CS 411 Fall 2025  >  Outline for October 15, 2025

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CS 411 Fall 2025
Outline
for October 15, 2025

Outline

Heaps & Heap Sort [L 6.4]

• Priority Queues
  • Associative structure with limited operations.
  • Key is called priority.
  • Create (insert) as in dictionary; other operations only performed on highest-priority item in container.
• Heaps
  • (Binary) Heap: common data structure for implementing a Priority Queue.
  • Heap = Binary Tree with Complete & Parental-Dominance properties.
  • Array representation: always used for a Binary Heap.
• Heap Operations
  • Heap Insert: Add new item as new final leaf. Trickle final leaf up. \(\Theta(\log n)\) time.
  • Heap Delete: Swap root & final leaf. Remove final leaf. Trickle root down. \(\Theta(\log n)\) time.
  • Fast In-place Heap Make
    • Could do \(n\) Heap Insert operations. \(\Theta(n\log n)\) time.
    • Better: Start from end and move toward beginning. Trickle each item down. Result: \(\Theta(n)\) time. [Wow!]
• Heap Sort
  • Note that Heap Delete puts the biggest item at the end. This is what each pass does in Selection Sort, but there it takes us linear time. Once a Heap is created, we can do this in logarithmic time.
  • Procedure:
    • Do in-place Heap Make.
    • Do Heap Delete operations until Heap is empty.
  • Analysis
    • Time: \(\Theta(n\log n)\).
    • Additional space required: \(\Theta(1)\). In-place!
    • Requirements on data: Must be random-access.
    • Stable: No.
    • Performance on nearly sorted data: \(\Theta(n\log n)\).
  • Introsort does Quicksort, keeping track of “recursion” depth [eliminated tail calls count, too!]. If this exceeds some bound [\(2\log_2 n\)?], then switch to Heap Sort.

Problem Reduction [L 6.6]

• Idea: Solve other problem to obtain solution to original problem
• Example: Reduction to graph problems
• Example: LCM reduced to GCD
  • Euclid’s Algorithm (a.k.a. Euclidean Algorithm)
  • Analysis
  • LCM via Euclid’s Algorithm