236801 seminar in computer science 1
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COMPUTER SCIENCE DEPARTMENT Technion - Israel Institute of Technology. 236801 Seminar in Computer Science 1. Two-Dimensional Patterns with Distinct Differences – Construction, Bounds, and Maximal Anticodes. Simon R. Blackburn, Tuvi Etzion , Keith M. Martin and Maura B. Peterson. Content. - PowerPoint PPT PresentationTRANSCRIPT
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236801Seminar in Computer Science 1
Two-Dimensional Patterns with Distinct Differences – Construction,
Bounds, and Maximal Anticodes
COMPUTER SCIENCE DEPARTMENTTechnion - Israel Institute of Technology
Simon R. Blackburn, Tuvi Etzion, Keith M. Martin and Maura B. Peterson
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Content
• Introduction• Grids & DDCs• Anticodes• DDCs - upper bounds• DDCs constructions• DDCs - lower bounds
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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).
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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.
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Grids
Both models are graphs who’s vertices are • In the square grid, neighbors are:
• In the hexagonal grid, neighbors are: 1, , , 1 , , 1 , 1,i j i j i j i j
2
,i j
,i j
, | , 1,0,1 , 0
1, 1 , 1, , , 1 , , 1 , 1, , 1, 1
i a j b a b a b
i j i j i j i j i j i j
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Grid (cont’)
2, ,
3 3
y yx y x
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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.
2 2
1 1 2 2 1 2 1 2, , ,d i j i j i i j j
1 1 2 2 1 2 1 2, , ,d i j i j i i j j
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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.
2
m
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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
,DD m r
,DD m r
* ,DD m r
* ,DD m r
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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
,x y S ,d x y r
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Examples of anticodes
a) Lee sphere (of radius 2) – anticode of radius 4b) bicentred Lee sphere (of radius 2) - anticode of radius 5c) quadricentred Lee sphere (of raidus 3) - anticode of radius 6
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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.
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• 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.
• 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.
• 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.
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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.
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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)
1
2
r
1
20 1, ,..., rA A A
Hexagonal maximal anticosed
iA
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• 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
iA
2 1 1 1 21
2 2 2
i i r i r i r rr i r i
0A
1
2
rA
Hexagonal maximal anticosed (cont’)
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• We will look at Euclidian maximal anticodes in R, then cut them with the different grids.
Euclidian maximal anticosed
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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
r
212
2w r O r
2 1
3 34/3
1 3
22m r r O r
,DD m r
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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.
* ,DD m r
2 14/33 3
5/3
3 3
2 2m r r O r
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• 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
2 O
2
2 1 / 24
w r
r
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• Theorem 15. If a DD(m, r) exists, then
• Theorem 18. If a DD*(m, r) exists, then
3
2/3 1/35/3
3
2 2m r r O r
5/6 3
2/3 1/34/34
3
22 3m r r O r
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DDCs constructions
• Welch construction• Periodic Golomb Construction• Folded ruler construction
(seen in previous lectures)
• Doubly periodic folding construction
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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.
a b
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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.
1 ,... mD a a A 2B
1 2i ia a
1 21 i i m
1 ,... mD a a A 2B
1 2i ia a1 21 i i m
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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 1 ,... mD a a
2B nZ
k
modta i j n
k n
/ . . , ,n g c d n n
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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 .
Ri j
2, iR j 1 R Rf i j R 1 2D a , a ,..., a
,f i j D
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The LeeDD Construction
The construction is actually “folding” the ruler along diagonals:
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• Theorem 26: The Lee sphere A of the LeeDD Construction is a , where
• Corollary 27: There exists a in which
20,1, ,..., 2 2m D R R
2
rm o r
,DD m r
,DD m r
The LeeDD Construction
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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.
1 2D a , a ,..., anZ
22 2 1n R R , 1f i j iR j R
,f i j D
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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.
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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.
', ' : ', 'i i j j i j S
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• 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.
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• 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.
2n R
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• 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
22
/ 2 2 sin 2' 2 / 2 2 sin 2
2cosS S R
2' / 2 / 2 2 sin 2 / 2 cosS n R R
2 cosn R
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• 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.
/ 2 2 sin 2 / cosR
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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.
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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.
20 0,i j Z
0 0, : 0 1C i k j k k t
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diagonally extended Lee sphere
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diagonally extended Lee sphere
• 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:
22 2 1R t R
22 2 1n R t R
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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 .
22 2n a R aR
2 21 / 2 2a R o R
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• 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
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Hexagonal sphere DDC
• We use the same technique here as in the square Euclidian case and “cut” the sphere with a diagonally extended Lee sphere:
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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.
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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.
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Euclidean distance in the hexagonal model
• We place the circle in the rectangle and cut them as before, we get:
2 / 3 0.79444r ( 1.58887)m r o r
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conclusion