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Heaps

This guide covers the essentials of array-based tree heaps, including the fundamental concepts of complete binary trees. Learn about max-heaps where each node's key is greater than its children, and min-heaps where it is lesser. The document details heap storage using vectors, insertion, and deletion algorithms, emphasizing the importance of reheapifying to maintain the heap's properties. Gain insight into the structure of heaps, where nodes are positioned based on their indices, enabling efficient data management.

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Heaps

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  1. Heaps CIS 237 – Data Structures

  2. Array Based Trees • Complete • full to h-1 • filled at level h from left to right • Stored as an vector, root in position 0 • For the node at position i • left child at 2i + 1 • right child at 2i + 2 • For a node at i the parent is at • For n nodes in the heap, the first leaf is at

  3. 78 15 32 5 10 20 0 1 2 3 4 5 78 15 32 5 10 20 Example

  4. Heap Definition • A binary tree • Complete • Max-Heap • the key value at each node is greater than its children • Min-Heap • the key value at each node is less than its children

  5. 78 v[0] 35 32 v[1] v[2] 5 10 20 19 v[3] v[4] v[5] v[6] 1 3 7 8 v[7] v[9] v[10] v[8] Heap Algorithms - Insert • Heap is a vector • Add by with push back • Restore heap order (reheapify)

  6. reheapify Algorithm • New value is pushed back and then moved to the appropriate location • v.push_back(value); • currentPos = last -1 • parentPos = (currentPos-1)/2 • target = v[last-1] • while (currentPos != 0) • 1. if (target > v[parentPos]) • 1. v[currentPos] = v[parentPos] • 2. currentPos = parentPos • 3. parentPos = (currentPos-1)/2 • 2. else • 1. break; //all moves done • 3. end if • end while • v[currentPos] = target

  7. 78 v[0] 35 32 v[1] v[2] 5 10 20 19 v[3] v[4] v[5] v[6] 1 3 7 8 v[7] v[9] v[10] v[8] Heap Algorithms – Delete • Root always deleted • Exchange root and last element • “Cut off” last element • Pushes new root down to proper place

  8. Adjust Algorithm • //Determine which child has the larger key • currentPos = 0 • target = heap[0] • childPos = 2 * currentPos + 1 • while (childPos <= last-1) //there is a left child • if childPos+1 <= last-1 then //there is a right child too • if v[childPos+1] > v[childPos] //right child bigger • childPos = childPos +1 //use the right child • end if • end if • ///Check for child being larger than parent….

  9. Delete continued • if v[childPos] > target //bigger than value so move up • v[currentPos] = v[childPos] • currentPos = childPos • childPos = 2 * currentPos + 1 • else • break //done • end if • end while • v[currentPos] = target

  10. Pop Heap temp = v[0] v[0] = v[last-1] adjustHeap(v, 0, last-1) return temp

  11. Heapifying a Vector • Start with the last non-leaf (position: n/2 -1) • adjustHeap(v, position, last-1) • Decrement position and adjust again

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