Lower Bounds for Comparison-Based Sorting Using Decision Trees

Introduction

• Focus: Lower bounds for comparison-based sorting algorithms using decision trees.
• Assumption: Familiarity with basic sorting algorithms, with emphasis on insertion sort.

Comparison-Based Sorting

• Definition: Sorting algorithms where outcomes of comparisons determine the algorithm's path.
• Examples: Insertion, bubble, selection, heap, merge, and quicksort.

Insertion Sort & Decision Trees

• Use of Decision Trees: Visual representation of element comparisons.
  • Non-leaf nodes: Comparisons.
  • Two children: Possible outcomes.
  • Leaves: Final outcomes.
• Case Study: Insertion sort on three elements:
  • Example inputs: 11, 13, 12 vs. 1, 3, 2 – same sorting steps.
  • Decision tree illustrates comparisons and outcomes.

Analysis of Sorting Algorithms via Decision Trees

• Insertion Sort: 2-3 comparisons for three elements, depending on input order.
• Bubble Sort: Always performs 3 comparisons.
• Selection Sort: Can have unstable outcomes due to redundant comparisons.
• Heap Sort: Contains useless and unstable comparisons.
• Merge Sort: Similar trees for 4 or fewer elements, different for 5+.
• Quicksort: Instability noted in decision tree.

Argument for Lower Bounds

• Comparison Count: Minimum 3 comparisons for sorting 3 items.
  • 6 permutations need separate leaves; each comparison has 2 outcomes.
  • Need a tree with height at least 3 for 3 items.
• General Case for n Items:
  • Permutations: n factorial possible permutations.
  • Tree Height (h): At least ( h \geq \log_2(n!) ).
  • Runtime: At least ( O(n \log n) ).

Conclusion

• Lower Bound: Any comparison-based sorting takes at least ( O(n \log n) ) time.
• Sufficient Algorithms: Merge sort and heap sort meet the lower bound.
• Further Studies: Linear sorts mentioned in other videos.

Final Remarks

• Update profiles with new knowledge on sorting algorithms.
• Good luck with further learning!