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Priority Queue

Priority Queue. A Priority Queue Set S is made up of n elements: x 0 x 1 x 2 x 3 … x n-1 Functions: createEmptySet() returns a newly created empty priority queue findMin(S) returns the minimum node with respect to ordering insert(S, x) returns S with x added

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Priority Queue

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  1. Priority Queue A Priority Queue Set S is made up of n elements: x0 x1 x2 x3 … xn-1 Functions: createEmptySet() returns a newly created empty priority queue findMin(S) returns the minimum node with respect to ordering insert(S, x) returns S with x added deleteMin(S) returns S with the minimum node removed isEmptySet(S) returns true if S is empty and false if it is not

  2. Homework 6 • Describe how to implement a priority queue that has a worst case findMin in O(1) time and insert and deleteMin in no more than O(lg n) time. You can assume that n is always less than 128. In other words, there is a max of 127 elements that can be stored in the queue. • Do the five Priority Queue functions.

  3. Priority Queue – Array List • Ordered Array List • findMin in constant time. • insert and deleteMin in O(n). • Unordered Array List • insert in constant time. • findMin and deleteMin in O(n).

  4. Priority Queue – Linked List • Ordered Linked List • findMin and deleteMin in constant time. • insert in O(n). • Unordered List • insert in constant time. • findMin and deleteMin in O(n).

  5. Priority Queue Trees • Binary Search Tree • Find can be more than O(lg n) if out of balance. • Insert and delete can be more than O(lg n). • AVL Tree • Find is O(lg n) time. • Insert and delete are O(lg n).

  6. Priority Queue Trees • AVL Tree with pointer to smallest • Find is O(1) time. • Insert and delete are O(lg n). • Works, but is way too complicated for the task • We need a simpler solution

  7. Binary Tree

  8. Binary Tree -- Array 0 1 11 2 3 4 5 6 7 8 9 10 2 * i + 1 is the left child 2 * i + 2 is the right child

  9. Binary Tree -- Array 0 1 11 2 3 4 5 6 7 8 9 10 2 * i + 1 is the left child 2 * i + 2 is the right child

  10. Binary Tree -- Array 0 1 11 2 3 4 5 6 7 8 9 10 2 * i + 1 is the left child 2 * i + 2 is the right child

  11. Priority Queue 4 5 7 9 12 10 17 21 11 25 15 13 14

  12. 4 5 13 7 9 12 10 17 21 11 25 15 14 Priority Queue 4 5 7 9 12 10 17 21 11 25 15 13 14

  13. createEmptySet() Declare an array of type node. Declare an int n that is the number of elements in the queue. node s[128] n = 0 return s

  14. isEmpty(s) isEmpty(s) return (n == 0)

  15. findMin(s) findMin(s) return s[0]

  16. insert(s, x) 4 5 7 9 12 10 17 21 11 25 15 13 14 insert(s, 19)

  17. insert(s, x) 4 5 7 9 12 10 17 19 21 11 25 15 13 14 s(n) = 19 n = n + 1

  18. insert(s, x) 4 5 7 9 12 10 17 21 11 25 15 13 14 insert(s, 6)

  19. insert(s, x) 4 5 7 9 12 10 17 6 21 11 25 15 13 14 s(n) = 6 n = n + 1

  20. insert(s, x) 4 5 7 9 12 10 6 17 21 11 25 15 13 14 m = parent-of(n-1) if s[n-1] < s[m] swap(s[n-1], s[m])

  21. insert(s, x) 4 5 6 9 12 10 7 17 21 11 25 15 13 14

  22. deleteMin(x) 4 5 7 9 12 10 17 21 11 25 15 13 14

  23. deleteMin(x) 14 5 7 9 12 10 17 21 11 25 15 13 4 n = n - 1 swap(s[0], s[n])

  24. deleteMin(x) 5 14 7 9 12 10 17 21 11 25 15 13 4 if s[0] < either of its children swap with the least of its children

  25. deleteMin(x) 5 9 7 14 12 10 17 21 11 25 15 13 4

  26. deleteMin(x) 5 9 7 11 12 10 17 21 14 25 15 13 4 return s[n]

  27. Dictionaries A Dictionary is a set S made up of n elements: x0 x1 x2 x3 … xn-1 that has the following functions. Functions: createEmptySet() returns a newly created empty set lookUp(S, k) returns the node in S that has the key k insert(S, x) returns S with x added delete(S, k) returns S with x (found using x.k) removed isEmptySet(S) returns true if S is empty and false if it is not

  28. A Node NameLast: Smart FirstName: Joe StudentNumber: 8 SSN: 123341112 Grade: 95

  29. Name Some Dictionaries

  30. Name Some Dictionaries • Lists • Binary Search Trees • AVL Trees

  31. Can we create a dictionary that takes constant time, in the worst case, to do lookUp, insert, and delete?

  32. A Node NameLast: Smart FirstName: Joe StudentNumber: 8 SSN: 123341112 Grade: 95

  33. Use the key as an index • Create an array large enough to hold all the key possibilities. • If we used SSN this would be: node s[1000000000] • With this array we could look up a node using SSN as the key in constant time. • Insert and delete will also be constant.

  34. Key Direct • Do we want to use an array of a billion elements to store 21 nodes? • Is there a way to sacrifice some loss in speed to reduce the size of the array? • Can we still maintain O(1) for the average time of findMin, deleteMin, and insert? • What will the worst case time be?

  35. Homework 7 • Describe how to implement a dictionary that has an average lookUp, insert, and delete time that is constant, but uses an array of no more than 2*n elements. • Do the five Dictionary functions.

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