PPT-Lecture 2: Concrete Models and Tight Upper and Lower Bounds

David Woodruff Carnegie Mellon University Theme Tight Upper and Lower Bounds Number of comparisons to sort an array Number of exchanges to sort an array Number of comparisons needed to find the largest and secondlargest elements in an array. CONSISTENT BEHAVIOUR Our new concrete range is compliant with Australian Standard AS1379 from cement that complies with Australian Standard AS3972 AS1379 COMPLIANT ENVISIA is a lower carbon concrete with excellent performance bene64257ts and plastic 1 Overview In this lecture we discuss the notion of lower bounds in particular for the problem of sorting We show that any deterministic comparisonbased sorting algo rithm must take 8486 log time to sort an array of elements in the worst case We th Our result is modular 1 We describe a carefullychosen dynamic version of set disjointness the multiphase problem and conjecture that it requires 84861 time per operation All our lower bounds follow by easy reduction 2 We reduce 3SUM to the multipha Indeed developing bounds on the per formance of procedures can give complementary insights By exhibiting fundamental limits of performance perhaps over restricted classes of estimators it is possible to guarantee that an a lgorithm we have developed 1 Overview In this lecture we discuss the notion of lower bounds in particular for the problem of sorting We show that any deterministic comparisonbased sorting algo rithm must take 8486 log time to sort an array of elements in the worst case We th Upper and Lower Bounds Colouring colouring of a graph is a map where with the property that whenever there is an edge with ends uv The elements of are called colours and the vertices of one colour form a colour class The chromatic number of den Reticulate Network of Multiple . Phylogenetic. Trees. Yufeng. . Wu. Dept. of Computer Science & Engineering. University of Connecticut, USA. ISMB 2010. 1. 1. 2. 3. 4. Keep. two . red. edges. Keep. 2 - . Calculations. www.waldomaths.com. Copyright © . Waldomaths.com. 2010, all rights reserved. Two ropes, . A. and . B. , have lengths:. A = . 36m to the nearest metre . B = . 23m to the nearest metre.. unseen problems. David . Corne. , Alan Reynolds. My wonderful new algorithm, . Bee-inspired Orthogonal Local Linear Optimal . Covariance . K. inetics . Solver. Beats CMA-ES on 7 out of 10 test problems !!. approximate membership. dynamic data structures. Shachar. Lovett. IAS. Ely . Porat. Bar-. Ilan. University. Synergies in lower bounds, June 2011. Information theoretic lower bounds. Information theory. Xiaoming. Sun and David P. Woodruff. Chinese Academy of Sciences and IBM Research-. Almaden. Streaming Models. Long sequence of items appear one-by-one. numbers, points, edges, …. (usually) . adversarially. Searching. : Given a large set of distinct keys, preprocess them so searches can be performed as quickly as possible. 1. CS 840 Unit 1: Models, Lower Bounds and getting around Lower bounds. Searching. dynamic data structures. Shachar. Lovett. IAS. Ely . Porat. Bar-. Ilan. University. Synergies in lower bounds, June 2011. Information theoretic lower bounds. Information theory. is a powerful tool to prove lower bounds, e.g. in data structures. Dagstuhl Workshop. March/. 2023. Igor Carboni Oliveira. University of Warwick. 1. Join work with . Jiatu. Li (Tsinghua). 2. Context. Goals of . Complexity Theory. include . separating complexity classes.