Binary Heap

Data Structure

A binary heap is a heap data struc­ture cre­ated using a binary tree.

Binary heap has two rules:

• Shape prop­erty: Binary Heap has to be com­plete binary tree at all lev­els except the last level.
• Heap prop­erty: All nodes are either greater than equal to (Max-Heap) or less than equal to (Min-Heap) to each of its child nodes.

Implementation:

• Use array to store the data.
• Start stor­ing from index 1, not 0.
• For any given node at posi­tion [i]:
  • its Left Child is at [2*i] if available.
  • its Right Child is at [2*i+1] if available.
  • its Par­ent Node is at [i/2] if avail­able.

Heap Operations

Insert

• Add the ele­ment at the bot­tom leaf of the Heap.
• Per­form the Bubble-Up operation: ~ O(log n)
  • Bubble-Up a.k.a up-heap, percolate-up, sift-up, trickle-up, heapify-up, or cascade-up
  • If inserted ele­ment is smaller than its par­ent node in case of Min-Heap OR greater than its par­ent node in case of Max-Heap, swap the ele­ment with its parent.
  • Keep repeat­ing the above step, if node reaches its cor­rect posi­tion, STOP.

Extract-Min OR Extract-Max Operation

• Take out the ele­ment from the root.( it will be min­i­mum in case of Min-Heap and max­i­mum in case of Max-Heap).
• Take out the last ele­ment from the last level from the heap and replace the root with the element.
• Per­form Sink-Down

Delete

• Find the index for the ele­ment to be deleted.
• Take out the last ele­ment from the last level from the heap and replace the index with this element.
• Per­form Sink-Down operation: ~ O(log n)
  • If replaced ele­ment is greater than any of its child node in case of Min-Heap OR smaller than any if its child node in case of Max-Heap, swap the ele­ment with its small­est child (Min-Heap) or with its great­est child (Max-Heap).
  • Keep repeat­ing the above step, if node reaches its cor­rect posi­tion, STOP.

See http://algorithms.tutorialhorizon.com/binary-min-max-heap/