Constructions for Difference Triangle Sets

An (n; k)-difference triangle set, or (n;k)-D1S, is a set X = fX i j1 i ng, where X i = fa ij j0 j kg; for 1 i n; are sets of integers called blocks, such that the differences aij 0 a ij for 1 i n and 0 j 6 = j 0 k, are all distinct and nonzero. An (n;k)-D1S is normalized if for 1 i n, we have 0 = a i0 < a i1 < 11 1 < a ik : All difference triangle sets considered in this correspondence are normalized. The scope of an (n;k)-D1S; X = fX i j1 i ng; is defined as m(X ) = max a a 2 n i=1 Xi :

The smallest scope possible for an (n;k)-D1S is m(n; k) = minfm(X)jX is an (n;k)-D1Sg: An (n;k)-D1S X is optimal if m(X ) = m(n; k): By counting differences, we easily obtain the trivial lower bound m(n; k) n k + 1 2 :

Better lower bounds can be found in the papers of Kløve [1], [2]. In particular, we have the following result [1, Theorem 2]. Theorem 1 (Kløve): For all n and k m(n; k) n k 2 0 2k p k + k + p k 4 : Given an (n; k)-D1S X; we can obtain an (n 0 1;k)-D1S by omitting from X the block containing the largest element. This operation is called a reduction. The operation of omitting the largest element from each block of X is called shortening, and this gives an (n;k 0 1)-D1S instead.

There is a restricted variant of difference triangle sets, called regular perfect systems of difference sets (see [3] and [4]), that is widely studied in combinatorial design theory. Let c be a positive integer. An (n;k)-D1S X = fX i j1 i ng; where X i = faijj0 j kg; is a regular (n;k + 1;c)-PSDS if n i=1 faij 0 a ij j0 j 0 < j kg = c; c + 1;111;c 0 1 + n k + 1 2 :

The existence of a regular (n; k + 1;c)-PSDS gives an (n;k)-D1S of scope c 0 1 + n k+1 2 :

Difference triangle sets have a number of interesting applications in data communications (see [2] and [5]). For all of these applications, difference triangle sets with small scopes are desirable. Hence, the determination of m(n; k) is of importance. Unfortunately, this is a rather difficult problem. The special case of determining m(1; k) is the well-known problem of finding Golomb rulers, which has continued to resist many attacks. Only recently was it shown that m(1; 18) = 246 [6]. In general, except for the following result (see, for example, [5]) on m(n; 1); m(n; 2); and m(n; 3); only finitely many values of m(n; k) are known.

Theorem 2: For n 1; we have m(n; 1) = n and m(n; 2) = 3n;

if n 0 or 1(mod4) 3n + 1; if n 2 or 3(mod4).

There are infinitely many values of n for which m(n; 3) = 6n:

The value of m(n; 1) is trivial. The value of m(n; 2) is folklore and is derived from the existence of Skolem and Langford sequences [7], [8]. The result on m(n; 3) follows from the results of Kotzig and Turgeon [9], and Rogers [10] on the existence of regular (n; 4;1)-PSDS. The following conjecture was made by Bermond [11].

Conjecture 1 (Bermond): For every n 4, we have m(n; 3) = 6n:

Bermond’s conjecture has been verified for 4 n 22 [12].

For each k 2 f1;2;3g; the above results indicate that there exists an n such that m(n; k) meets the trivial lower bound. However, this phenomenon cannot persist, as it was shown in [13], [14] that m(n; k) = n k+1 2 only if k 3; or k = 4 and n is an even integer at least six. The establishment of good bounds for m(n; k) is, therefore, of interest.

Our concern in this correspondence is on the constructive aspects of difference triangle sets. The aim is to provide combinatorial as well as algorithmic constructions for difference triangle sets of small scope, thereby improving some of the existing upper bounds on m(n; k): More information on difference triangle sets can be found in [1], [2], [5], [15], and [16].

In this section, if f and g are two nonnegative functions, we use the notation f g to mean that there is an absolute constant C such that f C g: A (v; k; n)-difference packing, or n-DP (v;k), is a set X = fXij1 i ng, where Xi = faijj1 j kg, for 1 i n, are sets of residues modulo v such that for 1 i; i 0 n; 1 j 6 = j 0 k, and 1 6 = 0 k; we have a ij 0 a ij a i 0 a i (mod v) only if i = i 0 ; j = j 0

; and = 0 : Difference packings and difference triangle sets are intimately related in many ways. In particular, the following observation is made by Chen, Fan, and Jin [16].

0018-9448/97$10.00 © 1997 IEEE Lemma 1 (Chen, Fan, and Jin): An n-DP (v; k) is an (n; k 0 1)-D1S, whose scope is at most v 0 1:

Furthermore, using Singer’s construction [17] of 1-DP(q 2 + q + 1;q + 1) for prime powers q and a technique of Colbourn and Colbourn [18], they constructed two infinite families of difference packings, one of which is given below.

Theorem 3 (Chen, Fan, and Jin): For any prime power q and prime n > q, there exists an n-DP(n(q 2 + q + 1);q + 1):

exists and equals one. Here, we show that the same conclusion holds even if one allows k to grow with n, provided that it does not grow too fast. The following result of Heath-Brown and Iwaniec [19] on differences between consecutive primes is useful. Theorem 4 (Heath-Brown and Iwaniec): Let p n denote the nth prime. Then Proof: Suppose n > k: Let p and q be the smallest prime at least n and k, respectively, such that p > q: Then Theorem 3 assures us of the existence of a p-DP(p(q 2 + q + 1);q + 1): This difference packing is a (p;q)-D1S by Lemma 1. Hence, by repeated shortening and reduction (if necessary), we obtain an (n; k)-D1S whose scope m is upper-bounded by p(q 2 + q +1): However, Theorem 4 implies that p(q 2 + q + 1) (1 + o(1))nk 2 :

(1)

For n = 1, we use Singer’s 1-DP (q 2 + q + 1;q + 1) and follow the same argument above. This is only feasible for small values of n and k: Nevertheless, the main advantage of exhaustive search is that it allows us to prove the nonexistence of (n; k)-D1S of certain scopes.

The existing results on difference triangle sets present few unknown values of m(n; k) that can be determined exactly with today’s technology. The determination of m(2; 7) is one of these possibilities. It is known that 61 m(2; 7) 73 (see [1]). We proved that m(2; 7) = 70 by employing a backtracking algorithm that ran for about a week on a network of 30 machines for undergraduate mathematics students at the University of Waterloo. The blocks of a (2;7)-D1S of scope 70 are given below: X1 = f0; 1;4;24;40;54;67;69g and X2 = f0; 6;11;18; 28;37;62;70g: We did not attempt to find all (2, 7)-D1S of scope 70.

In the following sections, we turn to faster heuristics for constructing difference triangle sets.

We define a partial (n; k)-D1S to be a set X = fXij1 i sg satisfying all of the following conditions:

1. s n:

2. jXij = ki + 1 k + 1, for 1 i s:

3. X i = fa ij j0 j k i g is such that 0 = a i0 < a i1 < 1 11 < a ik ; for 1 i s: 4) The differences a ij 0a ij , for 1 i s, and 0 j 6 = j 0 k i are all distinct and nonzero.

The trivial partial (n; k)-D1S is the partial (n; k)-D1S X = fX i j1 i ng such that X i = f0g for 1 i n: With the above definition, an (n;k)-D1S is a partial (n;k)-D1S X = fX i j1 i sg, where s = n and jXij = k + 1 for 1 i s:

Every partial (n; k)-D1S X = fX i j1 i sg has a representation by an n 2 (k + 1) array R = (rij);1 i n and 0 j k, where each cell is either empty or contains a nonnegative integer. The entries of the nonempty cells in row i of R are exactly the members of Xi: Let R(n; k) denote the n 2 (k +1) array with all the cells in the zeroth column containing zeros and all other cells empty. Then R(n; k) is an array representation for the trivial partial (n; k)-D1S:

The greedy algorithms we propose can be conveniently described in terms of these array-representations for difference triangle sets.

Our first algorithm, called the set-greedy algorithm, works as follows. We begin with R(n; k): At each stage of the algorithm, we pick the smallest i such that the ith row contains an empty cell. We place in this empty cell the smallest positive integer such that the resulting array remains a representation of a partial (n;k)-D1S: The algorithm terminates when the array contains no empty cells. The idea behind this algorithm is to fill in the empty cells of R(n; k) in a row-by-row manner. This suggests the following variant which fills in the empty cells of R(n; k) column-wise.

The transversal-greedy algorithm also starts with R(n; k): At each stage of the algorithm, we pick the smallest j such that the jth column contains an empty cell. We then fill in the first empty cell of this column with the smallest positive integer so that the resulting array remains a representation of a partial (n; k)-D1S: The algorithm terminates when the array contains no empty cells.

It is evident that both of the above algorithms terminate with an array representation of an (n; k)-D1S: Empirical computational results show that the transversal-greedy algorithm outperforms the set-greedy algorithm. There is also an interesting connection between the transversal-greedy algorithm and a certain combinatorial game introduced by Wythoff [20] in 1907. In Wythoff’s game, there are two players who play alternately. Initially, there are two piles of matches, r in each pile. A player may take an arbitrary number of matches from one pile or an equal number of matches from each of the two piles, but he must take at least one match. The player who takes the last match wins the game.

The position of a player is the pair (u; v) where u is the number of matches left in one pile and v is the number of matches left in the other, immediately after his/her move. Without loss of generality, we assume that u v: A player’s position is winning if no matter what his/her opponent does, the player can force a win. Define the numbers u i and v i recursively as follows:

3. u i+1 is the smallest positive integer distinct from the 2i integers u 1 ; u 2 ; 11 1;u i ; v 1 ; v 2 ; 111; v i :

Connell [21] has shown that the winning positions for Wythoff’s game are exactly those pairs (u i ; v i ), for i 1, together with (0; 0): Theorem 6: Let R = (r ij ) be the array-representation of an (n; 2)-D1S constructed by the transversal-greedy algorithm. Then (r i2 0 n 0 i; r i2 0 n), for 1 i n, are winning positions for Wythoff’s game.

Proof: We show that ri2 0n0i = ui : The proof is by induction on i: It is easy to see that after the transversal-greedy algorithm fills the cells of the first column of R(n; k), we have r i1 = i, for 1 i n: These generates also the n differences 1;2;111;n: Hence, the smallest integer that the first cell in the second column can receive is n + 2: Hence, r1;2 0 n 0 1 = n + 2 0 n 0 1 = 1 = u1: Now assume that for some j 1, the entries r i2 filled in by the transversal-greedy algorithm satisfy ri2 0 n 0 i = ui for 1 i j: Then, the set of differences to be avoided is D = f1; 2;111;ng [ fr i2 j1 i jg [ fr i2 0 ij1 i jg:

We consider how the transversal-greedy algorithm next determines r j+1;2 : Clearly, r j+1;2 is the smallest positive integer such that i) r j+1;2 6 2 D; and ii) r j+1;2 0 (j + 1) 6 2 D:

These conditions are satisfied if and only if rj+1;2 0n0(j+1)6 2D 0 where D 0 = fr i2 0 nj1 i j g [ fr i2 0 n 0 ij1 i jg:

The induction hypothesis implies that D 0 = fu i j1 i j g [ fv i j1 i jg and hence r j+1;2 0n0(j +1) = u j+1 : This completes the proof.

Corollary 2: The scope of the (n; 2)-D1S constructed by the transversal-greedy algorithm is b(5 + p 5)=2)nc:

Proof: Follows from Connell’s result [21] that

It follows that the (n; 2)-D1S constructed by the transversalgreedy algorithm is only about a factor of 1:21 worse than the optimal. Analysis of the scope of (n;k)-D1S constructed by the transversal-greedy algorithm, for any k 3; seems difficult.

We describe in this section some randomized heuristics that have been very effective in constructing difference triangle sets of small scope. These heuristics fit into a general framework. An (n;k)template is a subset of f1; 2;111;ng2f0;1;111;kg: There is a natural correspondence between a set of cells of an n 2 (k + 1) array and an (n;k)-template.

Step 1 : Let R be an array representation of any (n; k)-D1S:

Step 2 : Let T be a set of (n; k)-templates:

Step 3 : Repeat Step 4 to Step 6 N times:

Step 4 : Let s be the scope of the (n; k)-D1S represented by R:

Step 5 : Randomly choose (n; k)-template in T and delete the entries in the cells of R corresponding to that template.

Step 6 : Find all possible ways of filling in the empty cells of R using nonnegative integers no greater than s: Randomly choose one of these ways and fill in the empty cells of R accordingly:

The final difference triangle set constructed by the randomized heuristic have scope at most that of the initial difference triangle set, and strictly improves on the scope if at any stage of the algorithm, the cell containing the largest element of the array is emptied and this element is never used to refill any cell.

The sets of templates T that we find most effective in constructing difference triangle sets of small scope are the three listed below. 1) T 1 = ff(i; j)gj1 i n; 0 j kg; 2) T2 = ff(i;j)j0 j kgj1 i ng; 3) T 3 = ff(i; j i )j1 i ngj0 j i kg:

The templates T 1 ; T 2 ; and T 3 correspond to emptying single cells, emptying the cells in an entire row, and emptying one cell from each row in Step 5 of the randomized heuristic, respectively. Let us denote by i the randomized heuristic that uses T i : Naturally, 1 is the fastest. 2 and 3 are much slower but generally give better results when n and k are small. 2 is effective when n=k is large whereas 3 is effective when n=k is small. These heuristics work well when the initial array-representation of an (n; k)-D1S used in Step 1 is that constructed by the transversal-greedy algorithm. We used these three heuristics in combination, typically in the order 1 ; 2 ; 3 or 1; 3; 2; to obtain a number of improvements on the upper bounds for m(n; k): Instances of these improvements are given in Table I, where our improved upper bound is given above the best previous upper bound. The bounds in parentheses “( )” are due to Kløve [1], those in brackets “[ ]” are due to Chen, Fan, and Jin [16], and those in braces “f g” are due to Chen [15]. The blocks for difference triangle sets with the improved scopes are available from the authors.

One of the problems suggested by the results in Section II is the determination of the asymptotic behavior of m(n; k): Our results show that for f (n) satisfying lim sup n!1 f (n)=n < 1; we have lim n!1 m(n; f (n))=n(f (n)) 2 = 1: It would be interesting to know what happens if f (n) is allowed to grow at a faster rate.

We have also described algorithms that are used to construct difference triangle sets with the best known scopes for many intermediate values of n and k:

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