Divide & Conquer Order Statistics 2021-01-13 CSC373 Winter 2021 – Sam Toueg 1 Overview and Motivation 𝐴=𝑎!,𝑎”,𝑎#,…,𝑎$ (unsorted) • Find average of 𝐴: linear time • Find max/min of 𝐴: linear time • Find median of 𝐴: linear time ? ØIntuitively: median of 𝐴 = “middle element in sorted 𝐴” ØFormally: let Π be a […]
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Complexity NP-Complete Problems 2021-03-15 CSC373 Winter 2021 – Sam Toueg 1 Recall: Reducing SAT to Clique SAT Clique • Input: a CNF formula 𝜑 • Question: is 𝜑 satisfiable? • Input: A graph 𝐺 = (𝑉, 𝐸), an integer 𝑘 • Question: Does 𝐺 have a clique of size 𝑘? • SAT is “polynomially-reducible’’ to
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Last week recap • Divide & Conquer algorithms: ØMaster theorem ØFast integer mul8plica8on in 𝑂 𝑛!”#! $ [Karatsuba] ØFast matrix mul8plica8on in 𝑂 𝑛!”#! % [Strassen] ØFinding closest pair of points in R& in 𝑂 𝑛 log 𝑛 ØFinding 𝑘'( smallest element (in par8cular, median) o Randomized algorithm: 𝑂(𝑛) expected 8me o Determinis8c algorithm: 𝑂(𝑛)
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Divide & Conquer Master Theorem 2021-01-12 CSC373 Winter 2021 – Sam Toueg 1 Divide & Conquer • Divide & conquer algorithm: Ø divide problem of size 𝑛 into 𝑎 smaller subproblems of size 𝑛/𝑏 each Ø recursively solve each subproblem Ø combine the subproblem solutions into the solution of the original problem • Runtime 𝑛>1:
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Network Flow Some Applications 2021-03-03 CSC373 Winter 2021 – Sam Toueg 1 Bipartite Matching (brief recap of Monday’s lecture) 2021-03-03 CSC373 Winter 2021 – Sam Toueg 2 Bipartite Matching to Max Flow Lemma: ∃ matching of size ” in # ⟺ ∃ integral flow of value ” in F Corollary: Maximum Matching in # =
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CSCA48 – Unit 6 – Designing and Building Good Software Winter 2021 Learning Outcomes Unlike the previous units, the goal of this unit is not to get you to understand how to do things, but rather to give you a preview of some of the advanced concepts that you will encounter in courses beyond this
CSCA48 – Unit 5 – Graphs and Recursion Winter 2021 Learning Outcomes This unit expands on the materials we learned in previous units on linked lists and trees in order to discuss graphs and other generalized approaches to data structures. We will also cover recursion (which we have been implicitly using in previous units) and
CSCA48 – Unit 1 – Introduction to Programming in C Winter 2021 Learning Outcomes This unit will introduce you to the C programming language, and show you how to do most of what you can already to in Python in C. By the end of this unit, you will be able to: 1 • •
CSCA48 – Unit 2 – Data and Memory Winter 2021 Learning Outcomes This unit introduces the C memory model, a way of thinking about and tracing memory usage by C programs. It also covers fundamental data types and how to allocate memory for variables and arrays. By the end of this unit, you will be
CSCA48 – Unit 3 – Organizing, Storing, and Accessing Information Winter 2021 Learning Outcomes This unit will cover the basics of how to deal with data that doesn’t fit neatly into one of c’s pre-defined data types. By the end of this unit, you will be able to: 1 • • • • Use structs