1. ECE 250 Algorithms and Data Structures Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada ece.uwaterloo.ca dwharder@alumni.uwaterloo.ca © 2006-2013 by Douglas Wilhelm Harder. Some rights reserved. Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada ece.uwaterloo.ca dwharder@alumni.uwaterloo.ca © 2006-2013 by Douglas Wilhelm Harder. Some rights reserved. Disjoint sets

2. 2 Outline In this topic, we will cover disjoint sets, including: –A review of equivalence relations –The definition of a Disjoint Set –An efficient data structure A general tree –An optimization which results in Worst case O(ln(n)) height Average case O(  (n))) height Best case  (1) height –A few examples and applications

3. 3 Disjoint sets Definitions Recall the properties of an equivalence relation: – a ~ a for all a – a ~ b if and only if b ~ a –If a ~ b and b ~ c, it follows that a ~ c An equivalence relation partitions a set into distinct equivalence classes Each equivalence class may be represented by a single object: the representative object –Another descriptive term for the sets in such a partition is disjoint sets

4. 4 Disjoint sets Implicitly Defined Relations For example, big-  defines an equivalence class of functions which grow at the same rate –We choose a single function to represent the class e.g., we represent all functions with quadratic growth with n 2 Another example: partition the numbers from 1 to 20 according to the relation a ~ b if a and b share the same factors of 2 : {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, {2, 6, 10, 14, 18}, {4, 12, 20}, {8}, {16} These equivalence relations are implicitly defined by the relation

5. 5 Disjoint sets Explicitly Defined Disjoint Sets Alternatively, a partition or collection of disjoint sets may be used to explicitly define an equivalence relation: – a ~ b if and only if a and b are in the same partition For example, the 10 numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 can be partitioned into the three sets {1, 2, 3, 5, 7}, {4, 6, 9, 0}, {8} Therefore, 1 ~ 2, 2 ~ 3, etc.

6. 6 Disjoint sets Explicitly Defined Disjoint Sets Consider simulating a device and tracking the connected components in a circuit This forms an equivalence relation: a ~ b if a and b are connected http://www.morphet.org.uk/ferro/s6mono.html

7. 7 Disjoint sets Operations on Disjoint Sets There are two operations we would like to perform on disjoint sets: –Determine if two elements are in the same disjoint set, and –Take the union of two disjoint sets creating a single set We will determine if two objects are in the same disjoint set by defining a function which finds the representative object of one of the disjoint sets –If the representative objects are the same, the objects are in the same disjoint set

8. 8 Disjoint sets Implementation Given two elements a and b, they are in the same set if find( a ) == find( b ) What find returns is irrelevant so long as: –If a and b are in the same set, find( a ) == find( b ) –If a and b are not in the same set, find( a ) != find( b ) We will have find return an integer

9. 9 Disjoint sets Implementation Here is a poor implementation: –Have two arrays and the second array stores the representative object –Finding the representative object is  (1) –However, taking the union of two sets is  (n) It would be necessary to check each array entry

10. 10 Disjoint sets Implementation As an alternate implementation, let each disjoint set be represented by a general tree –The root of the tree is the representative object To take the union of two such sets, we will simply attach one tree to the root of the other Find and union are now both O(h)

11. 11 Disjoint sets Implementation Normally, a node points to its children: We are only interested in the root; therefore, our interest is in storing the parent

12. 12 Disjoint sets Implementation For simplicity, we will assume we are creating disjoint sets the n integers 0, 1, 2,..., n – 1 We will define an array parent = new size_t[n]; for ( int i = 0; i < n; ++i ) { parent[i] = i; } If parent[i] == i, then i is a root node –Initially, each integer is in its own set

13. 13 Disjoint sets Implementation We will define the function size_t Disjoint_set::find( size_t i ) const { while( parent[i] != i ) { i = parent[i]; } return i; } T find (n) = O(h)

14. 14 Disjoint sets Implementation Initially, you will note that find( i ) != find( j ) for i != j, and therefore, we begin with each integer being in its own set We must next look at the union operation –how to join two disjoint sets into a single set

15. 15 Disjoint sets Implementation This function is also easy to define: void set_union( size_t i, size_t j ) { i = find( i ); j = find( j ); if ( i != j ) { // slightly sub-optimal... parent[j] = i; } The keyword union is reserved in C++ T set_union (n) = 2T find (n) +  (1) = O(h)

16. 16 Disjoint sets Example Consider the following disjoint set on the ten decimal digits:

17. 17 Disjoint sets Example If we take the union of the sets containing 1 and 3 set_union(1, 3); we perform a find on both entries and update the second

18. 18 Disjoint sets Example Now, find(1) and find(3) will both return the integer 1

19. 19 Disjoint sets Example Next, take the union of the sets containing 3 and 5, set_union(3, 5); we perform a find on both entries and update the second

20. 20 Disjoint sets Example Now, if we take the union of the sets containing 5 and 7 set_union(5, 7); we update the value stored in find(7) with the value find(5) :

21. 21 Disjoint sets Example Taking the union of the sets containing 6 and 8 set_union(6, 8); we update the value stored in find(8) with the value find(6) :

22. 22 Disjoint sets Example Taking the union of the sets containing 8 and 9 set_union(8, 9); we update the value stored in find(8) with the value find(9) :

23. 23 Disjoint sets Example Taking the union of the sets containing 4 and 8 set_union(4, 8); we update the value stored in find(8) with the value find(4) :

24. 24 Disjoint sets Example Finally, if we take the union of the sets containing 5 and 6 set_union(5, 6); we update the entry of find(6) with the value of find(5) :

25. 25 Disjoint sets Optimizations To optimize both find and set_union, we must minimize the height of the tree –Therefore, point the root of the shorter tree to the root of the taller tree –The height of the taller will increase if and only if the trees are equal in height

26. 26 Disjoint sets Worst-Case Scenario Let us consider creating the worst-case disjoint set As we are always attaching the tree with less height to the root of the tree with greater height, the worst case must occur when both trees are equal in height

27. 27 Disjoint sets Worst-Case Scenario Thus, building on this, we take the union of two sets with one element –We will keep track of the number of nodes at each depth 1

28. 28 Disjoint sets Worst-Case Scenario Next, we take the union of two sets, that is, we join two worst-case sets of height 1: 1 2 1

29. 29 Disjoint sets Worst-Case Scenario And continue, taking the union of two worst-case trees of height 2: 1 3 1

30. 30 Disjoint sets Worst-Case Scenario Taking the union of two worst-case trees of height 3: 1 4 6 4 1

31. 31 Disjoint sets Worst-Case Scenario And of four: 1 5 10 5 1

32. 32 Disjoint sets Worst-Case Scenario And finally, take the union of two worst-case trees of height 5: –These are binomial trees 1 6 15 20 15 6 1

33. 33 Disjoint sets Worst-Case Scenario From the construction, it should be clear that this would define Pascal’s triangle –The binomial coefficients 1 1 6 1 5 1 4 15 1 3 10 1 2 6 20 1 3 10 1 4 15 1 5 1 6 1

34. 34 Disjoint sets Worst-Case Scenario Thus, suppose we have a worst-case tree of height h –We need the number of nodes and the average depth of a node –Using Maple > sum( binomial( h, k ), k = 0..h ); > sum( k*binomial( h, k ), k = 0..h ); we get: –Therefore, the average depth is – The height and average depth of the worst case are O(ln(n))

35. 35 Disjoint sets Best-Case Scenario In the best case, all elements point to the same entry with a resulting height of  (1) :

36. 36 Disjoint sets Average-Case Scenario What is the average case? Could it be any better than O(ln(n)) ? –is there something better? To answer this, I created a program which, given the integers from 0 to 2 n – 1 continued to randomly choose numbers until all entries were in a single large set –For each n, I did this multiple times and found the mean (average) height

37. 37 Disjoint sets Average-Case Scenario The resulting graph shows the average height of a randomly generated disjoint set data structure with 2 n elements This suggests that the average height of such a tree is o(ln(n)) –See reference [1] for a detailed analysis

38. 38 Disjoint sets Average-Case Scenario The actual asymptotic behaviour is O(  (n)) where  (n) is the inverse of the function A(n, n) where A(m, n) is the Ackermann function: The first values are: A(0, 0) = 1, A(1, 1) =3, A(2, 2) = 7, A(3, 3) = 61

39. 39 Disjoint sets Average-Case Scenario However, A(4, 4) = 2 A(3, 4) – 3 where A(3,4) is the 19729- decimal-digit number A(3, 4) = 2003 5299 3040 68 46 4 6 49 79 0 7 2 35 1 56025575044782547556975141926501697371089405955631145308950613088093334810103823434290726318182294938211881266886950636476154702916504187191635158796634721944293092798208430910485599057015931895963952486337236720300291696959215610876494888925409080591145703767520850020667156370236612635974714480711177481588091413574272096719015183628256061809 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7885651464670604298564781968461593663288954299780722542264790400616019751975007460545150060291806638271497016110987951336633771378434416194053121445291855180136575558667615019373029691932076120009255065081583275508499340768797252369987023567931026804136745718956641431852679054717169962990363015545645090044802789055701968328313630718997699153166679208958768572290600915472919636381673596673959975710326015571920237348580521128117458610065152598883843114511894880552129145775699146577530041384717124577965048175856395072895337539755822087777506072339445587895905719156733 Thus, A(4, 4) – 3, in binary, is 1 followed by this many zeros.... http://xkcd.com/207/

40. 40 Disjoint sets Average-Case Scenario Therefore, we (as engineers) can, in clear conscience, state that the average run-time is  (1) as there are no physical circumstances where the average depth could by anything more than 4

41. 41 Disjoint sets Optimizations Another optimization is that, whenever find is called, update the object to point to the root size_t Disjoint_set::find( size_t n ) { if ( parent[n] == n ) { return n; } else { parent[n] = find( parent[n] ); return parent[n]; } The next call to find(n) is  (1) ; the cost is O(h) memory

42. 42 Disjoint sets Lotto 6/49 problem How do you randomly select 6 numbers from 1-49 without ever selecting the same number twice? int array[49] = {1, 2, 3, 4,..., 49}; for ( int i = 0; i < 6; ++i ) { size_t index = mrand48() % (49 - i); std::cout << array[index]; array[index] = array[49 - i - 1]; }

43. 43 Disjoint sets Lotto 6/49 problem Let’s play Lotto 2/20: Pick a random number from 0 to 19, say 7: Move the object in the last entry to its Now pick a random number from 0 to 18, and so on 1234567891011121314151617181920 1234567 91011121314151617181920

44. 44 Disjoint sets Lotto 6/49 problem This is an easy way to randomly permute a list: –The randomly chosen entry is moved to the last position int array[N]; for ( size_t i = 0; i < N; ++i ) { array[i] = i; } for ( size_t rng = N; rng >= 2; --rng ) { // pick an entry from 0 to rng - 1 size_t index = mrand48() % rng; // swap it and the last entry--it's okay if index == rng - 1 std::swap( array[index], array[rng - 1] ); }

45. 45 Disjoint sets Implementation Summary We now have two exceptionally fast operations: both find and union will run in  (1) time on average, and O(ln(n)) in the worst-case scenario

46. 46 Disjoint sets Application: Image Processing One common application is in image processing Suppose you are attempting to recognize similar features within an image Within a photograph, the same object may be separated by an obstruction; e.g., a road may be split by –a telephone pole in an image –an overpass on an aerial photograph

47. 47 Disjoint sets Application: Image Processing Consider the following image of the author climbing up the Niagara Escarpment at Rattlesnake Point Suppose we have a program which recognizes skin tones

48. 48 Disjoint sets Application: Image Processing A first algorithm may make an initial pass and recognize five different regions which are recognized as exposed skin –the left arm and hand are separated by a watch Each region would be represented by a separate disjoint set

49. 49 Disjoint sets Application: Image Processing Next, a second algorithm may take sets which are close in proximity and attempt to determine if they are from the same person In this case, the algorithm takes the union of: –the red and yellow regions, and –the dark and light blue regions

50. 50 Disjoint sets Application: Image Processing Finally, a third algorithm may take more distant sets and, depending on skin tone and other properties, may determine that they come from the same individual In this example, the third pass may, if successful, take the union of the red, blue, and green regions

51. 51 Disjoint sets Application: Maze Generation Another fun application is in the generation of mazes Impress your (non-engineering) friends –They’ll never guess how easy this is...

52. 52 Disjoint sets Application: Maze Generation Here we have a maze which spans a 500 × 500 grid of squares where: –There is one unique solution –Each point can be reached by one unique path from the start Ref: Lance Hampton http://littlebadwolf.com/mazes/

53. 53 Disjoint sets Application: Maze Generation Zooming in on the maze, you will note that it is rather complex and seemingly random Ref: Lance Hampton http://littlebadwolf.com/mazes/

54. 54 Disjoint sets Application: Maze Generation Finding the solution is a problem for a different lecture –Backtracking algorithms We will look at creating the maze using disjoint sets Ref: Lance Hampton http://littlebadwolf.com/mazes/

55. 55 Disjoint sets Application: Maze Generation What we will do is the following: –Start with the entire grid subdivided into squares –Represent each square as a separate disjoint set –Repeat the following algorithm: Randomly choose a wall If that wall connects two disjoint set of cells, then remove the wall and union the two sets –To ensure that you do not randomly remove the same wall twice, we can have an array of unchecked walls

56. 56 Disjoint sets Application: Maze Generation Let us begin with an entrance, an exit, and a disjoint set of 20 squares and 31 interior walls

57. 57 Disjoint sets Application: Maze Generation First, we select 6 which joins cells B and G –Both have height 0

58. 58 Disjoint sets Application: Maze Generation Next we select wall 18 which joins regions J and O

59. 59 Disjoint sets Application: Maze Generation Next we select wall 9 which joins the disjoint sets E and J –The disjoint set containing E has height 0, and therefore it is attached to J

60. 60 Disjoint sets Application: Maze Generation Next we select wall 11 which joins the sets identified by B and H –H has height 0 and therefore we attach it to B

61. 61 Disjoint sets Application: Maze Generation Next we select wall 20 which joins disjoint sets L and M –Both are height 0

62. 62 Disjoint sets Application: Maze Generation Next we select wall 17 which joins disjoint sets I and N –Both are height 0

63. 63 Disjoint sets Application: Maze Generation Next we select wall 7 which joins the disjoint set C and the disjoint set identified by B –C has height 0 and thus we attach it to B

64. 64 Disjoint sets Application: Maze Generation Next we select wall 19 which joins the disjoint set K to the disjoint sent identified by L –Because K has height 0, we attach it to L

65. 65 Disjoint sets Application: Maze Generation Next we select wall 23 and join the disjoint set Q with the set identified by L –Again, Q has height 0 so we attach it to L

66. 66 Disjoint sets Application: Maze Generation Next we select wall 12 which joints the disjoint sets identified by B and I –They both have the same height, but B has more nodes, so we add I to the node B

67. 67 Disjoint sets Application: Maze Generation Selecting wall 15 joints the sets identified by B and L –The tree B has height 2 while L has height 1 and therefore we attach L to B

68. 68 Disjoint sets Application: Maze Generation Next we select wall 5 which joins disjoint sets A and F –Both are height 0

69. 69 Disjoint sets Application: Maze Generation Selecting wall 30 also joins two disjoint sets R and S

70. 70 Disjoint sets Application: Maze Generation Selecting wall 4 joints the disjoint set D and the disjoint set identified by J –D has height 0, J has height 1, and thus we add D to J

71. 71 Disjoint sets Application: Maze Generation Next we select wall 10 which joins the sets identified by A and B –A has height 1 while B has height 2, so we attach A to B

72. 72 Disjoint sets Application: Maze Generation Selecting wall 31, we union the sets identified by R and T –T has height 0 so we attach it to I

73. 73 Disjoint sets Application: Maze Generation Selecting wall 27 joins the disjoint sets identified by J and R –They both have height 1, but J has more elements, so we add R to J

74. 74 Disjoint sets Application: Maze Generation Selecting wall 8 joins sets identified by B and J –They both have height 2 so we note that J has fewer nodes than B, so we add J to B

75. 75 Disjoint sets Application: Maze Generation Finally we select wall 23 which joins the disjoint set P and the disjoint set identified by B –P has height 0, so we attach it to B

76. 76 Disjoint sets Application: Maze Generation Thus we have a (rather trivial) maze where: –there is one unique solution, and –you can reach any square by a unique path from the starting point

77. 77 Disjoint sets Application: Maze Generation You may also note that the average depth is 1.6 whereas the average depth of the worst-case disjoint tree is 2:

78. 78 Disjoint sets Application: Maze Generation For fun, the following C code generates mazes of arbitrary length: char M[2],A,Z,E=40,J[40],T[40];main(C){for(*J=A=scanf("%d",&C); -- E; J[ E] =T [E ]= E) printf("._"); for(;(A-=Z=!Z) || (printf("\n|" ), A = 39,C -- ) ; Z || printf (M ))M[Z]=Z[A-(E =A[J-Z])&&!C & A == T[ A] |6<<27<rand()||!C&!Z?J[T[E]=T[A]]=E,J[T[A]=A-Z]=A,"_.":" |"];} It does not use disjoint sets...

79. 79 Disjoint sets Application: Maze Generation $ gcc maze.c maze.c: In function ‘ main ’ : maze.c:1: warning: incompatible implicit declaration of built-in function ‘ scanf ’ maze.c:3: warning: incompatible implicit declaration of built-in function ‘ printf ’ $./a.out 30._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._ |_._.._| |_.._._... |.._| |_.._._._|_._. | |_. |_. | |_. | | |._|_. | | |_._._. |_._._|.._|_|. |._._._._._._._|._._._| |. |_. |._|_._._. | | | | |._| |.... |_._._|_| |_| | | |_._. | |...._|_| | |_.._._._|._._.._| |_._._._|_|_|_|_|.... | |._.._._|_.._|_| |_|._...._._. |_..._|_. | |._|. |_... |_| |_| |_|_._. |_._._._|._._|_. |_._|_|_| |_. |_._. |_| | | | | |_._|_.._|_|. |_| | | | |_.._.._.._._. |_...._. |_._|_. |_. |._. | | |_._. |_._. |_. |_._._| | |... |._|._|_. | |_| |_._|_| |._._| |_._| |_| | |_._..._._._|._.. |._._| | | |_|._|_. |._|_.._._._._|_._._._._.._._._| |._. |_._._._|_. |_|_|_.._|_| |_._|..._|_._._|._|._|_._._._..._._.._| |_._|_|._._._| |_. |_. | |_._._|. | |_|_._._|_._. |._..._._. | |_| | |_| | | | | |_..._.._|._|_._| |._. |_. |_._..... | |._|._. |._._._.._| | |.._|._|_. |._| |._.._.. |_|_._._. | | | | |._|_._| |_. | |_. |_._|_. | | |_._._|. |_|_| | |_. |_. |_|_..._| |_| |_|_| |_._|._|._| |_| |.._._._| | | |.._|_._._._._| |._|_._._|_|_._._. |_. | |_..._| |. |_|._._| |_.._| |_._. |_._. |. |_.._|._...._._._. | |._|_. |_| | | |_|_|_. |_. |._._._| |_.... |_._|_.. |_._| | |_|. |_._. | |_|_._. |._._| |._._._. | |_._| | | | |_|_| |_._._. |_|._.._._|._|._._| | |...._._._._|_| | |._|._._| | | |._...._|_._._|_. | | |_._|_._._|_._|_| |_|_|_. | |. | |._. | |_.._|_. | |_|_. |_|_.. |_. |_._|_._._.._._...._|_._._| |._._|._|_. | | | |_. |_. | |_..._|._|._._._.. | | | |_._. |_|_| |_... |_| | |_|.._|_._.. |._._| |_. |_. |._|. | | | |_|_.._._._|_._|.. |_| |. |._|._|_| |_._. |_|_._. | |_._._|_|._|_|._._| |_. |.._._._|_._|_| |_._|_| |._|_._._.._. |_| | |_. | |_._._. |_| |_._._._|. | | |_|_._. |..._| |._|.._._|_._. | |_..._|._| |_.._._|_. | |_.._._|_. |_. |_.._. | |_|_.._| |_|_. | |_.._|._. |._._| | |_._._. |. | | |_| |_. |_. |_|_.._|_|_._.._|._. | | | | |._|_._| |_._._| | |. |._._|_._.._._|_.._._|._|.. |_. | |._. |._._|_. |_._.._| | | |_. | | |._|._._..._| |._._._. | |_| | | |_._. |_._|._.._._._._._._|_._. |_. | | |._|._. | |._|_._|_... | | |_|._. |_._._. |_._|_.._|_..._|. |._._| | | |._._|_|_|_._.._._|_|_|_. | |._. |_._| | |_|_. | |_._.._|_._._|_|._| | |_|_|_._._|_._._._._._._._._._._._._._|_|_._._._._._._._._._._|_._._._._._._._|

80. 80 Disjoint sets Application: Maze Generation The ECE 250 web site has a  (mn) algorithm for generating an m × n maze: http://ece.uwaterloo.ca/~dwharder/aads/Algorithms/Maze_generation The actual maze generation code is quite short: Disjoint_sets rooms( m*n ); int number_of_walls = 2*m*n - m - n; bool is_wall[number_of_walls]; Permutation untested_walls( number_of_walls ); for ( int i = 0; i 1 ) { int wall = untested_walls.next(); int room[2]; find_adjacent_rooms( room, wall, n ); if ( rooms.find( room[0] ) != rooms.find( room[1] ) ) { is_wall[wall] = false; rooms.set_union( room[0], room[1] ); } }

81. 81 Disjoint sets Summary In this topic, we have covered disjoint sets –Equivalence relations and the definition of a disjoint set –An efficient data structure A general tree –An optimization which results in Worst case O(ln(n)) height Average and best cases  (1) height –A few examples and applications

82. 82 Disjoint sets References [1]Cormen, Leiserson, and Rivest, Introduction to Algorithms, McGraw Hill, 1990, Ch.22, pp.440-461. [2]Weiss, Data Structures and Algorithm Analysis in C++, 3 rd Ed., Addison Wesley, Ch.8, pp.315-337.

83. 83 Disjoint sets References Wikipedia, http://en.wikipedia.org/wiki/Disjoint_set_(data_structure) [1]Cormen, Leiserson, and Rivest, Introduction to Algorithms, McGraw Hill, 1990, Ch.22, pp.440- 461. [2]Weiss, Data Structures and Algorithm Analysis in C++, 3 rd Ed., Addison Wesley, Ch.8, pp.315- 337. These slides are provided for the ECE 250 Algorithms and Data Structures course. The material in it reflects Douglas W. Harder’s best judgment in light of the information available to him at the time of preparation. Any reliance on these course slides by any party for any other purpose are the responsibility of such parties. Douglas W. Harder accepts no responsibility for damages, if any, suffered by any party as a result of decisions made or actions based on these course slides for any other purpose than that for which it was intended.