1. 236801 Seminar in Computer Science 1 Two-Dimensional Patterns with Distinct Differences – Construction, Bounds, and Maximal Anticodes COMPUTER SCIENCE DEPARTMENT Technion - Israel Institute of Technology Simon R. Blackburn, Tuvi Etzion, Keith M. Martin and Maura B. Peterson

2. Content Introduction Grids & DDCs Anticodes DDCs - upper bounds DDCs constructions DDCs - lower bounds

3. Introduction Goal – determine the maximal number of dots that can be placed on a two dimensional grid such that: 1. the lines connecting two dots are different in either their length or their slope (avoid duplicate key pairs). 2.The distance between any two dots is at most r (communication range constraints).

4. Introduction (cont’) We consider 2 models for the grid – the square module and the hexagonal module. In every model we consider 2 distance functions (ways to measure r). In the square model we consider the Euclidian, and Manhattan distance. In the hexagonal model we consider the Euclidian, and hexagonal distance.

5. Grids Both models are graphs who’s vertices are In the square grid, neighbors are: In the hexagonal grid, neighbors are:

6. Grid (cont’)

7. Distance functions Euclidian – Manhattan – Hexagonal – the minimal number of hexagons between the 2 dots (the shortest path between the 2 vertices on the graph) The “natural” distance is the Euclidian distance, but both the Manhattan and Hexagonal distances are easy to work with and are good approximations.

8. DDC types All types of DDCs (of size m with radius r) satisfies: Any 2 dots in the configuration are at a distance of at most r apart. All the differences between pairs of dots are distinct either in length or slope.

9. DDCs types The DDCs differ in their grid and/or distance function: - square grid with Euclidian distance - square grid with Manhatten distance - Hexagonal grid with Euclidian distance - Hexagonal grid with Hexagonal distance

10. Anticodes An anticode of diameter r in a two dimentional grid (square or hexagonal) is a set S of points, such that for each pair of points we have where the distance can be Manhattan, hexagonal or Euclidian. An anticode is optimal if there exists no other anticode of the same diameter which is bigger. An anticode is maximal if it is not contained in a bigger anticode of the same diameter

11. Examples of anticodes a)Lee sphere (of radius 2) – anticode of radius 4 b)bicentred Lee sphere (of radius 2) - anticode of radius 5 c)quadricentred Lee sphere (of raidus 3) - anticode of radius 6

12. The following two results provide an obvious connection between DDCs and anticodes. Lemma 1. Any anticode S of diameter r is contained in a maximal anticode S′ of diameter r. Corollary 2. A DD(m, r) is contained in a maximal anticode of (Euclidean) diameter r. The same statement holds for all other types of DDCs when the appropriate distance measure is used.

13. A Lee sphere of radius R is the shape in the square model, which consists of one point as centre and all positions of Manhattan distance at most R from this centre.Lee sphere A bicentred Lee sphere of radius R is the shape in the square model, which consists of two centre points and all positions of Manhattan distance at most R from at least one point of this centre.bicentred Lee sphere A quadricentred Lee sphere of radius R is the shape in the square model, which consists of four centre points and all positions of Manhattan distance at most R − 1 from at least one point of this centre.quadricentred Lee sphere

14. Square maximal anticosed Theorem 3. For even r there are two different types of maximal anticodes of diameter r in square grid: the Lee sphere of radius r/2 and the quadricentred Lee sphere of radius r/2. For odd r there is exactly one type of maximal anticode of diameter r in the square grid: the bicentred Lee sphere of radius (r−1)/2.

15. Theorem5. There are exactly different types of maximal anticodes of diameter r in the hexagonal grid, namely the anticodes: is recived by “cutting” the square which corners are (0,0), (0,r), (r,r), (r,0) with the lines: y = x - i and y = x - i + r (after shifting the palin to the square grid) Hexagonal maximal anticosed

16. the number of grid points in is: The smallest anticode is, which is a triangle The largest anti-code is, which in the even case is a hexagonal sphere Hexagonal maximal anticosed (cont’)

17. We will look at Euclidian maximal anticodes in R, then cut them with the different grids. Euclidian maximal anticosed

18. DDCs - upper bounds Lemma 8. Let r be a non-negative integer. Let A be an anticode of Manhattan diameter r in the square grid. Let be a positive integer such that, and let w be the number of Lee spheres of radius that intersect A non- trivially. Then Theorem 9. If a exists then

19. Theorem 11. If a exists then Corollary 12. hexagonal sphere of radius R with one dot in each line exist for only a finite number of values of R.

20. Lemma 13. Let be a positive integer, and let S be a (Euclidean) circle of radius in the plane. Then the number of points of the square grid contained in S is Lemma 14. Let r be a non-negative integer. Let A be an anticode in the square grid of Euclidean diameter r. Let be a positive integer such that, and let w be the number of circles of radius whose centers lie in the square grid and that intersect A non- trivially. Then

21. Theorem 15. If a DD(m, r) exists, then Theorem 18. If a DD*(m, r) exists, then

22. DDCs constructions Welch construction Periodic Golomb Construction Folded ruler construction (seen in previous lectures) Doubly periodic folding construction

23. Remainder A Golomb ruler of order m is a set S of integers with |S| = m having the property that all differences a−b (for a, b ∈ S, with ) are distinct. Let A be an array of dots in the square grid, and let η and κ be positive integers. We say that A is doubly periodic with period (η, κ) if A(i, j) = A(i+η, j) and A(i, j) = A(i, j + κ) for all integers i and j. We define the density of A to be d/(ηκ), where d is the number of dots in any κ×η sub-array of A.

24. Doubly periodic folding construction Definition 5. Let A be an abelian group, and let be a sequence of m distinct elements of A. We say that D is a - sequence over A if all the sums with are distinct. Lemma 22. A subset is a - sequence over A if and only if all the differences with are distinct in A.

25. Let n be a positive integer and be a -sequence in. Let and k be integers such that. Let A be the square grid. For any integers i and j, there is a dot in A(i, j) if and only if for some t. Theorem24: Let A be the array of the Doubly Periodic Folding Construction. Then A is a doubly periodic DDC of period and density m/n. Doubly periodic folding construction

26. The LeeDD Construction Let r be an integer, and define R = ⌊ r/2 ⌋. Let be a ruler of length n. Define. Let A be the Lee sphere of radius R centered at (0, 0), so A has the entry A(i, j) if. We place a dot in A(i, j) if and only if.

27. The LeeDD Construction The construction is actually “folding” the ruler along diagonals:

28. Theorem 26: The Lee sphere A of the LeeDD Construction is a, where Corollary 27: There exists a in which The LeeDD Construction

29. Doubly Periodic LeeDD Construction Let r be an integer, R = ⌊ r/2 ⌋, and let be a B2-sequence over, where. Let mod n. Let A be the square grid. For each two integers i and j, there is a dot in A(i, j) if and only if Theorem28: The array A constructed in the LeeDD Construction is doubly periodic with period (n,n) and density μ/n. The dots contained in any Lee sphere of radius R form a DDC.

30. Euclidian distance DDC construction Now that we have a DDC with Manhatten distance we want to construct a DDC with Euclidian distance, with the optimal m. Let R = ⌊ r/2 ⌉, and let S be the set of points in the square grid that are contained in the Euclidean circle of radius R about the origin. Our goal is to show the best possible DD(m,r) construction contained in S.

31. Let S be a shape (a set of positions) in the square grid. We are interested in finding large DDCs contained in S, where (for example) S is an anticode. We write (i, j) + S for the shifted copy of S. Let A be a doubly periodic array. We say that A is a doubly periodic S-DDC if the dots contained in every shift (i, j) + S of S form a DDC.

32. Lemma 30: Let A be a doubly periodic S-DDC, and let S′ ⊆ S. Then A is a doubly periodic S′- DDC. Theorem31: Let S be a shape, and let A be a doubly periodic S-DDC of density δ. Then there exists a set of at least ⌈ δ|S| ⌉ dots contained in S that form a DDC.

33. To motivate our better construction, we proceed as follows. We find a square of side n where that partially overlaps our circle: Will be determined later.

34. Now we construct a DDC in the squere with density 1/n (Costas array). Define S’ as the intersection of S with the square. By theorem 31 we get that their exists a DDC contained in S’ with at least dots since

35. By maximizing (selecting n to determine ) we get μR dots, where μ ≈ 1.61589. Thus we get a DDC with (μ/2)r-o(r) ≈ 0.80795r dots.

36. Hexagonal constructions We now turn back to the hexagonal grid. We wish to construct the best DDCs for the hexagonal and Euclidian distances. We will build a DDC in a hexagonal sphere, then use this construction to build a DDC in an Euclidian circle.

37. diagonally extended Lee sphere For positive integers R and t, an (R, t)- diagonally extended Lee sphere is a set of positions in the square grid defined as follows. Let, and define:. Then an (R, t)- diagonally extended Lee sphere is the union of the Lee spheres of radius R with centers lying in C.

38. diagonally extended Lee sphere

39. An (R, t)-diagonally extended Lee sphere contains exactly positions by choosing we can generalize the doubly periodic LeeDD construction by continuing folding along the diagonals of the rectangle. This yields the following corollary:

40. diagonally extended Lee sphere Corollary 29: Let a be positive, and let n be an integer such that. Consider the array A constructed using the doubly periodic LeeDD Construction. Then A is a doubly periodic array with density μ/n. The dots contained in any (R, ⌊ aR ⌋ )-diagonally extended Lee sphere form a DDC. There exists a family of B2 sequences so that A has density at least.

41. If we shift an hexagonal sphere to the square grid, we get the shape: This shape is similar to a diagonally extended Lee sphere. Hexagonal sphere DDC

42. We use the same technique here as in the square Euclidian case and “cut” the sphere with a diagonally extended Lee sphere:

43. Hexagonal sphere DDC After the same process as before we can construct a DDC with: m = (μ/2)r − o(r) ≈ 0.79444r (μ ≈ 1.58887). This is done by optimizing a.

44. Euclidean distance in the hexagonal model The technique here is again the same. This time we use a rotated rectangle, which is transformed into a diagonally extended Lee sphere.

45. Euclidean distance in the hexagonal model We place the circle in the rectangle and cut them as before, we get:

46. conclusion