1. Priority Queue What is that? Implementation with linked list with O(n) behaviour The Heap (O(log(n)) An implementation using an array Your mission …

2. Q.insert(e) e = Q.removeMin() Q.size() Q.isEmpty() Q.min() Priority queue Store a collection of prioritized elements Elements are comparable Allow insertion of an element Can only remove the element with highest priority Comes first in order We present 3 implementations

3. Priority queue Example applications of a priority queue Dispatching processes in a computer Hospital waiting lists Standby passengers for a flight Queuing at call centres …

4. Store a collection of prioritized elements Elements are comparable Allow insertion of an element Can only remove the element with highest priority Comes first in order Priority queueHow can we compare objects? Two examples on comparing things … …

5. An exampleHow can we compare objects? This is a Vertex

6. An exampleHow can we compare objects? This is a Comparator for vertices

7. An exampleHow can we compare objects? Using the comparator to sort an array of Vertex

8. An exampleHow can we compare objects? Another example: a Car

9. An exampleHow can we compare objects? This is a CarComparator

10. An exampleHow can we compare objects? Using the CarComparator

11. unsorted list We might use a linked list To insert we add to the front of the list To find the minimum we must iterate over entire the list To remove the minimum we must find the minimum and remove it Maintain a counter of number of elements in the list MethodTime sizeO(1) isEmptyO(1) insertO(1) removeMinO(n) minO(n) Implementing a priority queue with an unsorted list

12. sorted list We might use a linked list The list is maintained in non-decreasing order To insert we scan to find position and splice in (see below) To find the minimum we deliver the first element in the list To remove the minimum we return and remove the first element Implementing a priority queue with an sorted list MethodTime sizeO(1) isEmptyO(1) insertO(n) removeMinO(1) minO(1)

13. An alternative the heap

14. heap a heap H is a binary tree

15. heap a heap H is a binary tree H is a complete binary tree

16. heap a heap H is a binary tree H is a complete binary tree Fill up level d before moving to level d+1 each level but the last must be full in last level fill from left to right

17. heap a heap H is a binary tree H is a complete binary tree Fill up level d before moving to level d+1 each level but the last must be full in last level fill from left to right

18. heap a heap H is a binary tree H is a complete binary tree Fill up level d before moving to level d+1 each level but the last must be full in last level fill from left to right

19. heap a heap H is a binary tree H is a complete binary tree Fill up level d before moving to level d+1 each level but the last must be full in last level fill from left to right Not a heap!

20. heap a heap H is a binary tree H is a complete binary tree heap order property is maintained

21. heap a heap H is a binary tree H is a complete binary tree heap order property is maintained Given a node v (not the root) the parent of v is less than or equal to v

22. heap a heap H is a binary tree H is a complete binary tree heap order property is maintained 4 5 915 6 207 Given a node v (not the root) the parent of v is less than or equal to v

23. heap a heap H is a binary tree H is a complete binary tree heap order property is maintained 4 5 915 6 207 Given a node v (not the root) the parent of v is less than or equal to v A heap H with n nodes has height O(log(n))

24. Example: adding to a heap

25. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 11 20

26. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 11 20 Insert 8

27. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 11 8 20 Insert 8

28. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 11 8 20 Insert 8 8 is greater than parent (7)... done

29. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 20

30. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 20 Insert 2

31. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 20 2 Insert 2

32. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 20 2 Insert 2 2 is less than parent 20

33. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 20 2 Insert 2 2 is less than parent 20... swap!

34. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 2 20 Insert 2

35. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 2 20 Insert 2

36. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 2 20 Insert 2 2 is less than parent 6

37. heap Example: adding to a heap 4 5 6 15 1625 9 1412 7 118 2 20 Insert 2 2 is less than parent 6... swap!

38. heap Example: adding to a heap 4 5 2 15 1625 9 1412 7 118 6 20 Insert 2 2 is less than parent 6... swap!

39. heap Example: adding to a heap 4 5 2 15 1625 9 1412 7 118 6 20 Insert 2

40. heap Example: adding to a heap 4 5 2 15 1625 9 1412 7 118 6 20 Insert 2 2 is less than parent 4

41. heap Example: adding to a heap 4 5 2 15 1625 9 1412 7 118 6 20 Insert 2 2 is less than parent 4... swap!

42. heap Example: adding to a heap 2 5 4 15 1625 9 1412 7 118 6 20 Insert 2 2 is less than parent 4... swap!

43. heap Example: adding to a heap 2 5 4 15 1625 9 1412 7 118 6 20 Insert 2

44. heap Example: adding to a heap 2 5 4 15 1625 9 1412 7 118 6 20 Insert 2 Done!

45. heap Example: adding to a heap 2 5 4 15 1625 9 1412 7 118 6 20

46. Example: removal from a heap NOTE: it is a heap in a different state

47. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 1113 20

48. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 1113 20 Save off top of heap

49. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 1113 20 Save off top of heap 4

50. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 1113 20 Copy last item in heap to top of heap 4

51. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 11 13 20 Copy last item in heap to top of heap 4

52. heap Example: removal from a heap 4 5 6 15 1625 9 1412 7 11 13 20 Copy last item in heap to top of heap 4

53. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 13 20 Copy last item in heap to top of heap 4

54. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 13 20 Delete last item in heap 4

55. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 20 4

56. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 20 Compare current node with its children 4

57. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 20 Compare current node with its children 4

58. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4

59. heap Example: removal from a heap 13 5 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4

60. heap Example: removal from a heap 5 13 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4

61. heap Example: removal from a heap 5 13 6 15 1625 9 1412 7 11 20 4

62. heap Example: removal from a heap 5 13 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4 Compare current node with its children

63. heap Example: removal from a heap 5 13 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4

64. heap Example: removal from a heap 5 13 6 15 1625 9 1412 7 11 20 If greater then swap with smallest child 4

65. heap Example: removal from a heap 5 9 6 15 1625 13 1412 7 11 20 If greater then swap with smallest child 4

66. heap Example: removal from a heap 5 9 6 15 1625 13 1412 7 11 20 4

67. heap Example: removal from a heap 5 9 6 15 1625 13 1412 7 11 20 4 Compare current node with its children

68. heap Example: removal from a heap 5 9 6 15 1625 13 1412 7 11 20 4 If greater then swap with smallest child

69. heap Example: removal from a heap 5 9 6 15 1625 13 14 12 7 11 20 4 If greater then swap with smallest child

70. heap Example: removal from a heap 5 9 6 15 1625 13 14 12 7 11 20 4 If greater then swap with smallest child

71. heap Example: removal from a heap 5 9 6 15 1625 12 14 13 7 11 20 4 If greater then swap with smallest child

72. heap Example: removal from a heap 5 9 6 15 1625 12 1413 7 11 20 4

73. 4 heap Example: removal from a heap 5 9 6 15 1625 12 1413 7 11 20 Return result

74. heap Example: removal from a heap 5 9 6 15 1625 12 1413 7 11 20 Done

75. upheap bubbling: when we add to the heap downheap bubbling: when we remove from the heap What we just saw Add and remove are O(log(n)) processes

76. An implementation of a Heap data structure

78. Number the vertices as follows

79. An implementation of a Heap data structure 1 23 4 89 5 1011 6 1213 7 1415 Number the vertices as follows

80. An implementation of a Heap data structure 1 23 4 89 5 1011 6 1213 7 1415 Number the vertices as follows Note: parent of node i is i/2....

81. An implementation of a Heap data structure 1 2 3 4 89 5 1011 6 1213 7 1415 Number the vertices as follows Note: parent of node i is i/2....

82. An implementation of a Heap data structure 1 23 4 89 5 10 11 6 1213 7 1415 Number the vertices as follows Note: parent of node i is i/2....

83. An implementation of a Heap data structure 1 2 3 4 89 5 1011 6 1213 7 1415 Number the vertices as follows Note: left child of i is i×2

84. An implementation of a Heap data structure 1 2 3 4 89 5 1011 6 1213 7 1415 Number the vertices as follows Note: right child of i is (i×2) +1

85. An implementation of a Heap data structure 1 23 4 89 5 1011 6 1213 7 1415 Represent as a one dimensional array

86. An implementation of a Heap data structure 1 23 4 89 5 1011 6 1213 7 1415 Represent as a one dimensional array S 0123456789101112131415 S

87. An implementation of a Heap data structure 1 23 4 89 5 1011 6 1213 7 1415 Represent as a one dimensional array S ** 0123456789101112131415 S To simplify implementation we do not use S[0]

88. An implementation of a Heap data structure ** 0123456789101112131415 Require two integer variables, last and capacity where last is initially 0 In our example capacity is 15 Represent as a one dimensional array S S

89. An implementation of a Heap data structure ** 0123456789101112131415 Represent as a one dimensional array S S last: 0 capacity: 15

90. An implementation of a Heap data structure Consider the following heap H

91. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 16 Consider the following heap H S last: 12 capacity: 15 **498172650161969329355 0123456789101112131415

92. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **498172650161969329355 0123456789101112131415 H.add(6)

93. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 6 16 S last: 12 capacity: 15 **4981726501619693293556 0123456789101112131415 S[last+1] = 6

94. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 6 16 S last: 13 capacity: 15 **4981726501619693293556 0123456789101112131415 last++

95. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 6 16 S last: 13 capacity: 15 **4981726501619693293556 0123456789101112131415 upheapBubble(13)

96. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 6 16 S last: 13 capacity: 15 **4981726501619693293556 0123456789101112131415 upheapBubble(13)

97. An implementation of a Heap data structure 4 98 17 1969 26 3293 50 55 6 16 S last: 13 capacity: 15 **4981726501619693293556 0123456789101112131415 upheapBubble(13) S[6] > S[13] ?

98. An implementation of a Heap data structure 4 98 17 1969 26 3293 6 55 50 16 S last: 13 capacity: 15 **4981726616196932935550 0123456789101112131415 upheapBubble(13) swap S[6] S[13]

99. An implementation of a Heap data structure 4 98 17 1969 26 3293 6 55 50 16 S last: 13 capacity: 15 **4981726616196932935550 0123456789101112131415

100. An implementation of a Heap data structure 4 9 8 17 1969 26 3293 6 55 50 16 S last: 13 capacity: 15 **4981726616196932935550 0123456789101112131415 upheapBubble(6)

101. An implementation of a Heap data structure 4 9 8 17 1969 26 3293 6 55 50 16 S last: 13 capacity: 15 **4981726616196932935550 0123456789101112131415 upheapBubble(6)

102. An implementation of a Heap data structure 4 9 8 17 1969 26 3293 6 55 50 16 S last: 13 capacity: 15 **4981726616196932935550 0123456789101112131415 upheapBubble(6) S[3] > S[6] ?

103. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 upheapBubble(6) swap S[3] S[6]

104. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415

105. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 upheapBubble(3)

106. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 upheapBubble(3)

107. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 upheapBubble(3) S[1] > S[3] ?

108. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 Done!

109. Removal from the heap H

110. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 H.remove()

111. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 Save S[1] 4

112. An implementation of a Heap data structure 4 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **4961726816196932935550 0123456789101112131415 S[1] = S[last] 4

113. An implementation of a Heap data structure 50 9 6 17 1969 26 3293 8 55 50 16 S last: 13 capacity: 15 **50961726816196932935550 0123456789101112131415 S[1] = S[last] 4

114. An implementation of a Heap data structure 50 9 6 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **509617268161969329355 0123456789101112131415 last-- 4

115. An implementation of a Heap data structure 50 9 6 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **509617268161969329355 0123456789101112131415 4 downHeapBubble(1)

116. An implementation of a Heap data structure 50 96 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **509617268161969329355 0123456789101112131415 4 downHeapBubble(1) findMin(S[2],S[3])

117. An implementation of a Heap data structure 50 96 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **509617268161969329355 0123456789101112131415 4 downHeapBubble(1) findMin(S[2],S[3])

118. An implementation of a Heap data structure 50 9 6 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **509617268161969329355 0123456789101112131415 4 downHeapBubble(1) swap(S[1],S[3])

119. An implementation of a Heap data structure 6 9 50 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **695017268161969329355 0123456789101112131415 4 downHeapBubble(1) swap(S[1],S[3])

120. An implementation of a Heap data structure 6 9 50 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **695017268161969329355 0123456789101112131415 4 downHeapBubble(1)

121. An implementation of a Heap data structure 6 9 50 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **695017268161969329355 0123456789101112131415 4 downHeapBubble(3) findMin(S[6],S[7])

122. An implementation of a Heap data structure 6 9 50 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **695017268161969329355 0123456789101112131415 4 downHeapBubble(3) findMin(S[6],S[7])

123. An implementation of a Heap data structure 6 9 50 17 1969 26 3293 8 55 16 S last: 12 capacity: 15 **695017268161969329355 0123456789101112131415 4 downHeapBubble(3) swap(S[3],S[6])

124. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 4 downHeapBubble(3) swap(S[3],S[6])

125. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 4 downHeapBubble(3)

126. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 4 downHeapBubble(3) findMin(S[12])

127. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 4 downHeapBubble(3) No swap

128. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 4 downHeapBubble(3) No swap

129. An implementation of a Heap data structure 6 9 8 17 1969 26 3293 50 55 16 S last: 12 capacity: 15 **698172650161969329355 0123456789101112131415 Return result 4

130. MethodTime sizeO(1) isEmptyO(1) insertO(log(n)) removeMinO(log(n)) minO(1) heap

131. Have a look at priority queue as given in Java distribution

133. … based on a priority heap

134. Different method names … add rather than insert

135. Exercise 4 (assessed) Implement the Heap Use it for sorting Your mission, should you choose to accept it …

136. fin