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Intro to the Heap data structure

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• Definition: complete binary tree
  • Complete binary tree:

    Complete binary tree = a binary tree where all levels, except possibly the last level are completely filled with nodes Furthermore: the last level has all its nodes to the left side

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  • Examples of complete binary trees:

    A perfert binary tree is a complete binary tree: The last level does not need to be filled up: All the nodes must be packed on the left side !!!

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  • These are not complete binary trees:

    Levels are not not completely filled: Nodes on the last level are not packed to the left:

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• Heap: another kind a binary tree
  • Heap:

    Heap = a complete binary tree where the value in every node x is the smallest value in the subtree rooted at node x

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  • Example: the following complete binary tree is a heap

    because:

    Node 1 has the smallest value in the subtree rooted at node 1: Node 5 has the smallest value in the subtree rooted at node 5: Node 6 has the smallest value in the subtree rooted at node 6:

    So:

    the value in every node x is the smallest value in the subtree rooted at node x

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  • The following complete binary tree is not a heap:

    Because:

    The following subtree rooted at node 11 has the smallest value stored in a node that is not the root node of the subtree:

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• Priority queue
  • Priority:

    Priority = a value that states the rank of urgency that a certain task must be executed A job with a higher priority is given a smaller ranking value

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• Representing a priotity using a heap
  • Fact:

    The root node in a heap always has the smallest value This follows directly from the definition of a heap !!!

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  • Representing a priority queue using a heap:

    The job with the highest priority will always be stored in the root node of a heap In fact: The heap data structure is often used to store jobs that need to be ranked by their priorities

    That's why a heap is sometimes called a priority queue

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    We will learn (study) algorithm used to:

    insert a new value into a heap After inserting, the resulting structure must still be a heap delete a node (value) from a heap After deleting, the resulting structure must still be a heap