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Advanced Algorithms (COMPSCI 224), Lecture 1

Harvard University・45 minutes read

The CS224 Advanced Algorithms course led by Gelani Nelson and TF Jeffrey includes scribing, psets, and a final project, with key components due on specific dates and submission guidelines outlined on the course website. The course delves into various data structures and sorting algorithms, exploring topics like the Word RAM model, Van Emde Boas data structures, and efficient predecessor queries to optimize time and space complexities in algorithm design.

Insights

  • Faster sorting algorithms exist beyond n root log n, such as o of n log log n deterministic and o of n square root log log n expected time randomized, with the pursuit of a linear time sorting algorithm still an open question.
  • The VB tree data structure offers o of log log U time complexity for predecessor and insertion operations, utilizing hash tables for cluster IDs and pointers to non-empty clusters, ensuring linear space complexity through efficient storage and charging of minimum elements.

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Recent questions

  • What is the CS224 course breakdown?

    Components include scribing, psets, and final project.

  • How are psets and final projects submitted?

    Psets via email, final project proposal due by October 30th.

  • What is the focus of the Word RAM model?

    Allows various operations like integer arithmetic and bit shifting.

  • How does the VB tree handle predecessor queries?

    Involves constant time extraction and recursive cluster searches.

  • What is the space complexity of the dictionary problem solution?

    Linear space complexity, denoted as Theta of n.

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Summary

00:00

CS224 Advanced Algorithms Course Overview

  • CS224 Advanced Algorithms course introduction by Gelani Nelson with TF Jeffrey
  • Contact via email at cs224df14staff@cs.harvard.edu
  • Fill out a yellow sheet circulating in class
  • Course website available, Google Gelani Nelson for access
  • Course components: scribing (10%), psets (60%), final project (30%)
  • Scribing involves note-taking during lectures, using a provided template
  • Psets and final project details on the website
  • Final project proposal due by October 30th, project due last day of reading period
  • Psets to be submitted via email, with page limits to avoid excessive content
  • Students may take turns being graders, with scribing and grading being first come first serve tasks.

25:47

Efficient VB Tree Implementation for Sorting

  • Pre-processing time is a concern in creating data structures, especially in the static case.
  • Faster sorting algorithms than n root log n exist, such as o of n log log n deterministic and o of n square root log log n expected time randomized.
  • The possibility of achieving a linear time sorting algorithm in this model remains an open question.
  • The Word RAM model allows for various operations like integer arithmetic, bitwise operations, and bit shifting.
  • The Van Emde Boas data structure is based on divide and conquer principles, recursively defined with a universe size parameter.
  • VB trees consist of square root U VB data structures on each universe of size square root U, along with a top structure and a minimum element.
  • Practical implementation in an object-oriented programming language involves specific fields and structures for VB data.
  • Insertion into the VB tree involves binary representation of elements, cluster checks, and summary updates.
  • The predecessor query in the VB tree involves constant time extraction and recursive cluster searches.
  • The time complexity for predecessor and insertion operations in the VB tree is o of log log U, optimizing the process.

57:11

Efficient Data Structures for Constant Time Queries

  • The hash table keys will be cluster IDs, and the values will be pointers to corresponding non-empty clusters.
  • The space complexity of this scheme is linear, denoted as Theta of n, due to charging the cost of storing cluster IDs and pointers to minimum elements.
  • Each minimum element is charged exactly once, ensuring linear space usage.
  • The dictionary problem involves storing key-value pairs, with constant query time and expected constant insertion or deletion.
  • A dynamic dictionary with linear space and constant query time is achievable, with high probability.
  • The Y fast tree data structure can support the same bounds as Venkatesan de trees, using indirection to eliminate extra space.
  • A bit array of length U can be used for predecessor queries, with a perfect binary tree structure improving efficiency.
  • Binary search on monotone bits in the tree allows for log log U query time.
  • Storing the tree as an array or with ancestor pointers enables constant time access to ancestors for efficient predecessor queries.

01:24:52

Optimal Fusion and Venom de Boaz Trees

  • The time complexity of the algorithm discussed is W, as inserting an item requires changing W elements in the tree, while the space complexity is linear. Query time remains at log log U, with insertions involving inserting into a balanced BST and splitting super items every W steps at a cost of log log W.
  • Fusion trees and Venom de Boaz trees are optimal for linear or nearly linear space data structures, with a paper showing that no better alternative exists, combining the strengths of both tree types.
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