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Relations (2)

Relations (2). Rosen 6 th ed., ch . 8. Closures of Properties.

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Relations (2)

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  1. Relations (2) Rosen 6thed., ch. 8

  2. Closures of Properties • Let R be a relation on a set A. R may or may not have some property P, such as reflexivity, symmetry, or transitivity. If there is a relation S with property P containing R such that S is a subset of every relation with property P containing R, then S is called the closure of R with respect to P. • For any property X, the “X closure” of a set A is defined as the “smallest” subset of A that has the given property.

  3. Reflexive Closure • Reflective closure of R is R Δ, where Δ = {(a, a) | a  A} is the diagonal relation on A. • EXAMPLE: What is the reflexive closure of the relation R = {(1, 1), (1, 2), (2, 1), (3, 2)} on the set A = {1, 2, 3}? Solution: RΔ = {(1, 1), (1, 2), (2, 1), (3, 2)}  {(1, 1), (2, 2), (3, 3)} = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 2), (3, 3)}

  4. Reflexive Closure Cont. • EXAMPLE: What is the reflexive closure of the relation R = {(a, b) | a < b} on the set of integers? Solution: RΔ = {(a, b) | a < b}  {(a, a) | a Z} = {(a, b) | a ≤ b}.

  5. Symmetric Closure • R R- is the symmetric closure of R, where R- = {(b, a) | (a, b)  R}. • EXAMPLE: What is the symmetric closure of the relation R = {(a, b) | a > b} on the set of positive integers? Solution: RR- = {(a, b) | a > b}  {(b, a) | a < b} = {(a, b) | a ≠ b}.

  6. A case of transitivity closure • Consider the relation R = {(1, 3), (1, 4), (2, 1), (3, 2)} on the set {1, 2, 3, 4}. This relation is not transitive since (1, 2), (2, 3), (2, 4), and (3, 1) is not in R. • Adding (1, 2), (2, 3), (2, 4), and (3, 1), We get a relation {{(1, 3), (1, 2), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2)}. Is this relation transitive? => NO. • The relation contains (3, 1) and (1, 4) but does not contain (3, 4) • Constructing the transitive closure of a relation is more complicate than constructing either the reflective or symmetric closure.

  7. Paths in Directed Graphs • DEFINITION: A path from a to b in the directed graph G is a sequence of edges (x0, x1), (x1, x2), (x2, x3),…, (xn-1, xn) in G, where n is a nonnegative integer, and x0 = a and xn= b, that is, a sequence of edges where the terminal vertex of an edge is the same as the initial vertex in the next edge in the path. • This path is denoted by x0, x1, x2, … , xn-1, xn and has length n. We view the empty set of edges as a path from a to a. A path of length n ≥ 1 that begins and ends at the same vertex is called a circuit or cycle.

  8. Paths in Directed Graphs Cont. • Which of the following are paths in the directed graph shown below: a, b, e, d; a, e, c, d, b; b, a, c, b, a, a, b; d, c; c, b, a; e, b, a, b, a, b, e? What are the lengths of those that are paths? Which are circuit? • Paths: a, b, e, d(l=3); b, a, c, b, a, a, b(l=6); d, c(l=1); c, b, a(l=2); e, b, a, b, a, b, e(l=6) • Circuits: b, a, c, b, a, a, b; e, b, a, b, a, b, e a b c d e

  9. Paths in Directed Graphs Cont. • THEOREM: Let R be a relation on a set A. There is a path of length n, where n is a positive integer, from a to b if and only if (a, b)  Rn. R R R2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  10. Transitive Closure • DEFINITION: Let R be a relation on a set A. The connectivity relation R* consists of the pairs (a, b) such that there is a path of length at least one from a to b in R. • Because Rn consists of the pairs (a, b) such that there is a path of length n from a to b, it follows that R* is the union of all the sets Rn. In other words,

  11. Transitive Closure Cont. • EXAMPLE: R이 전 세계의 모든 사람들의 집합에 대한 관계이고 a가 b를 만난 적이 있으면 (a, b)를 포함한다고 하자. Rn (n은 1보다 큰 양의 정수)은 무엇을 의미하는가? 또한 R*는 무엇을 의미하는가? • 관계 R2는 (a, c)  R이고 (c, b)  R 인 사람 c가 있으면, 즉, a가 c를 만난 적이 있고c가 b를 만난 적이 있는 사람 c가 존재하면, (a, b)를 포함한다. • 비슷하게,Rn은 a가 x1을 만난 적이 있고, x1이 x2를 만난 적이 있고, … xn-1이 b를 만난 적이 있는 사람들 x1, x2, …, xn-1이 존재하는 (a, b)들로 구성된다. • 관계 R*는 어떤 순차(sequence) 내의 각 사람이 순차 내의 다음 사람을 만난 적이 있는, a에서 시작하여 b로 끝나는 사람들의 순차가 존재하면 (a, b)를 포함한다.

  12. Transitive Closure Cont. • THEOREM: The transitive closure of a relation R equals the connectivity relation R*. • THEOREM: Let MR be the zero-one matrix of the relation R on a set with n elements. Then the zero-one matrix of the transitive closure R* is

  13. Example of Transitive Closure. • EXAMPLE: Find the zero-one matrix of the transitive closure of the relation R where

  14. Example Cont. • Because ⊙ = and ⊙ =

  15. Simple Transitive Closure Alg. A procedure to compute R* with 0-1 matrices. procedure transClosure(MR: n x n 0-1 matrix) A := B := MR; for i := 2 to n begin A := A⊙MR; B := B  A {join}endreturn B {Alg. takes Θ(n4) time} {note A represents Ri}

  16. Warshall’sAlgorithm • 앞서 설명한 R*알고리즘의 시간 복잡도는 O(n4)이다. 따라서 좀 더 빠른 시간 안에 R*를 구할 수 있는 알고리즘이 필요했으며, 1960년에 Warshall이 새로운 알고리즘을 발표한다. • 내부 정점(internal vertex) • 만약a, x1, x2, …, xm-1, b가 경로(path)이면, 내부 정점은 x1, x2, …, xm-1이다. 즉, 맨 앞과 맨 뒤가 아닌 경로 상의 임의의 위치에 놓인 정점이 내부 정점이다.

  17. Warshall’sAlgorithm • Warshall의 알고리즘은 일련의 0-1 행렬을 구하는 것에 기초한다. 이들 행렬은 W0, W1, … Wn이며 W0은 MR(주어진 관계 R의 0-1 행렬)이고, 에서 는 모든 내부 정점이 집합 {v1, v2, … vk} (앞 부분 k개의 정점)의 원소인 vi에서 vj로의 경로가 존재하면 1이고, 그렇지 않으면 0이다. • 이렇게 해서 Wn(n은 관계 R이 정의된 집합의 원소의 수)을 구하면 Wn = MR*이다.

  18. Warshall’sAlgorithm • EXAMPLE: Let R be the relation with directed graph shown below. Let a, b, c, d be a listing of the elements of the set. Find the matrices W0, W1, W2, W3, and W4. The matrix W4 is the transitive closure of R. Solution: v1 = a, v 2 = b, v3 = c, v4 = d라 하자. W0 는 주어진 관계의 행렬이므로, W0 = . a b c d

  19. Warshall’sAlgorithm • W1 은 내부 정점으로v1 = a만을 포함하는 vi에서 vj로의 경로가 있으면 (i, j) 위치를 1로 한다. 길이 1인 이전의 경로들은 내부 정점이 없으므로 여기서도 모두 1이다. • 이제, b에서 d로 가는 경로 b, a, d가 허용되므로, W1 = . (이전 단계에서 (i, a) 위치와 (a, j)가 모두 1이었던 (i, j)들의 위치를 1로 한다.) a b c d

  20. Warshall’sAlgorithm • W2 는 내부 정점으로v1 = a와 v2 = b 만을 포함하는 vi에서 vj로의 경로가 있으면 (i, j) 위치를 1로 한다. b로 향하는 간선(edge)이 없으므로, 변화가 없다. W2 = . a b c d

  21. Warshall’sAlgorithm • W3 은 내부 정점으로v1 = a,v2 = b와 v3 = c 만을 포함하는 vi에서 vj로의 경로가 있으면 (i, j) 위치를 1로 한다. • d에서 a로 가는 경로 d, c, a와 d에서 d로 가는 경로 d, c, d가허용되므로, W3 = . (이전 단계에서 (i, c) 위치와 (c, j)가 모두 1이었던 (i, j)들의 위치를 1로 한다.) a b c d

  22. Warshall’sAlgorithm • W4 는 d까지 포함하여 모든 정점을 내부 정점으로 허용한다. 따라서, vi에서 vj로의 경로가 있는 모든 (i, j) 위치를 1로 한다 W4 = . a b c d

  23. Warshall’s Algorithm • Uses only Θ(n3) operations! Procedure Warshall(MR : rank-n 0-1 matrix) W := MR for k := 1 to n for i := 1 to n for j := 1 to nwij := wij (wik  wkj)return W {this represents R*} wij = 1 means there is a path from i to j going only through nodes ≤k

  24. Equivalence Relations • An equivalence relation (e.r.) on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive. • E.g., = itself is an equivalence relation.

  25. Equivalence Relation Examples • “Strings a and b are the same length.” • “Integers a and b have the same absolute value.” • “Real numbers a and b have the same fractional part (i.e., a− b Z).” • “Integers a and b have the same residue modulo m.” (for a given m>1)

  26. Equivalence Classes • Let R be any equiv. rel. on a set A. • The equivalence class of a, [a]R :≡ { b | aRb } (optional subscript R) • It is the set of all elements of A that are “equivalent” to a according to the eq.rel. R. • Each such b (including a itself) is called a representative of [a]R.

  27. Equivalence Class Examples • “Strings a and b are the same length.” • [a] = the set of all strings of the same length as a. • “Integers a and b have the same absolute value.” • [a] = the set {a, −a} • “Real numbers a and b have the same fractional part (i.e., a − b Z).” • [a] = the set {…, a−2, a−1, a, a+1, a+2, …} • “Integers a and b have the same residue modulo m.” (for a given m>1) • [a] = the set {…, a−2m, a−m, a, a+m, a+2m, …}

  28. Partitions • A partition of a set A is the set of all the equivalence classes {A1, A2, … } for some e.r. on A. • The Ai’s are all disjoint and their union = A. • They “partition” the set into pieces. Within each piece, all members of the set are equivalent to each other.

  29. An example of Partition • EXAMPLE: What are the partition of the integers arising from congruence modulo 4? Solution: [0]4 = {…, -8, -4, 0, 4, 8, …} [1]4 = {…, -7, -3, 1, 5, 9, …} [2]4 = {…, -6, -2, 2, 6, 10, …} [3]4 = {…, -5, -1, 3, 7, 11, …} These classes are disjoint, and every integer is in exactly one of them. They form a partition.

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