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Priority Queues

Priority Queues. Sections 6.1 to 6.5. Vector Representation of Complete Binary Tree. v[k].parent == v[(k-1)/2] v[k].lchild == v[2*k+1] v[k].rchild == v[2*k+2]. R. root. l. r. ll. lr. rl. rr. 0. 1. 2. 3. 4. 5. 6. Vector Representation of Complete Binary Tree.

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Priority Queues

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  1. Priority Queues Sections 6.1 to 6.5

  2. Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R root l r ll lr rl rr

  3. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R

  4. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l

  5. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l r

  6. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l r ll

  7. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l r ll lr

  8. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l r ll lr rl

  9. 0 1 2 3 4 5 6 Vector Representation of Complete Binary Tree • v[k].parent == v[(k-1)/2] • v[k].lchild == v[2*k+1] • v[k].rchild == v[2*k+2] R l r ll lr rl rr

  10. Priority Queue deleteMax Note: STL provides deleteMax. But book provides deleteMin. We’ll follow the deleteMax strategy.

  11. int main() { priority_queue<int> Q; Q.push(1); Q.push(4); Q.push(2); Q.push(8); Q.push(5); Q.push(7); assert(Q.size() == 6); assert(Q.top() == 8); Q.pop(); assert(Q.top() == 7); Q.pop(); assert(Q.top() == 5); Q.pop(); assert(Q.top() == 4); Q.pop(); assert(Q.top() == 2); Q.pop(); assert(Q.top() == 1); Q.pop(); assert(Q.empty()); } Priority Queue Usage

  12. STL Priority Queues • Element type with priority • <typename T> t • Compare & cmp • Associative queue operations • Void push(t) • void pop() • T& top() • void clear() • bool empty()

  13. Priority Queue Implementation • Implement as adaptor class around • Linked lists • Binary Search Trees • B-Trees • Heaps • This is what we’ll study • O( logN ) worst case for both insertion and delete operations

  14. Partially Ordered Trees • Definition: A partially ordered tree is a tree T such that: • There is an order relation >= defined for the vertices of T • For any vertex p and any child c of p, p >= c

  15. Partially Ordered Trees • Consequences: • The largest element in a partially ordered tree is the root • No conclusion can be drawn about the order of children

  16. Heaps • Definition: A heap is a partially ordered, complete, binarytree • The tree is completely filled on all levels except possibly the lowest 4 root 3 2 1 0

  17. Heap example • Parent of v[k] = v[(k-1)/2] • Left child of v[k] = v[2*k+1] • Right child of v[k] = v[2*k + 2] A B C D E F G H I J

  18. The Push-Heap Algorithm • Add new data at next leaf • Repair upward • Repeat • Locate parent • if tree not partially ordered • swap • else • stop • Until partially ordered

  19. The Push Heap Algorithm • Add new data at next leaf 7 6 5 4 3

  20. The Push Heap Algorithm • Add new data at next leaf 7 6 5 4 3 8

  21. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 7 6 5 4 3 8

  22. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 7 6 5 4 3 8

  23. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 7 6 8 4 3 5

  24. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 7 6 8 4 3 5

  25. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 7 6 8 4 3 5

  26. The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if tree not partially ordered • swap • else • stop 8 6 7 4 3 5

  27. The Pop-Heap Algorithm • Copy last leaf to root • Remove last leaf • Repeat • find the larger child • if not partially ordered tree • swap • else • stop • Until tree is partially ordered

  28. The Pop Heap Algorithm • Copy last leaf to root 8 6 7 4 3 5

  29. The Pop Heap Algorithm • Copy last leaf to root 5 6 7 4 3 5

  30. The Pop Heap Algorithm • Remove last leaf 5 6 7 4 3

  31. The Pop Heap Algorithm • Repeat • find the larger child • if tree not partially ordered • swap • else • stop 5 6 7 4 3

  32. The Pop Heap Algorithm • Repeat • find the larger child • if tree not partially ordered • swap • else • stop 5 6 7 4 3

  33. The Pop Heap Algorithm • Repeat • find the larger child • if tree not partially ordered • swap • else • stop 7 6 5 4 3

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