Heaps, Heapsort and Priority Queues

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Lecture 8

• Heaps, Heapsort and Priority Queues

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Heap

• A binary tree satisfying heap property data in
  parent larger than data in children. Examples.
• Implemented efficiently as an array fast access
  time. Space n. Access O(1).

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Heap

• Data at parent is larger than data at children
• Tree is binary, balanced, filled left

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Operations on Heaps

• Insert/remove element from heap trickle up/down.
  
• Time height of tree O(log n)

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Delete Max

• Remove root, replace with last element.

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• Trickle down to readjust heap

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HeapSort

• Heapsort Variation on Selection Sort, with
  maximum computed efficiently at each step using a
  heap.
• Step 1 create a heap. In O(n) time.
• Then iterate as in Selection Sort. Remove root
  (max). Readjust tree (move last key to root,
  trickle down) in O(log n). Continue until all are
  removed (in sorted order).
• Time O(n log n) because removing/readjusting
  takes log n.

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Analysis

• Create a heap insert an element O(log n). Total
  O(n log n).
• Analysis can be improved to O(n) time (read
  handout, ch. 7.3) start with elements in the
  arary, heapify for each element. Use nr of
  elements at height h, lt n/2h1
• Total time S h n/2h1 O(n)

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Applications Priority Queues

• Priority Queues data structure to allow for
  efficient finding of the minimum/maximum, after
  initial preprocessing, and maintain it in O(log
  n) time. Done with heaps.
• Used in Operating Systems, to maintain priority
  queues of processes.

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Homework

• From handout, ex. 7.1-1,2,3,4,5,6.
• 7.2-1,2
• 7.3-1,3
• 7.4-1,2,3.
• 7.5-1,2
• Extra credit 7.5-6