, 19, 4]. Using hamming distane as an approximation to how well a word and the Watson Crik omplement of a seond word will bind, suh a requirement an be ahieved in part by designing a set of n words suh that the Hamming distane between any pair in the set is large. There has been extensive work done in designing sets of words with this and other non-interation onstraints [5, 6, 10 13, 16 19,, 3]. While the Hamming distane onstraint is important for appliations requiring that no pair of words in a ode hybridize, new appliations are emerging for whih hybridization between different words in a ode word set is desirable and neessary. That is, there is growing need for the effiient design of DNA odes suh that the hybridization between any two words in the ode is determined by an input matrix speifying whih strands should bond and whih should not. Aggarwal et al. [, 3] have shown that a tile self assembly system that uses a set of glues that bind to one another aording to a given input matrix, rather than only binding to themselves, greatly redues the number of distint tile types required to assemble ertain shapes. Effiient algorithms for designing sets of DNA strands whose pairwise hybridization is determined by an input matrix may permit implementation of suh tile effiient self-assembly systems. Further, Tsaftaris et al. [0, 1] have reently proposed a tehnique for applying DNA omputing to digital signal proessing. Their sheme involves designing a set of equal length DNA strands, indexed from 1 to n, suh that the melting temperature of the duplex formed by a word and the Watson Crik omplement of another word is proportional to the differene of the indies of the words. Thus, this is again an example in whih it is desirable to design a set of DNA words suh that different words have varying levels of distintness from one another. Given an input graph G, we onsider the problem of onstruting a labeling suh that the Hamming distane between labels of adjaent verties is small, the Hamming distane between non-adjaent verties is large, and there is a separation of at least γ between the small Hamming distane and the large Hamming distane. Breuer et al.[8, 9] first studied this problem for the speial ase of γ = 1 and ahieved labels of size O(Dn) for general graphs, where D is the size of the highest degree node in the graph. By ombining graph deompositions with odes similar in spirit to Hadamard odes from oding theory, we get a labeling of length O(γ ˆD + ˆDn) where ˆD is the smaller of the size of the maximum degree node in G and its omplement. We then explore more sophistiated graph deompositions to ahieve new bounds that are a funtion of the number of edges in G. We also onsider the lass of trees and forests and provide various reursive algorithms that ahieve poly-logarithmi length labels for degree bounded trees. Our forest algorithms also make use of probabilisti bounds from traditional word design to use randomization to redue label

3 Table 1: Summary of our results for word length and time omplexity for flexible word design. Lower Bound Word Length Upper Bound Time Complexity General Graphs O(γ ˆD + ˆDn) O(γ ˆDn + ˆDn ) (Mathing Algorithm) Theorem 14 Theorem 14 General Graphs O( γ ˆm n + γ ˆmn ) O( γ ˆm n 3 + γ ˆmn 4 ) (StarDestroyer) Theorem 15 Theorem 15 ( { } ) ( { γ + n, Weighted Graphs Ω(γ + n) O min γn + n } ), γw (Multi-graph based) n + γw n O min γw n 3 + γw n 4 Theorem Theorem 31 Theorem 31 Weighted Graphs (Weighted Mathing) Weighted Graphs (Hybrid Algorithm) Don t are edges O ( min { i w id i γ + i d in, Dw ργ + Dρn } ) O( min { n ( i w id i γ + i d in), n (Dw ργ + Dρn) Theorems 33 and 34 Theorems 33 and 34 O( 3 W γn(γ + n)(πγ + n)) O(n 3 W γn(γ + n)(πγ + n)) Theorem 35 Theorem 35 ( { } γ D + Dn, ) ( { O min γ Dn + Dn γ m n + γ mn O min } ), n γ m n + γ mn Theorem 37 Theorem 37 Forests O(γD log n) O(nγD log n) (Node Reursion) Theorem 18 Theorem 18 Forests Ω(γ + log n) ( ) ( min{t, γd log t} + γd log f n min{t, γd log t} + n γd log ) f O + max{γd log( n ), 0} O γd +n max{γd log ( n ), 0} γd (Leaf Reursion) Theorem 3 Theorem Theorem Forests O(γD log f + max{γd log( n ), 0}) γd O(nγD log f + max{γd log ( n ), 0}) γd (Randomized) Theorem 3 Theorem 3 Rings O(γ log n) O(nγ log n) Theorem 5 Theorem 9 Theorem 9 Speial Rings with O(γk log n k ), where k n 1 O(nγk log n k ) Distane Queries Ω(γk + log n) Theorem 7 Theorem 7 Paths with Theorem 5 O(γk + γk log n k ) O(n (γk + γk log n k )) Distane Queries Theorem 8 Theorem 8 } ) G : input graph (V, E) G : omplement of G γ : Hamming distane separation n : number of verties in G m : number of edges in G ˆm : smaller of the number of edges in G or G m : smaller of the number of near edges and the number of far edges W : the sum of weights of all the edges d i : highest degree of any vertex with respet to weight w i edges D : highest degree of any vertex in G ˆD : smaller of the highest degree of any vertex in G or G D : smaller of maximum near-degree and maximum far-degree : maximum weighted degree t : number of trees in the forest f : maximum number of leaves in any tree in the input forest k : range of distane queries π : maximum weight of an edge ρ : number of distint weights of edges 3

4 length. We further onsider some important generalizations to the basi flexible word design problem. First, we initiate the study of extending our problem from adjaeny queries to distane queries. For this we apply basi properties of hyperube traversal to provide effiient labelings for rings and paths and disuss how this problem and lass of graphs may have appliations to DNA signal proessing. We also onsider a version of flexible word design for weighted graphs and provide multiple general ase algorithms for the problem. Finally, we disuss graphs that have don t are edges whih make no requirement for the Hamming distane between two nodes. Our results are summarized in Table 1. Paper Layout: In Setion, we introdue basi notation and tools and formulate the problem. In Setion 3, we desribe tehniques for obtaining a restrited type of labeling for speial graphs. In Setion 4, we desribe how to ombine speial graph labelings to obtain algorithms for labeling general graphs. In Setion 5, we present reursive algorithms for labeling forests. In Setion 6, we generalize our problem to distane labeling and provide solutions for rings and paths. In Setion 7, we provide algorithms for solving the weighted version of flexible word design. In Setion 8, we present results for a generalization to graphs with don t are edges. In Setion 9, we onlude with a disussion of future researh diretions. Preliminaries Let S = s 1 s... s l denote a length l bit string with eah s i {0, 1}. For a bit s, the omplement of s, denoted by s, is 0 if s = 1 and 1 if s = 0. For a bit string S = s 1 s... s l, the omplement of S, denoted by S, is the string s 1, s... s l. For two bit strings S and T, we denote the onatenation of S and T by S T. For a graph G = (V, E), a length l labeling of G is a mapping σ : V {0, 1} l. Let deg(g) denote the maximum degree of any vertex in G and let Ḡ = (V, Ē) denote the omplement graph of G. A γ-labeling of a graph G is a labeling σ suh that there exist integers α and β, β α γ, suh that for any u, v V the Hamming distane H(σ(u), σ(v)) α if (u, v) E, and H(σ(u), σ(v)) β if (u, v) / E. We are interested in designing γ-labelings for graphs whih minimize the length of eah label, length(σ). Problem 1 (Flexible Word Design Problem). Input: Graph G; integer γ Output: A γ-labeling σ of G. Minimize l = length(σ). Throughout this paper, for ease of exposition, we will assume the Hamming distane separator γ is a power of. For general graphs in Setions 3, 4, 7, and 8 we also assume the number of verties n in the input graph is a power of. These assumptions an be trivially removed for these ases. For the remaining setions, in partiular Setion 6 whih deals with distane queries on rings and paths, we do not make this assumption as the extension to all n is non-trivial. Theorem. The required worst ase label length for general n node graphs is Ω(γ + n). Proof. For graph labeling supporting adjaeny queries, it is well known [15] that a lass of Ω(n1+ɛ ) n-vertex graphs must ontain a member that requires labels of size Ω(n ɛ ). As there are Ω(n) n node graphs, we get a lower bound of Ω(n) on label length. Speifi to our problem, we further observe that any graph other than the omplete graph must have Hamming distane at least γ for some pair of nodes, whih then requires length at least γ. Thus, there must exist some n node graph requiring label length at least max{γ, n} 1 (γ + n) = Ω(γ + n). 4

5 Theorem 3. The required worst ase label length for n node forests is Ω(γ + log n). Proof. Consider the n node graph for whih there are no adjaent nodes. To satisfy γ 1, eah label must be distint, yielding a lower bound of log n. Further, eah node must have a Hamming distane of at least γ, and thus must have length at least γ for a total length of at least max{γ, log n} = Ω(γ + log n). An important tool that we will make use of repeatedly in our onstrutions is the following equidistant ode whih is similar to the Hadamard odes from oding theory. The key property of this ode is that it yields short words suh that every pair of strings in the ode has the exatly the same Hamming distane between them. The Equidistant Code. To define the Equidistant ode we first define a speial type of string. Consider a binary string S of length l and an integer γ with l a multiple of γ suh that the w th size γ word of S is either all 1 s or all 0 s. Then, define f γ (S) to be the length l binary string with w th size γ word being γ 0 s followed by γ 1 s if the w th word of S is 0 s, and γ 1 s followed by γ 0 s if the w th word of S is 1 s. For a given n and γ, eah powers of with γ n, define the n element equidistant ode ED(n, γ) reursively as follows. 1. Define ED( 1, γ) to be the word ode with S 1 onsisting of γ 0 s and S onsisting of γ 1 s.. For i 1, define ED( i+1, γ) as follows. Let ED( i, γ) = S 1, S,..., S i. For j = 1 to i+1, let S j = S j ( γ i 0 s) if j i and S j = f γ i (S j i) ( γ i 1 s) if j > i. ED( i+1, γ) is then S 1, S i+1. It is straightforward to argue indutively that ED(n, γ) is well defined for all powers of with γ n. We will also use a slight variant of this ode suh that the length of eah word is exatly twie γ. Balaned Equidistant Code. For a given n and γ, eah powers of with γ n, define the n element balaned equidistant ode ED B (n, γ) by appending γ n 0 s to eah S i ED(n, γ). Theorem 4. Consider the odes ED(n, γ) and ED B (n, γ) for n and γ powers of and γ n. The following properties hold. 1. For any S ED(n, γ), length(s) = γ γ n.. For any non-equal S i, S j ED(n, γ) (or any S i, S j ED B (n, γ)), H(S i, S j ) = γ. 3. For any non-equal S i, S j ED(n, γ), H(S i, S j ) = γ γ n. 4. For any S ED B (n, γ), length(s) = γ. 5. Let F B (n, γ) = ED B (n, γ) \ {A 1,... A r } {A 1,..., A r} for an arbitrary subset {A 1,... A r } of ED B (n, γ). Then, properties 1 and 4 still hold for F B (n, γ). 6. The odes ED(n, γ) and ED B (n, γ) an be omputed in time O(n γ). Proof of 1. We an show this for ED( i+1, γ) by indution on i. For i = 0, γ γ n = γ, whih is equal to the length by definition. For i 1, the length l for ED( i+1, γ), by definition, is equal to the length l for ED( i, γ) plus γ, whih, by indutive hypothesis, is γ γ + γ = γ γ. i i i Proof of. Consider the following ases. i+1 Case 1: i, j i. In this ase the result follows by indutive hypothesis. 5

6 Case : i, j > i. Sine H(X, Y ) = H(f γ (X), f γ (Y )), the result follows by indutive hypothesis. Case 3: i i, j > i. Note that for two length l strings X and Y, H(X, f γ (Y )) = l. Thus, by part 1 of the theorem, H(S i, S j ) = (γ γ )/ + γ = γ. i Proof of 3. This follows immediately from parts and i 1. Proof of 4. This follows from 1 and the definition of ED b (n, γ). Proof of 5. This follows from and 4. Proof of 6. It is straightforward to ompute ED(n, γ) in Θ(γn) time given ED( n, γ). Thus, suh an algorithm ahieves time T (n) = Θ(γn) + T ( n ) for a given n. The solution to this reurrene is Θ(γn). 3 Exat Labelings for Speial Graphs In onstruting a γ-labeling for general graphs, we make use of a more restritive type of satisfying label alled an exat labeling, as well as an inverted type of labeling. Suh labelings an be ombined for a olletion of graphs to obtain labelings for larger graphs. We onsider two speial types of graphs in this setion, mathings and star graphs, and show how to obtain short exat labelings for eah. These results are then applied in Setion 4 by algorithms that deompose arbitrary graphs into these speial subgraphs effiiently, produe exat labelings for the subgraphs, and then ombine the labelings to get a labeling for the original graph. Definition 5 (Exat Labeling). A γ-labeling σ of a graph G = (V, E) is said to be exat if there exist integers α and β, β α γ, suh that for any two nodes u, v V it is the ase that H(σ(u), σ(v)) = α if (u, v) E, and H(σ(u), σ(v)) = β if (u, v) / E. A labeling that only satisfies H(σ(u), σ(v)) = α if (u, v) E, but H(σ(u), σ(v)) β if (u, v) / E is alled a lower exat labeling. Definition 6 (Inverse Exat Labeling). A labeling σ of a graph G = (V, E) is said to be an inverse exat labeling for value γ if there exist integers α and β, β α γ, suh that for any two nodes u, v V it is the ase that H(σ(u), σ(v)) = α if (u, v) / E, and H(σ(u), σ(v)) = β if (u, v) E. Thus, the differene between an exat γ-labeling and a γ-labeling is that an exat labeling requires the Hamming distane between adjaent verties to be exatly α, rather than at most α, and the distane between non-adjaent nodes to be exatly β, rather than at least β. An inverse exat labeling is like an exat labeling exept that it yields a large Hamming distane between adjaent nodes, rather than a small Hamming distane. We are interested in exat γ-labelings beause the exat γ-labelings for a olletion of graphs an be onatenated to obtain a γ-labeling for their union. We define an edge deomposition as follows. Definition 7 (Edge Deomposition). Graphs G 1,..., G r where G i = (V i, E i ) are said to form an edge deomposition of graph G = (V, E) if: 1. V = V i for all i.. E = i E i. Theorem 8. Consider a graph G with edge deomposition G 1,... G r. For eah G i let σ i be a labeling of G i with length length(σ i ) = l i. Consider the labeling σ(v) = σ 1 (v) σ (v) σ r (v) defined by taking the onatenation of eah of the labelings σ for eah vertex in G. Then the following hold. 1. If eah σ i is an exat γ i -labeling of G i with thresholds α i and β i, then for γ = min{γ i } the labeling σ(v) is a γ-labeling of G with thresholds α = β i γ and β = β i. 6

7 . If eah σ i is an inverse exat γ i -labeling of G i with thresholds α i and β i, then for γ = min{γ i } the labeling σ(v) is a γ-labeling of the omplement graph G with thresholds α = r i=1 α i and β = r i=1 α i + γ. Proof of 1. Consider an arbitrary pair of verties (u, v). If (u, v) / E, then learly H(σ(u), σ(v)) = r i=1 β i = β. On the other hand, if (u, v) E, then for at least one j we know that (u, v) E j. Thus, we get that H(σ(u), σ(v)) r β i β j + α j. i=1 And sine β j α j γ j γ, we get that H(σ(u), σ(v)) r β i β j + β j γ = i=1 r β i γ = α. i=1 Proof of. Consider an arbitrary pair of verties (u, v). If (u, v) is an edge in G, then (u, v) is not an edge in G, and so H(σ(u), σ(v)) = r i=1 α i = α. On the other hand, if (u, v) / E, then (u, v) E, and so for at least one j we know that (u, v) E j. Thus, we get that H(σ(u), σ(v)) r α i α j + β j i=1 r α i + γ = β. i=1 We now disuss how to obtain exat and inverse exat labelings for speial lasses of graphs. For the lasses of graphs we onsider, it is surprising that we are able to ahieve the same asymptoti label lengths for exat labelings as for inverse exat labelings. In Setion 4 we disuss algorithms that deompose general graphs into these lasses of graphs, obtain exat or inverse exat labelings, and then ombine them to obtain a satisfying labeling from Theorem Mathings A graph G = (V, E) is said to be a mathing if eah onneted omponent ontains at most two nodes. To obtain an exat labeling for a mathing we use Algorithm 1 MathingExat. To obtain an exat inverse mathing, we use Algorithm InverseMathingExat. The performane of these algorithms is analyzed in Lemmas 9 and 10. Algorithm 1 MathingExat(G, γ) 1. Let γ = max(γ, n ). Generate ED(n, γ ).. Assign a distint string from ED(n, γ ) to eah lique of G. That is, apply the labeling σ suh that for eah v V, σ(v) ED(n, γ ) and σ(v) = σ(u) iff (v, u) E. 3. Output σ. Lemma 9. Algorithm 1 MathingExat(G, γ) obtains an exat γ-labeling with α = 0, β = max(γ, n ), and length = O(γ + n), in run time O(γn + n ). Proof. The exat γ-labeling and length properties follow from parts and 1 respetively of Theorem 4. The run time is also learly linear in the output size given part 6 of Theorem 4. 7

8 Algorithm InverseMathingExat(G, γ) 1. Let γ = max(γ, n ). Generate EDB (n, γ ).. Arbitrarily index the edges of E. For i = 1 to E, onsider the i th edge (u i, v i ): (a) Set σ(u i ) S i. (b) Set σ(v i ) S i. 3. For eah vertex v V that is not adjaent to an edge in E, set σ(v) S i for some S i in ED B (n, γ ) that has not yet been assigned. 4. Output σ. Lemma 10. Algorithm InverseMathingExat(G, γ) obtains an exat inverse labeling with α = max(γ, n ), β = max(γ, n ), and length = O(γ + n), in run time O(γn + n ). Proof. The bound on length follows from part 4 of Theorem 4, the exat γ-labeling from parts and 4, and the run time from part Star-graphs A graph is a star graph if there exists a vertex suh that all edges in the graph are inident to. For suh a graph, let A be the set of all verties that are not adjaent to and let B be the set of verties that are adjaent to. Algorithm 3 StarExat obtains an exat γ-labeling for a star graph G. (In fat, it ahieves an exat γ-labeling. The length of the generated labeling ould be redued by half by tightening this but is left for ease of explanation.) Figure 1 provides an example of the labeling assigned by StarExat. Algorithm 4 InverseStarExat obtains an exat inverse γ-labeling for a star graph G. Algorithm 3 StarExat(G, γ) 1. Let γ = max(γ, n ). Let x = min(γ, n ). Arbitrarily index the n verties of V as v 1, v,..., v n with = v n.. Set the first (n 1)γ bits of σ() to be 0 s. 3. For eah vertex v i, set the first (n 1)γ bits to be all 0 s exept for the i th size γ word whih is set to all 1 s. 4. Append γ 1 s to σ(a) for eah a A and γ 0 s to σ(b) and σ() for eah b B. 5. For eah v i A or v i = append x opies of S i to σ(v i) where S i is the i th string in ED(γ, n). 6. For eah v i B append x opies of S i ED(γ, n) to σ(v i ). 7. Output σ. Lemma 11. Algorithm 3 StarExat(G, γ) obtains an exat γ-labeling for a Star graph G with α = γn, β = γ + γn and length = O(γn), in run time O(γn ). Proof. We will first show the stated word length is ahieved, followed by the stated α and β values, and finally the stated run time omplexity. (Length) Step 1 of StarExat does not append any length to strings. Steps, 3, and 4 append a total of exatly nγ bits to eah of the n output strings. From Theorem 4, eah S i from ED(n, γ ) has length γ γ n. Thus, steps 5 and 6 together append exatly x (γ γ n ) < xγ = max{ n, γ} min{ n, γ} = γn. The total length is thus length(σ) < nγ = O(nγ). 8

9 Algorithm 4 InverseStarExat(G, γ) 1. Let γ = max(γ, n ). Let x = min(γ, n ). Arbitrarily index the n verties of V as v 1, v,..., v n with = v n.. Set the first γ bits of σ() to be 0 s. 3. For eah vertex v i B, set the first γ bits of σ(v i ) to be 1 s. 4. For eah vertex v i A, set the first γ bits of σ(v i ) to be 1 s and the seond γ bits of σ(v i ) to be 0 s. 5. For eah v i B or v i = append x opies of S i to σ(v i) where S i is the i th string in ED(γ, n). 6. For eah v i A append x opies of S i ED(γ, n) to σ(v i ). 7. Output σ. A B A B γ 1's (n-)γ 0's γ 1's S 1 S S 3 S 4 S 4 S 5 S 5 S 6 S 6 S 7 S 8 X opies S 1 S S 3 S 7 S S 1 S S 3 S 4 S 4 S 5 S 5 S 6 S 6 S 7 S 8 S 1 S S 3 S 7 S 8 (a) (b) Figure 1: (a) A star graph and (b) its orresponding exat labeling. (α and β) To show that the Hamming distane between eah pair of strings satisfies the α and β requirement, onsider two arbitrary verties v, u V. There are five ases to onsider. Case 1: v, u A. From steps 3 and 4 a distane of γ is obtained. From step 5 an additional distane of exatly x times the Hamming distane between two distint words from ED(n, γ ) is added, whih is exatly x γ = γn from Theorem 4. This yields a total Hamming distane of γ + γn. Case : v, u B. The analysis is the same as for ase 1. Case 3: v A, u B. Step 3 provides a Hamming distane of exatly γ. Step 4 adds an additional γ distane. By part 3 of Theorem 4 steps 5 and 6 add a Hamming distane of exatly x (γ γ n ). To finish the alulation, onsider two subases. 1. γ n. In this ase x (γ γ n ) = n (γ γ n ) = γn H(u, v) = γ + γn.. γ < n. In this ase x (γ γ n ) = γ ( n 1) = γn H(u, v) = γ + γn. γ. The total Hamming distane is thus γ. The total Hamming distane is thus Case 4: v A, u =. Steps, 3, and 4 reate a Hamming distane of exatly γ. By Theorem 4, step 5 adds another x γ = γn Hamming distane for a total of γ + γn. Case 5: v B, u =. Steps, 3, and 4 reate a Hamming distane of exatly γ. Steps 5 and 9

10 6 add an additional x (γ γ n ). As shown previously in ases 3.1 and 3., this value is always equal to γn γ, yielding a total Hamming distane of exatly γn. (Time Complexity) The time omplexity of Steps 1, and 4 are learly dominated by that of step 3 whih takes Θ(n γ). By Theorem 4 steps 5 and 6 eah take O(γn) time to ompute ED(n, γ). The rest of steps 5 and 6 are linear to the number of total bits appended for a total of O(n x γ ) = O(n γ). Finally, Step 7 is bounded by the size of the output O(n γ) whih is assured by the bound on the length of the labeling proven above. The total time omplexity is thus O(n γ). Lemma 1. Algorithm 4 InverseStarExat(G, γ) obtains an exat inverse γ-labeling for a Star graph G with α = γn, β = γ + γn and length = O(γn), in run time O(γn ). Proof. The run time omplexity and word length analysis is essentially the same as for Lemma 11. To show the γ-labeling is ahieved, onsider two arbitrary verties v, u V and the following five ases. Case 1: v, u A. σ(u) and σ(v) are idential exept for the appendage from step 6 whih yields, by Theorem 4 part, a Hamming distane of x γ = γn = α. Case : v, u B. By Theorem 4 parts and 5, the Hamming distane is exatly x γ = γn = α. Case 3: v A, u B. From steps 3 and 4 a Hamming distane of exatly γ is obtained. By part 3 of Theorem 4 steps 5 and 6 add an additional distane of exatly x(γ γ n ) = γn γ, for a total distane of exatly γn = α. Case 4: v A, u =. The analysis is idential to that of Case 3. Case 5: v B, u =. Steps and 3 reate a Hamming distane of exatly γ. Step 5, by parts and 5 of Theorem 4, add an additional xγ = γn distane. The total is thus γ + γn = β. 4 Labeling General Graphs To obtain a γ-labeling for a general graph, we deompose either the graph or its omplement into a olletion of star and mathing subgraphs. We then apply Lemmas 9 and 11 or Lemmas 10 and 1 to obtain exat or exat inverse labelings for these subgraphs, and then apply Theorem 8 to obtain a γ-labeling for the original graph. We first onsider Algorithm 5 MathingDeomposition that deomposes the general graph G into a olletion of mathings. 4.1 Mathing Deomposition Lemma 13. An edge deomposition of a graph G = (V, E) into maximal mathings ontains Θ(D) graphs where D is the maximum degree of any vertex in G. Proof. For any vertex, sine eah mathing ould ontain only one of its inident edges, the number of mathings in a deompositions is Ω(D). For the upper bound, we will prove that the maximum number of maximal mathings in the deomposition of a graph G is at most D 1. Suppose we remove a maximal set of edges whih form a mathing in suessive rounds of deomposition. We will argue that eah edge (x, y) E will be inluded in some deomposition within D 1 rounds. Sine eah round of deomposition omputes a maximal mathing M, (x, y) / M only if either x or y is an endpoint of an edge in M. In the former ase, one of the edges inident to x is inluded in M. Sine x is inident to at most D edges, this ase an our in at most D 1 rounds. Similarly, the latter ase an our in at most D 1 rounds. Therefore, the two ases an our in at most D rounds. Consequently, in D 1 rounds, (x, y) is inluded in some deomposition. 10

11 Algorithm 5 MathingDeomposition(G, γ) yields a γ-labeling σ of G with length(σ) = O(D (γ+ n)). For dense graphs whose verties are all of high degree, MathingDeomposition(G, γ) an be modified to deompose the omplement graph G into maximal mathings and apply the routine InverseMathingExat to obtain a length bound of O(D (γ + n)) where D is the maximum degree of any vertex in G. We thus get the following result. Algorithm 5 MathingDeomposition(G = (V, E), γ) 1. Deompose G into maximal mathings M 1,..., M r.. For eah mathing M i, generate an exat labeling σ i = MathingExat(M i, γ). 3. For every vertex v, σ(v) = σ 1 (v) σ (v) σ r (v). 4. Output σ. Theorem 14. For any graph G and γ, there exists a γ-labeling σ of G with length(σ) = O( ˆDγ+ ˆDn) that an be omputed in time omplexity O(γ ˆDn + ˆDn ) where ˆD is the smaller between the size of the maximum degree vertex in G and the maximum degree vertex in G. Proof. Let D denote the size of the maximum degree node in G and D denote the size of the maximum degree node in G. To obtain suh a labeling, apply Algorithm 5 MathingDeomposition if D D. If D > D, apply a modified version of Algorithm 5 MathingDeomposition to the omplement graph G in whih step assigns an inverse exat labeling to eah mathing aording to Algorithm InverseMathingExat. Thus, by Theorem 8 a γ-labeling is obtained for G in either ase. To bound the length, note that the labeling for eah mathing has length O(γ + n) by Lemmas 9 and 10. Further, by Lemma 13 the total number of mathings is at most ˆD, yielding a total label length of O( ˆDγ + ˆDn). The bound on run time follows from Lemmas 9 and Hybrid Deomposition (Star Destroyer) The next algorithm for obtaining a γ-labeling for a general graph adds the star deomposition to the basi mathing algorithm. The intuition is that, from Theorem 14, the mathing algorithm may perform poorly even if there are just a few very high and very low degree verties in the graph. In that ase, we an use the Star labeling to effiiently get rid of either the high or the low degree verties, and then finish off with the mathing algorithm. Theorem 15. For any graph G and γ, Algorithm 6 StarDestroyer(G, γ) yields a γ-labeling σ of G with length(σ) = O( γ ˆm n + γ ˆmn ) in time omplexity O( γ ˆm n 3 + γ ˆmn 4 ) where ˆm = min{ E, E 1 }. Proof. It is straightforward to see that the run time is linear in the output size. To show that the output labeling σ is a γ-labeling, we observe that σ(v) for eah vertex is the onatenation of labelings from StarExat from step and MathingExat from step 3. From Lemmas 9, 11, and Theorem 8, this thus yields a γ-labeling of G. To bound the length of the words, note that the while loop of step an only exeute at most ˆm x times. Thus, from Lemma 11, the length of eah string σ(v) after step is at most O( ˆmγn ). After x step, we know the degree of the graph is bounded by x, and thus know from Theorem 14 that the added length from step 3 is bounded by O(x (γ + n)). By hoosing x = two terms a total length of length(σ) = O( ˆmγn x ˆmγn ˆm+n to balane these + x (γ + n)) = O( γ ˆm n + γ ˆmn ) is ahieved. 11

12 Algorithm 6 StarDestroyer(G = (V, E), γ) 1. If E > E, then set INV true and G (V, E). Else, set INV false and G (V, E).. Let m = min{ E, E }. Set x m γn m +n. 3. While v V suh that deg(v) > x among edges in E Do: (a) Let T denote the star graph onsisting of vertex set V and the subset of E onsisting of all edges inident to v. (b) If INV= false, then ompute the exat labeling σ = StarExat(T, γ). () Else, ompute the exat inverse labeling σ = InverseStarExat(T, γ). (d) For eah u V, set σ(u) σ(u).σ (u) (e) Remove all edges inident to v from G. 4. If INV= false, then set σ = MathingDeomposition((V, E ), γ). 5. Else set σ = InverseMathingDeomposition((V, E ), γ). 6. For eah vertex v V, set σ(v) = σ(v).σ (v). 7. Output σ. 5 Trees and Forests In this setion we onsider input graphs that are trees or forests and show that we are able to obtain substantially smaller labelings than what is possible for general graphs. For a olletion of trees with a speial type of γ-labeling, we show how to ombine the olletion into a single speial γ-labeled tree. Thus, using reursive separators for trees we provide a reursive algorithm for tree labeling that ahieves a length of O(γD log n) where D is the largest degree node in the tree. We then show how to improve this bound with a more sophistiated algorithm that assigns labels effiiently to paths as a base ase, and reurses on the number of leaves in the tree rather than the number of nodes to ahieve a length of O(γD log f + max{γd log( n γd ), 0}) where f is the number of leaves in the tree. Note that this seond bound is always at least as good as the first, and for trees with few leaves but high γ, is better. For example, onsider the lass of graphs onsisting of log n length n log n paths, eah onneted on one end to a single node v. The number of nodes in this graph is n = n + 1, the highest degree node has degree D = log n, and the number of leaves is f = log n. For γ = n log n, the first bound yields l = O(n log n) while the seond yields l = O(n log log n). 5.1 Combining Trees To make our reursive algorithms work, we need a way to take labelings from different trees and effiiently reate a labeling for the tree resulting from ombing the smaller trees into one. To do this, we will make use of a speial type of γ-labeling. Definition 16 (Lower Bounded Labeling). A γ-labeling σ is said to be a lower bounded γ- labeling with respet to α a, α b, and β, α a α b < β, β α b γ if for any two nodes v and u, α a H(σ(v), σ(u)) α b if v and u are adjaent, and H(σ(v), σ(u)) β if they are not adjaent. Given a olletion of lower bounded labelings for trees, we an ombine the labelings into a new lower bounded labeling with the same parameters aording to Lemma 17. See Figure for an illustration of Algorithm 7 CombineTrees. For the rest of this setion, we will be dealing with 1

13 a parameter D whih will be an upper bound on the maximum degree value of the input graph suh that D + 1 is a power of greater than S S S S S S S S S S4 Figure : Given a lower bounded γ-labeling for four trees as shown, Algorithm CombineTrees first shifts the labeling so that the label for the root of eah of the trees is all 0 s. Then, S i ED(D + 1, γ(d +1) ) is appended to the label of eah node in the i th tree. To the entral node is appended the omplement of a final distint string SD +1. Lemma 17 (Combining Trees). Consider a vertex v in a tree T of degree t. Suppose for eah of the t subtrees of v we have a orresponding length at most l lower bounded γ-labeling σ i with β = γ(d +1), α b = β γ, and α a = γ for some D max{t, }, D + 1 a power of. Then, Algorithm 7 CombineTrees(T, v, {σ i } t i=1 ) omputes a lower bounded γ-labeling with the same α a, α b, and β values and length l l + γd. Proof. Let T 1,... T t denote the t subtrees of v, σ denote the output labeling from CombineTrees(T, v, {σ i } t i=1 ), and l denote the largest label size from the t labelings σ i. First note that the resultant length of σ is l plus the length of the words in ED(D + 1, γ(d +1) ) = γd (Theorem 4), whih yields a total length of l = l + γd. To show that CombineTrees preserves the lower bounded γ-labeling, onsider two arbitrary verties x and y in T. By a ase analysis we will show that the Hamming distane between σ(x) and σ(y) always satisfies the speified lower bounded labeling. Without loss of generality, it is suffiient to onsider the following 3 ases. Case 1: x and y are both in the tree T i for some i. In this ase, the Hamming distane between σ(x) and σ(y) has not hanged, and thus the preservation of the lower bounded labeling follows by assumption. Case : x is in T i, y is in T j, i j. As S i is appended to x and S j to y, the Hamming distane between σ(x) and σ(y) by Theorem 4 is at least β, whih is onsistent with the lower bounded labeling as x and y annot be adjaent. Case 3: x is in T i, and y = v. Case 3.1: x is adjaent to y = v. In this ase, σ(x) and σ(y) are equal (all 0 s) exept for the final appendage of S i to σ(x) and SD +1 to σ(y). The Hamming distane is thus β β D +1 = β γ by Theorem 4 whih is equal to α b, thus satisfying the lower bounded labeling. Case 3.: x is not adjaent to y = v. By assumption, σ i attains the speified lower bounded labeling. Thus, σ i (x) must have distane at least α a from the root v i of T i. And as the 13

14 Algorithm 7 CombineTrees(T = (V, E), v, {σ i } t i=1 ) Input: 1. A degree t vertex v in tree T with t D.. An α a = γ, α b = γ(d 1), β = γ(d +1) lower bounded γ-labeling σ i for eah subtree of v. Output: An α a = γ, α b = γ(d 1), β = γ(d +1) lower bounded γ-labeling of T. 1. For eah labeling σ i, append 0 s suh that length(σ i ) = max i=1...t {length(σ i )}.. For eah of the hild trees T 1,... T t of v, do (a) Let v i be the vertex in T i adjaent to v and let v i,j denote the value of the j th harater of σ i (v i ). For eah u T i, u v i, invert the j th harater of σ i (u) if v i,j = 1. (b) Set σ i (v i ) to all 0 s. 3. Let σ(v) be max i=1...t {length(σ i )} 0 s onatenated with S t+1 ED(D + 1, γ(d +1) ). Let σ(u) = σ i (u) for eah u T i. 4. For i = 1 to t (a) For eah u T i, σ(u) σ(u) S i for S i ED(D + 1, γ(d +1) ). 5. Output σ. labeling σ i is shifted in Step of the algorithm so that the label of v i is idential to that of v = y, the first part of σ i (x) with out the appended S i must have Hamming distane at least α a from σ(y). Finally, the strings S i and SD +1 add an additional distane of β γ (Theorem 4), for a total distane of at least α a + β γ = β. 5. Node Based Reursion Define a node separator for a graph to be a node suh that its removal leaves the largest sized onneted omponent with at most n verties. Given Lemma 17 and the well known fat that every tree has a node separator, we are able to label a tree by first finding a separator, then reursively labeling the separated subtrees using lower bounded labeling parameters α a = γ, α b = γ(d 1), and β = γ(d +1) for D = O(D). Sine it is trivial to obtain a lower bounded labeling satisfying suh α a, α b, and β for a onstant sized base ase tree, we an obtain a O(γD log n) bound on length of labelings for trees. We an then extend this to a t tree forest by reating t length γ(d +1) log t length strings suh that eah pair of strings has Hamming distane at least γ(d +1) and appending a distint string to the nodes of eah forest. This yields the following result. Theorem 18 (Node Reursive Forest). Given a forest F with maximum degree D, and an integer γ and a γ-labeling of F with length O(γD log n) an be onstruted in time O(nγD log n). Proof. Let D be the smallest integer greater or equal to the maximum degree of any node in the input graph suh that (D + 1) is a power of greater than. Now, assuming at first that F is a tree, onsider the simple reursive algorithm for omputing a lower bounded γ-labeling with β = γ(d +1), α b = β γ, and α a = γ. As a base ase, it is simple to attain the desired lower 14

15 bounded labeling for a graph that is a path of only one or two nodes. For larger graphs, we simply find a node separator, whih an be done in time O(n), reursively solve for the subtrees of the separator, and ombine using Lemma 17. By Lemma 17, the inrease in length of the labeling for eah ombination of trees is O(γD ), yielding a total length of O(γD log n) = O(γD log n). To extend this to a t tree forest, we first apply the desribed algorithm to eah tree of the forest, and then append a distint string to eah tree from a set of words suh that eah pair of distint words has a Hamming distane of at least β. This an be done trivially by attahing length O(β log t) = O(γD log n) strings, whih does not inrease the length omplexity or run time omplexity beyond what is ahieved for trees. (Time Complexity) We bound the work done for eah node as follows. Sine eah node has a label of length l, for every run of the CombineTree algorithm the work done on a node in the input to the algorithm is O(l). Sine the size of the tree a node belongs to is at least doubled when the tree is ombined by Algorithm CombineTree, the number of times any node partiipates in CombineTree is O(log n). Therefore the total work done for any node is O(l log n) whih gives a total runtime of O(nl log n) = O(nγD log n). 5.3 Leaf Based Reursion We now desribe a more sophistiated reursive algorithm for labeling trees that makes use of reursion based on the number of leaves in the tree rather than the number of nodes. The key idea is that we an effiiently label paths using a ombination of reursion for large paths and equidistant odes for shorter paths. Thus, we reursively break down the tree into trees with smaller numbers of leaves until we reah the base ase of a path for whih we employ our effiient path labeling algorithm. Path Labeling. As a base ase for our reursive algorithm, aording to Lemma 17, we want to be able to produe a short lower bounded γ-labeling for a path graph with β = γ(d +1), α b = γ(d 1), and α a γ for any given D. When alled from the tree algorithm, D will be on the order of the maximum degree of any node in the input tree. The Path algorithm will ahieve α a = β γ to satisfy the desired onstraints. The reason for this hoie of α a is that it is a power of, whih is neessary for our algorithm. The basi struture of the Path algorithm is that it uses reursion based on node separators and Lemma 17 until the path is suffiiently short. Then, a labeling based on the equidistant ode is used. Reursive Algorithm 8 Path ahieves the following result. Lemma 19. For D 3, D + 1 a power of, and path P, Algorithm 8 Path(P, γ, D ) generates a lower bounded γ-labeling σ of P with α a = γ(d +1) 4, α b = γ(d 1), β = γ(d +1), and length(σ) = O(max{γD log( n γd ), γd }) in time O(n (max{γd log ( n γd ), γd })). Proof. For paths of length at most γ(d + 1) 1, Step 1 assigns eah node a label of length O(γD) from Theorem 4. The Hamming distane between two adjaent nodes is the Hamming distane between two words in the equidistant ode ED(γ(D + 1), γ(d +1) 4 ) whih is γ(d +1) 4. The Hamming distane between non adjaent nodes is twie the Hamming distane between two words in ED(γ(D + 1), γ(d +1) 4 ) whih is γ(d +1). Therefore, the speified lower bounded γ-labeling is ahieved for the base ase. For longer paths with n nodes, Algorithm 8 Path breaks the path into two paths of length n and finds the labeling for eah reursively. The labeling for the original path is then found using Algorithm 7 CombineTree whih inreases the label length by O(γD ). The depth of reursion is 15

16 Algorithm 8 Path(P = v 1,... v n, γ, D ) 1. If n γ(d + 1) 1 then (a) Compute S 1,... S γ(d +1) ED(γ(D + 1), γ(d +1) 4 ). (b) For i = 1 to γ(d + 1) 1 do i. σ(v i 1 ) S i.s i ii. σ(v i ) S i.s i+1 () σ(v γ(d +1) 1) S γ(d +1).S γ(d +1) (d) Output σ.. Else (a) Let P 1 = v 1,..., v n 1, P = v n +1..., v n. (b) σ 1 Path(P 1, γ, D ), σ Path(P, γ, D ). () Output CombineTrees(P, v n, {σ 1, σ }). O(log n γd ) and using Lemma 17 the inrease in length of the labeling for eah ombination of trees is O(γD ), yielding a total length of O(max{γD log( n γd ), γd }). Analysis of time omplexity is similar to that of Theorem 18. Eah node has a label of length O(max{γD log( n γd ), γd }) and partiipates in CombineTree O(log( n γd ) times. Therefore the total runtime is O(n (max{γd log ( n γd ), γd log( n γd })) Leaf Reursive Tree Algorithm The leaf reursive tree algorithm reursively redues the number of leaves in the tree until the input is a simple path, for whih Algorithm Path an be used. For a tree T with f leaves, a leaf separator is a node suh that its removal redues the largest number of leaves in any of the remaining onneted omponents to at most f + 1. We start by observing that every tree must have a leaf separator. Lemma 0. Every tree has a leaf separator. Proof. The proof is by onstrution. Consider an arbitrary tree T with f leaves. Removing any node u in T ould lead to at most one onneted omponent with more than f + 1 leaves. Let that omponent be alled the offending omponent of u. For any node whih is not a leaf separator with the offending omponent having g leaves and h nodes, we will find a node whih is either a leaf separator or whose offending omponent has at most g leaves and h 1 nodes. Sine any omponent with at most f + nodes ould have at most f + 1 leaves, repeating this proess will eventually find a leaf separator. Consider an arbitrary node u in T. Let the adjaent nodes to u in T be v 1,..., v r. Consider the r trees T 1,..., T r obtained by removing u from T where T i is the tree ontaining v i. If all these trees have at most f +1 leaves we are done. Otherwise, note that only one of them an have more than f + 1 leaves. Without loss of generality, let T 1 be the tree with g leaves where g > f + 1 leaves. Now onsider removing v 1 from T. Of the trees obtained by this removal, note that the tree ontaining u will have at most f + 1 leaves and of the rest at most one ould have greater than f + 1 leaves. Let that tree be T. Note that number of leaves in T is at most g and the number of nodes in T is stritly less than that in T 1. 16

17 Note that a leaf separator always redues the number of leaves in a tree unless there are only leaves, in whih ase the tree is a path whih an be handled aording to Lemma 19. Having removed a leaf separator and reursively solved for the sub trees, we an then apply Lemma 17 to finish the labeling. The details of the algorithm are given as follows. Here, the input parameter D is the smallest integer suh that D + 1 is a power of and D is at least the degree of the highest degree node in the tree, or 3 in the ase of an input path. Algorithm 9 Tree(T = (V, E), γ, D ) 1. If Deg(T ), then output Path(T, γ, D ).. Else (a) Find a leaf separator v for T. (b) For eah of the hild trees T 1,... T t of v, σ i Tree(T i, γ, D ). () Output CombineTrees(T, v, {σ i } t i=1 ). Theorem 1 (Trees). Consider a tree T with f leaves and integer D = j 1 max{deg(t ), 3}. Then, Algorithm 9 Tree(T, γ, D ) omputes a length O(γD log f+max{γd log( n γd ), 0}) γ-labeling of T in time omplexity O(n (γd log f + max{γd log ( n γd ), 0})). Proof. This follows from Lemma 17, Lemma 19 and Lemma 0. To extend this result to a forest of trees T 1 T t, we an use Tree(T i, γ, D ) for eah individual tree. We an then append a distint string from a set of t strings to eah tree suh that the distane between any two strings is at least β = γ(d +1). Deterministially we an ahieve suh a set of strings trivially using additional length O(γD log t) where D = deg(t ). Alternately, we an use elements of ED(t, β) for an additional length of O(t+γD). These approahes yield the the following theorem. Theorem (Leaf Reursive Forest). There exists a deterministi algorithm that produes a length O(min{t, γd log t}+γd log f +max{γd log( n γd ), 0}) γ-labeling for an input forest F in time omplexity O(n (min{t, γd log t}+γd log f +max{γd log ( n γd ), 0})), where D = deg(f ), f is the largest number of leaves in any of the trees in F, and t is the number of trees in F. Proof. To ahieve this for a given forest F, we an first label eah tree in F using Theorem 1. We an then append strings of 0 s to make all labels of equal length. This yields label length O(γD log f+max{γd log( n γd ), 0}). Next, to obtain a γ-labeling for F, we an append a distint string S i from a list S 1,... S t to eah of the t trees in F suh that eah distint S i and S j has hamming distane at least γd. A satisfatory list of strings S i of length O(γD log t) an be obtained trivially. Alternately, a set of length O(γD + t) an be obtained by using the equidistant ode aording to Theorem 4. We thus get a total length bound of O(min{t, γd log t} + γd log f + max{γd log( n γd ), 0}). The time omplexity bound follows from Theorems 1 and 4. Alternately, we an use randomization to append shorter strings to eah tree and avoid an inrease in omplexity. Kao et al.[16] showed that with high probability, a set of n uniformly generated random binary strings has Hamming distane at least x between any two words with high probability for words of length at least 10(x + log n). Thus, we an produe a γ-labeling for a forest by first finding a labeling for eah tree, making the length of the labels equal, and finally piking a random string of length 10(β + log n) for eah tree and appending the string to eah of the nodes in the tree. We thus get the following result. 17

18 Theorem 3 (Randomized Forest). There exists a randomized algorithm that produes a length O(γD log f +max{γd log( n 1 γd ), 0}) γ-labeling for an input forest F with probability at least 1 n+, γ in time omplexity O(n (γd log f + max{γd log( n γd ), 0})), where D = deg(f ), and f is the largest number of leaves in any of the trees in F. Proof. This is ahieved in exatly the same fashion as in Theorem exept that the set S = {S 1,..., S t } an be onstruted using randomized algorithms as desribed in [16] whih yields words of length O(γD + log t) in O(t(γD + log t)) time. Note that both the γd and log t terms of the added length are absorbed by the length omplexity for building a single tree. 6 Distane Labeling, Rings and Paths In this setion we initiate the study of distane labelings for graphs using Hamming distane by generalizing the flexible word design problem. We show how this problem an be solved effiiently for a ertain lass of ring (rings are paths suh that the first and last verties of the path are adjaent) by applying some basi properties of hyperube traversal. We then show how this an be extended to obtain effiient distane labelings for all paths as well as a γ-labeling for any ring. We also note that even distane labeling for simple graphs suh as paths may already have use in the appliation of DNA omputing to digital signal proessing. Definition 4 ((k, γ)-labeling). A labeling σ of a graph G is said to be a (k, γ)-labeling if there exist non-negative integers α 1,..., α k, β 1,..., β k suh that β i α i γ and for eah pair of nodes u, v in G, if the distane d =dist(u, v) k, then β d 1 H(σ(u), σ(v)) α d, and if d > k, then β k H(σ(u), σ(v)). A (k, γ)-labeling is said to be lower exat if H(σ(u), σ(v)) = α d for d k. The goal of distane labeling is then to take as input a graph G and integers γ and k and output a (k, γ)-labeling of G. Note that for k = 1, this is the standard flexible word design problem. For the ase of rings and paths, we have the following lower bound. Theorem 5. The required label length for (k, γ)-labelings for both the lass of n-vertex rings and the lass of n-vertex paths is Ω(γk + log n). Proof. For n 3 eah label must be distint, thus yielding a bound of Ω(log n). Further, given k n for rings and k n for paths there must be two nodes of distane at least k. The labels for these verties must have Hamming distane, and thus length, of at least Ω(kγ) yielding a total bound of Ω(kγ + log n). Our algorithms in this setion make use of the following basi property onerning the traversal of nodes in a hyperube. Define a Hamiltonian yle of a graph to be a path suh that eah vertex is visited exatly one, and the first and last vertex visited are adjaent. Lemma 6. For every degree d hyperube H d and even integer q = j, there exists a size q subgraph H of H d suh that H has a Hamiltonian yle p. Further, given d and q, suh a path p an be omputed in time O( d ). Proof. To show this, onsider the algorithm HyperHamiltonian(H d, q) to ompute suh of length q Hamiltonian path H from a degree d hyperube H d. The orretness of HyperHamiltonian an be shown by indution on d. The base ase of d = 1 is lear. Indutively assume that HyperHamiltonian is orret for d 1. We then get by indutive hypothesis that a i is adjaent to a i+1 and b i is adjaent to b i+1 for i = 1 to q 1. And by design a q is adjaent to b q. The output is thus a Hamiltonian path of Hd. 18