DE BRUIJN SEQUENCES FOR FIXED-WEIGHT BINARY STRINGS

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1 DE BRUIJN SEQUENCES FOR FIXED-WEIGHT BINARY STRINGS FRANK RUSKEY, JOE SAWADA, AND AARON WILLIAMS Abstract. De Bruijn sequences are circular strings of length 2 n whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length ( n w) for the binary strings of length n with weight (number of s) equal to w. In this case, each fixed-weight string can be encoded by its first n bits since the final bit is redundant. For this reason, we construct circular strings of length ( n ) ( w + n w ) whose length n substrings are the binary strings of length n with weight w or w. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of n bits are generated in constant amortized time (CAT) while using O(n log n)-space. This article s results were also used to create de Bruijn sequences for binary strings of length n with a specified maximum weight. Key words. de Bruijn sequences, universal cycles, FKM algorithm, necklaces, Lyndon words, cool-lex order, middle-levels, shift Gray code. Introduction. All strings in this paper are binary. Let B(n) denote the set of strings with length n. A de Bruijn sequence for B(n) (or simply a de Bruijn sequence) is a circular string of length 2 n that contains each string in B(n) exactly once as a substring. De Bruijn sequences are also known as de Bruijn cycles. A de Bruijn sequence for B(3) appears in Figure., where the substrings are read clockwise from 2 o clock and allow wrap-around. substrings B(3):,,,,,,, Fig... A de Bruijn sequence for B(3). One can prove that de Bruijn sequences exist for B(n) by using an Eulerian cycle in its associated de Bruijn graph (see Section 2). However, this standard proof does not directly lead to an efficient method for constructingan individual de Bruijn sequencedue to the exponentialsize ofthe associatedgraph(2 n nodes and 2 n directed arcs). A fundamental question is determining the computational complexity of generating a specific de Bruijn sequence. Perhaps the most famous construction that leads to an efficient algorithm is for the lexicographically least de Bruijn sequence, which Knuth calls the grand-daddy [5]. De Bruijn sequences have many applications including dynamic connections in overlay networks (Fraigniaud and Gauron [7]), genomics (Alekseyev and Pezner []), and software calculation of the ruler function in computer words (Knuth [6], Leiserson, Prokop, and Randall [7]). De Bruijn sequences also appear in textbooks on discrete mathematics (Graham, Knuth, and Patashnik []). Generalizations and variations have been investigated, most famously under the name universal cycles (Chung, Graham and Diaconis [4]). Interested readers can refer to the Generalizations of de Bruijn Cycles and Gray Codes proceedings [4]. Our paper gives a new variation of de Bruijn sequences that restricts the weight (number of s) of each string. Let B w (n) denote the set of length n strings with fixed-weight w and let B u l (n) denote the set of length n strings with weight-range l,l+,...,u having a specified lower-bound l and upper-bound u. In general, if L is a subset of B(n), then a de Bruijn sequence for L is a circular string of length L containing Department of Computer Science, University of Victoria, PO Box 3 STN CSC, Victoria BC, V8W 3N4, Canada ruskey@cs.uvic.ca. Research supported in part by an NSERC discovery grant. School of Computer Science, University of Guelph, 27 Reynolds, Guelph ON, NG 2W, Canada jsawada@uoguelph.ca. Research supported in part by an NSERC discovery grant. School of Mathematics and Statistics, Carleton University 25 Colonel By Drive, Ottawa ON, KS 5B6, Canada haron@uvic.ca

2 each string in L exactly once as a substring. Strictly speaking, de Bruijn sequences for B w (n) only exist in trivial cases when w {,,n,n}. For example, the circular strings of length ( 4 2) = 6 containing and are,, and but none are de Bruijn sequences for B 2 (4). However, we can take advantage of a simple fact: The last bit of each string in B w (n) is redundant. That is, each α B w (n) is completely determined by its first n bits. For this reason, we say that a de Bruijn sequence for B w w (n ) is a fixed-weight de Bruijn sequence for B w (n). The circular string in Figure.2 is a fixed-weight de Bruijn sequence for B 3 (5). Its substrings of length four include each string in B 3 2 (4) exactly once; appending the missing bit extends each substring to a unique string in B 3 (5). In general, the shorthand sequence of a fixed-weight de Bruijn sequence for B w (n) is its circular sequence of substrings of length n and the longhand sequence is obtained by appending the missing bit to each string in the shorthand sequence so that each resulting string has weight w. shorthand B 3 2(4):,,,,,,,,, longhand B 3 (5):,,,,,,,,, Fig..2. A fixed-weight de Bruijn sequence for B 3 (5). Our main result is a construction of fixed-weight de Bruijn sequences for any B w (n). A subsequent analysis shows that our cool-daddy de Bruijn sequences can be created efficiently, with successive blocks of n bits being generated in amortized O()-time while using only O(n log n)-space(sawada and Williams [25, 26]). This is an improvement over algorithms that construct universal cycles one symbol at a time (compare Ruskey and Williams [22] to Ruskey, Williams, and Holroyd [2, 3] for an example of this improvement in the context of permutations). The space measurement is also important since certain algorithms for generating universal cycles use exponential space. The mathematical foundation for our construction uses a general result involving binary bubble languages and cool-lex order (Ruskey, Sawada, and Williams [2]). Our paper is organized as follows: Section 2 covers de Bruijn graphs, Section 3 describes the granddaddy de Bruijn sequence, Section 4 modifies the aforementioned construction, Section 5 covers cool-lex order, Section 6 gives the cool-daddy construction, and Section 7 discusses open problems. We mention that articles in this research area often use the term density when referring to the number of s in a binary string. This includes work by the authors of this article ([2, 25, 26]) and by other authors (Buck and Wiedemann [2], Wang and Savage [29], and Ueda [28]). However, it is more natural to interpret the density of a length n binary string with w copies of as the fraction w/n. For this reason, we use the term weight in this article. 2. de Bruijn Graphs. Thede Bruijn graph forb(n)isadirectedgraphwhosenodesetisb(n ). For each node α = a a n and x {,} there is an arc labeled x that is directed from α to β = a 2 a n x. Each arc represents a unique string αx B(n). The de Bruijn graph for B(4) is illustrated in Figure 2.. Fig. 2.. The de Bruijn graph for B(4). More generally, the de Bruijn graph for L B(n) is a directed graph G(L) whose nodes are the length 2

3 n prefixes and suffixes of the strings in L. There is an arc labeled x {,} from α = a a n to β = a 2 a n x if αx L. Again, each arc represents a unique string αx L. We are interested in de Bruijn graphs for L = B w w (n). (a) (b) Fig Two de Bruijn graphs (a) G(B 2 (4)), and (b) G(B 3 2 (4)). A directed graph is Eulerian if it has a directed cycle that includes each arc exactly once. It is wellknown that a directed graph is Eulerian if and only if it is balanced (every node has the same number of incoming and outgoing arcs) and strongly connected (there is a directed path from any node to any other node). Furthermore, Eulerian cycles in G(L) are in one-to-one correspondence with de Bruijn sequences for L. For example, Figure 2.2 (a) shows that G(B 2 (4)) is not strongly connected, and this provides an alternate proof that there are no de Bruijn sequences for B 2 (4). The connection between de Bruijn sequences and de Bruijn graphs can be found in de Bruijn s paper for B(n) [5]; also see his note on the history of these observations [6]. Table 2. illustrates the connection between an Eulerian cycle in Figure 2.2 (b) and the fixed-weight de Bruijn sequence in Figure.2. Table 2. (i) and Figure 2.2 (b) also illustrate that the node set of G(B w w (n)) is B w w 2(n ). Eulerian cycle Substrings nodes arcs shorthand longhand B 3 (3) B() B 3 2(4) B 3(5) (i) (ii) (iii) (iv) Table 2. (i) The nodes along an Eulerian cycle in the de Bruijn graph G(B 3 2 (4)) from Figure 2.2 (b), (ii) arc labels on this Eulerian cycle, and the fixed-weight de Bruijn sequence for B 3 (5) in Figure.2, (iii) its shorthand sequence, and (iv) its longhand sequence. The remainder of this section shows that G(B w w (n)) is Eulerian. Since G(B w w (n)) is directed, we shorten directed path to path. Lemma 2.. G(B w w (n)) is balanced. Proof. The node set of G(B w w (n)) is B w w 2(n ). Each α B w (n ) has in- and out-degree 2, and each α B w 2 (n ) B w (n ) has in- and out-degree. The fact that many nodes in G(B w w (n)) have out-degree contributes to the difficulty of proving that it is strongly connected. As a specific example, a maximum length shortest path in G(B 9 8 (8)) is illustrated 3

4 length 3 length 85 (a) (b) Fig Shortest path from to in (a) G(B(8)) and (b) G(B 9 8 (8)). Nodes are read top-to-bottom with black and white squares respectively for and. in Figure 2.3 (b) and has length 85. In contrast, Figure 2.3 (a) illustrates that trivial paths of length at most n exist between each pair of nodes in the original de Bruijn graph G(B(n)). To prove that G(B w w (n)) is strongly connected we repeatedly apply the following lemma. Lemma 2.2. Let α = a a n and β = b b n be distinct nodes in G(B w w (n)). If i is the smallest integer such that a i b i, then there exists a path from α to some node with prefix b b i. Proof. We assume a i = since the a i = case is similar. Note that α has three possible weights since the node set of G(B w w (n)) is B w w 2(n ). For each weight we provide a valid path (labeled by the arcs) that ends at a node with prefix b b i. If α has weight w 2, then this path suffices:,a,...,a i,,a i+,...,a n. If α has weight w, then this path suffices:,a,...,a i,,a i+,...,a n. If α has weight w, then there exists i < j n such that a j =. (Otherwise, α = a a i n i has weight w, so α s prefix a a i has weight w. However, this implies β s prefix a a i has weight w +, which contradicts β B w w 2 (n ).) First we find a path from α to γ = a a j a j+ a n as follows:,a,...,a j,,a j+,...,a n. Since γ has weight w and has the same prefix of length i as α, we can complete our path by applying the path from the w case. Through repeated application of this lemma we obtain the following corollary. Corollary 2.3. G(B w w (n)) is strongly connected. We obtain the following theorem from Lemma 2. and Corollary 2.3. Theorem 2.4. G(B w w (n)) is Eulerian. Theorem 2.4 implies the existence of fixed-weight de Bruijn sequences. Section 6 strengthens this result by providing an explicit construction. 3. The FKM Algorithm. While de Bruijn graphscanbe used toprovethat de Bruijn sequencesexist, we are instead interested in efficiently constructing individual de Bruijn sequences. Martin [8] examined this issue in 934, and proposed a simple backtracking approach that builds a de Bruijn sequence for B(n) one bit at a time. In fact, by slightly modifying his presentation, the de Bruijn sequence he creates is the lexicographically least for each value of n. Unfortunately, Martin s approach is algorithmically infeasible since it requires exponential space. Fredericksen, Kessler and Maiorana [9, 8] discovered a direct method the FKM algorithm for constructing the lexicographically least de Bruijn sequence for B(n). Describing their method requires the introduction of several basic concepts. Given distinct strings α = a a n and β = b b m, α is less than β in lexicographic order if there exists an i such that a a i = b b i and either i = n or a i+ < b i+. The sequence of a set of strings L listed in lexicographic order is denoted lex(l). A necklace is a string in its lexicographically least rotation. That is, α = a a 2 a n is a necklace if a j a j+ a n a a 2 a j α for all j. The set of necklaces in 4

5 B(n) and B w (n) are denoted N(n) and N w (n) respectively. The aperiodic prefix of string α is its shortest prefix whose repeated concatenation yields α. That is, the aperiodic prefix of α = a a 2 a n is the shortest γ = a a 2 a k such that γ n/k = α, where exponentiation denotes repeated concatenation. The aperiodic prefix of α is denoted by ρ(α). If ρ(α) n/k = α, then the number of distinct rotations of α is k; we say that α is aperiodic if k = n and is periodic otherwise. A Lyndon word is an aperiodic necklace. The set of Lyndon words in B(n) and B w (n) are denoted L(n) and L w (n), respectively. The FKM algorithm [8] produces a circular string fkm(n) that is the concatenation of the Lyndon words whose length divides n in lexicographic order. That is, fkm(n) = l l 2 l m where lex = l,l 2,...,l m. (3.) j nl(n/j) Figure 3. illustrates fkm(6) with(a) showing the Lyndon words and(b) showing their circular concatenation with as a visual separator (Figure 3. (c) and (d) are explained in Section 4). The surprising connection between lexicographic order, the FKM algorithm, and de Bruijn sequences is given in Theorem 3.. Theorem 3. ( Grand-daddy [8]). fkm(n) is a de Bruijn sequence for B(n). Lyndon words L(),L(2),L(3),L(6) lexicographic order de Bruijn sequence fkm(6) necklaces N(6) aperiodic prefixes (a) (b) (c) (d) Fig. 3.. Concatenating the Lyndon words of length, 2, 3, 6 in lexicographic order in (a) gives the grand-daddy de Bruijn sequence fkm(6) in (b). This construction can also be obtained by concatenating the aperiodic prefix in (d) of the necklaces of length 6 in (c). The de Bruijn sequence in Theorem 3. is the lexicographically least de Bruijn sequence for each B(n). A careful analysis by Ruskey, Savage, and Wang [9] proved that each successive bit in fkm(n) can be generated in amortized O()-time while using O(n)-space. In fact, their algorithm visits successive blocks of n bits in this time and space complexity. Unfortunately, fixed-weight de Bruijn sequences are not created by restricting the FKM algorithm to the appropriate fixed-weight Lyndon words. To make this observation precise, let fkm w (n) = l l 2 l m where lex j gcd(w,n) L w/j (n/j) = l,l 2,...,l m. (3.2) Figure 3.2 illustrates that fkm 4 (8) is not a fixed-weight de Bruijn sequence. In particular, fkm 4 (8) has invalid substrings such as / B 4 3(7), and repeated substrings such as. Although fkm w (n) is not a fixed-weight de Bruijn sequence, it does have the correct length of ( n w). To understand why this is true, observe that if α B w (n) has k distinct rotations, then the rotations of α 5

6 Lyndon words necklaces aperiodic L 4(8) fkm 4(8) N 4(8) prefixes lexicographic order (a) (b) (c) (d) Fig Concatenating the Lyndon words of length 2, 4, 8 and weight, 2, 4 respectively in lexicographic order in (a) does not give a fixed-weight de Bruijn sequence fkm 4 (8) in (b). The substring is repeated, and the substring is invalid. This construction can also be obtained by concatenating the aperiodic prefix in (d) of the necklaces of length 8 and weight 4 in (c). will contribute k bits to fkm w (n). Since fkm w (n) has the correct length, we will consider rearranging its constituent Lyndon words in Section Necklace-Prefix Algorithm. In this section we reformulate the FKM algorithm and then provide a simple generalization. Instead of describing fkm(n) as the concatenation of Lyndon words whose length divides n, it can be described as the concatenation of the aperiodic prefixes of the necklaces of length n. That is, fkm(n) = ρ(η ) ρ(η 2 ) ρ(η m ) where lex(n(n)) = η,η 2,...,η m. (4.) To see why the concatenations in (3.) and (4.) are identical, simply observe that ρ(η i ) = l i. The fixedweight variant of fkm(n) can be similarly described as follows fkm w (n) = ρ(η ) ρ(η 2 ) ρ(η m ) where lex(n w (n)) = η,η 2,...,η m. (4.2) These two restatements of the FKM algorithm are illustrated in Figure 3. (c)-(d) and 3.2 (c)-(d). The advantage of (4.2) over (3.2) is that lexicographic order can be replaced by other previously developed orders of fixed-weight necklaces. For the remainder of this article, a necklace-prefix algorithm refers to the concatenation ofthe aperiodic prefixesof N w (n) arrangedin some order. The reasoningat the end of Section 3 explains why the necklace-prefix algorithm produces circular strings of the correct length. There are two previously developed orders for fixed-weight necklaces [29, 28]. However, in both cases the necklace-prefix algorithm does not produce a fixed-weight de Bruijn sequence due to invalid strings. This fact is explained by the following lemma, which provides a necessary condition on the weight of prefixes in consecutive aperiodic necklaces. Lemma 4.. Suppose L is an ordering of N w (n) that contains consecutive aperiodic necklaces α = a a n and β = b b n. If there exists j such that j i= j a i i= 6 b i / {,},

7 then applying the necklace-prefix algorithm to L will not result in a fixed-weight de Bruijn sequence for B w (n) due to invalid substrings. Proof. Observe that γ = a j+ a n b b j is an invalid substring since its weight can be computed as follows: Σγ = w Σ(a a j )+Σ(b b j ) / {w,w }. The invalid substring in Figure 3.2 is explained by Lemma 4. using α =, β =, and j = 6. The lemma suggests that the necklace-prefix algorithm should be applied to orders that do not significantly change the weight of each prefix. Such an ordering is discussed in the next section. 5. Cool-lex Order. This section discusses the cool-lex order for fixed-weight binary strings and necklaces. The reverse order for fixed-weight necklaces is used in the next section. Cool-lex order is a shift Gray code for B w (n), meaning that the ( n w) fixed-weight binary strings are ordered so that successive strings differ by a single shift. If α = a a 2 a n, then a shift from the jth position to the ith position with i < j causes the substring a i a i+ a j to be replaced by a j a i a i+ a j. In other words, the symbol a j is removed and then reinserted somewhere to the left in position i; the intermediate symbols accommodate this shift by moving one position to the right. This operation is denoted by shift α (j,i), which we shorten to shift(j,i) when the initial string is clear. There is a very simple rule for cyclically creating the cool-lex order of B w (n) one string at a time: If α B w (n) and k is the length of its longest prefix of the form, then the next string in cool-lex order is shift(min(k+2, n), ). This rule was discovered by Ruskey and Williams [2], although in our discussion all bits are complemented with respect to its original presentation. By convention, the last string in the cool-lex order of B w (n) is n w w. Table 5. (a)-(b) illustrates the cool-lex order of B 4 (8) and the shifts according to this rule. cool(b 4 (8)) Gray code cool(n 4 (8)) Gray code case condition reverse shift(3,) shift(4,) (5.b) a s+t+2 = shift(4, ) shift(6, 3) (5.c) shift(5, ) shift(5, 3) (5.c) shift(6,) shift(5,) (5.b) a s+t+2 = shift(3, ) shift(7, 3) (5.c) shift(5, 2) (5.c) shift(7, ) shift(3, ) (5.b) β / L shift(8, ) shift(5, ) (5.b) β / L shift(8, ) shift(7, ) (5.b) β / L shift(8, ) shift(8, 3) (5.a) (a) (b) (c) (d) (e) (f) (g) Table 5. (a) Cool-lex orders for B 4 (8) and (b) the shifts that generate this order. The cool-lex order of N 4 (8) appears in (c), along with the corresponding shifts according to (5.) in (d)-(f). Column (g) gives the reverse order of (c). Given L B w (n) let cool(l) represent the order of strings in L according to the cool-lex order of B w (n). Recently, it was shown that cool(n w (n)) is also a shift Gray code [2]. Furthermore, the following rule cyclically creates the order one string at a time [2]. Table 5. (c)-(f) illustrates the cool-lex order of N 4 (8) along with the shifts and cases according to this rule. Condition 5.b is restated in a slightly simplified form since n is the only necklace ending in. 7

8 Cool-lex Gray code for Necklaces [2] Let α = s t γ N w (n) where s,t > and γ is empty or begins with. The necklace following α in cool-lex order is denoted next(α) and is obtained from α by the following shift shift(s+t, i+) if γ = ǫ (5.a) next(α) = shift(s+t+,) if a s+t+2 = or β / N w (n) (5.b) shift(s+t+2, i+) otherwise (5.c) where β = shift α (s+t+2,s+t+), and i is the minimum value such that i s i t γ N w (n). In [26] it is proventhat cool(n w (n)) can be generatedin constant amortizedtime. Reverse cool-lex order is cool-lex order with the relative order of the strings reversed (see Table 5. (c) and (g)). The advantage of reverse cool-lex order is that it satisfies Lemma 4.. We complete this section with two results. Lemma 5.. [2] If α is a necklace, then swapping its first (if it exists) to yields another necklace. Lemma 5.2. If α is a periodic necklace, then next(α) is an aperiodic necklace. Proof. There are two cases. If a s+t+2 =, then next(α) = shift(s+t+,) by (5.b). If a s+t+2 =, then β = shift(s+t+2,s+t+) / N w (n) and so next(α) = shift(s+t+,) by (5.b). Therefore, s+ is a prefix of next(α) so it is aperiodic, since this is its only s+ substring. 6. Cool-Daddy de Bruijn sequences. Let C w (n) denote the result of applying the necklace-prefix algorithm to the reverse cool-lex order of the necklaces of length n and weight w. That is, C w (n) = ρ(η ) ρ(η 2 ) ρ(η m ) where cool(n w (n)) = η m,η m,...,η. (6.) Figure 6. illustrates that C 4 (8) is a fixed-weight de Bruijn sequence for B 4 (8), and this section proves this result in general. To simplify our presentation, we define an additional circular string D w (n) as the concatenation of the necklaces of length n and weight w without first reducing each necklace to its aperiodic prefix. In other words, D w (n) concatenates the necklaces in their entirety, regardless of whether they are periodic or aperiodic. That is, D w (n) = η η 2 η m where cool(n w (n)) = η m,η m,...,η. (6.2) Thelengthof D w (n)exceeds ( n w),soitssubstringsoflengthn mustcontainrepeatedstringsinb w w (n ). Although it has repeated strings, Theorem 6. proves that D w (n) does not miss any strings. In other words, D w (n) s substrings include each string in B w w (n ) at least once. Theorem 6.2 then completes our main result by proving that C w (n) contains each string in B w w (n) exactly once. In these proofs we let prev(α) denote the necklace before α in cool-lex order. That is, next(prev(α)) = α. Theorem 6.. The circular string D w (n) contains each string in B w w (n ) as a substring. Proof. Every string in B w w (n ) can be written as pq such that qxp N w (n) with x {,}. Our goal is to demonstrate a necklace α N w (n) such that pq is a substring of next(α) α, and thereby a substring of D w (n). Specifically, we will provide α with prefix q such that next(α) has suffix p. As a special case, if p or q is empty then clearly we can let α = qxp. If q has prefix s t where s,t >, then α = qxp suffixes (next(α) is obtained from (5.b) or (5.c)). Otherwise, since qxp is a necklace, we can assume that q = s t where s > and t. For this remaining case we consider the two possible values for x separately and assume that the longest prefix of the form in qp is i j where i,j >. 8

9 cool-lex order necklaces aperiodic fixed-weight de Bruijn sequence N 4(8) prefixes C 4(8) (a) (b) (c) Fig. 6.. Concatenating the aperiodic prefix in (b) of the necklaces of length 8 and weight 4 in reverse cool-lex order in (a) creates the cool-daddy fixed-weight de Bruijn sequencec 4 (8) in (c). The substrings of the cycle are the ( 8 4) = 7 strings in B 4 3 (7). Assume x =. If p = k, then α = qxp suffices (next(α) is obtained from (5.a)). Otherwise, consider two cases depending on t. t : Transpose the first to in qp to obtain α, which is a necklace by Lemma 5. (next(α) is obtained from (5.c)). Note that the first must occur after q, and hence α has prefix q. t = : If qp = i j then α = qp suffices (next(α) is obtained from (5.a)); otherwise obtain α by inserting x = into position i+j +2 (after the first ) of qp (next(α) is obtained from (5.c)). Assume x =. Again we consider two cases depending on t. t : Obtain α by inserting x = into qp as far right as possible up to position i+j + so that the resulting string is a necklace (next(α) is obtained from (5.b)). Note that the will be inserted after q since qp is a necklace. t = : If it is possible to insert x = past the first in qp to obtain a necklace, then apply α as described when t. Otherwise, construct α so that next(α) = qp. Observe that α has prefix q and next(α) is obtained by (5.b). Theorem 6.2. The circular string C w (n) is a fixed-weight de Bruijn sequence for B w (n). Proof. Since C w (n) hasthecorrectlengthof ( n w), weneedonlyshowthateverystringinb w w (n)appears as a substring in C w (n). From Theorem 6., this means that we need only show that every substring in D w (n) of length n is also a substring in C w (n). For this reason, let us consider an arbitrary periodic necklace N w (n) of the form γ k where γ is the aperiodic prefix. Since consecutive necklaces cannot both be periodic by Lemma 5.2, we must show that each length n substring of next(γ k ) γ k prev(γ k ) is also a substring of next(γ k ) γ prev(γ k ). This can be verified by applying the iterative cool-lex rules and considering two cases for γ where s,t > and ω is non-empty: γ = s t next(γ k ) γ prev(γ k ) = s t γ k 2 γ s t γ = s t ω next(γ k ) γ prev(γ k ) = γ k γ prev(γ k ). From this illustration, it should be clear in both cases that each length n substring in next(γ k ) γ k prev(γ k ) is also a substring of next(γ k ) γ prev(γ k ). 7. Summary and Open Problems. This paper provides an explicit fixed-weight de Bruijn sequence. It is constructed by concatenating the aperiodic prefixes of fixed-weight necklaces in reverse cool-lex order. 9

10 An algorithm in [26] shows that this fixed-weight de Bruijn sequence can be generated efficiently, with successive blocks of n bits being generated in amortized O()-time while using only O(n log n)-space. In addition to these results, we also investigated the de Bruijn graph G(B w w (n)). We conclude with additional observations and natural open problems:. Can specific constructions of weight-range de Bruijn sequences for B u l (n) with l < u be generated efficiently? Theorem6.2 provesthe answeris yes for l = u. Morespecifically, C w (n+) provides an explicit construction since its substrings of length n are precisely B u l (n) for l = w and u = w. Sawada, Stevens, and Williams [24] have solved the l = case. Their construction efficiently glues together copies of C i (n+) for i = w,w 2,w 4,... (and inserts a single to create the substring n when the maximum-weight u is even). This construction was recently extended to include all cases where u l is odd by Stevens and Williams [27] by using the necklace-prefix algorithm and a natural generalization of cool-lex order from B w (n) to B u l (n). The remaining open case is when u l is even, where the weight-range {l,l+,...,u} contains an odd number of values. 2. The set B w w (n) with n = 2w is known as the middle-levels. A well-known open problem is to determine if there is a Hamming distance Gray code for the middle-levels (see Savage and Winkler [23]). Our results have provided a universal cycle for the middle-levels. 3. The shorthand sequence of B w w (n ) appears in a single-track order when obtained from a fixed-weight de Bruijn sequence for B w (n) (see Hiltgen et al [] for single-track Gray codes). For example, see Table 2. (iii). Which other sets of binary strings have single-track orders? 4. The longhand sequence of B w (n) appears in a special cyclic order when obtained from a fixedweight de Bruijn sequence for B w (n): Successive strings differ by the prefix-rotation σ n or σ n. For example, see Table 2. (iv). It was proven that σ 2 and σ n cannot be used to create a Gray code for B w (n) by Cheng [3]. The sufficiency of σ n and σ n, and the insufficiency of σ 2 and σ n, are two special cases of a general question asked in [2]: Which sets of σ i are necessary and sufficient for generating a (cyclic) Gray code for B w (n)? 5. What is the maximum and minimum number of σ n that can be used in a (σ n,σ n ) Gray code for B w (n)? How many of each operation are used when B w (n) is ordered according to C w (n)? The analogous maximization problem has been solved for permutations by a natural construction [3]. 6. The grand-daddy de Bruijn sequence for B(n) is the first de Bruijn sequence for B(n) in lexicographic order. The cool-daddy de Bruijn sequences for B w (n) are neither the first nor last in lexicographic order or cool-lex order. For example, from Figure.2 is bracketed by the fixed-weight de Bruijn sequences and in both lexicographic and cool-lex order. What is the first fixed-weight de Bruijn sequence for B w (n) in lexicographic order, and can it be constructed directly without backtracking? 7. The necklace-prefix algorithm creates de Bruijn sequences when using lexicographic order, and creates fixed-weight de Bruijn sequences when using reverse cool-lex order. Are these orders special in this respect or are there many orders with these properties? 8. Removing the last redundant symbol of each string is also used in constructing universal cycles for permutations [22, 2, 3]. Similarly, shorthand representations are natural for any fixed-content language in which the frequency of each symbol is fixed. Which other fixed-content languages have shorthand universal cycles? A final open problem is to determine the diameter (length of the longest shortest path) of the de Bruijn graph for B w w (n), or more generally Bw u (n). For small values of n and w we computed the diameter of

11 G(B w w (n)) in Table 7., as well as pairs of nodes that achieve the maximum diameter for each n. (Table 7. assumes w n/2 since G(B w w (n)) and G(Bn w n w+ (n)) are isomorphic.) A conjecture appears below. n w = n/2,...,n (α,β) (, ) 6 9 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Table 7. Diameter of G(B w w (n)) for n 8. The (α,β) pairs give strings at maximum distance for each n. Conjecture 7.. The de Bruijn graph G(B w w (n)) has maximal diameter ( ) n+ 2 /2 when w = n/2 for (n mod 4) 3 and w = n/2 otherwise. Moreover, this maximal diameter is obtained by the nodes x y and a b c d for x = n 2, y = n 2, a = n 4, b = n+3 4, c = n+2 4, and d = n Acknowledgments. The authors thank Glenn Hurlbert and Garth Isaak for helpful discussions regarding de Bruijn graphs at canadam 29. REFERENCES [] Max A. Alekseyev and Pavel A. Pevzner. Colored de Bruijn graphs and the genome halving problem. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 4():98 7, January 27. [2] M. Buck and D. Wiedemann. Gray codes with restricted density. Discrete Mathematics, 48(2 3):63 7, 984. [3] Yongxi Cheng. Generating combinations by three basic operations. Journal of Computer Science and Technology, 22(6):99 93, 27. [4] F. Chung, P. Diaconis, and R.L. Graham. Universal cycles for combinatorial structures. Discrete Mathematics, :43 59, 992. [5] N.G. de Bruijn. A combinatorial problem. Koninkl. Nederl. Acad. Wetensch. Proc. Ser A, 49: , 946. [6] N.G. de Bruijn. Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of 2n zeros and ones that show each n-letter word exactly once. T.H. Report 75-WSK-6, Technological University Eindhoven, pages. [7] Pierre Fraigniaud and Philippe Gauron. D2B: A de Bruijn based content-addressable network. Theoretical Computer Science, 355():65 79, 26. [8] H. Fredericksen and I. J. Kessler. An algorithm for generating necklaces of beads in two colors. Discrete Mathematics, 6:8 88, 986. [9] H. Fredericksen and J. Maiorana. Necklaces of beads in k colors and kary de Bruijn sequences. Discrete Mathematics, 23(3):27 2, 978. [] R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete Mathematics. Adison Wesley, 994. [] A.P. Hiltgen, K.G. Paterson, and M. Brandestini. Single-track Gray codes. IEEE Transactions on Information Theory, 42(5):555 56, Sept [2] A. Holroyd, F. Ruskey, and A. Williams. Faster generation of shorthand universal cycles for permutations. In COCOON 2: The 6th Annual International Computing and Combinatorics Conference, volume 696 of Lecture Notes in Computer Science, pages , Nha Trang, Vietnam, 2. Springer-Verlag. [3] A. Holroyd, F. Ruskey, and A. Williams. Shorthand universal cycles for permutations. Algorithmica, 22, to appear.

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