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An introduction to algorithms, including examples and pseudo-code. It covers topics such as greedy algorithms, optimization, and designing an algorithm. The document also discusses the properties of an algorithm and how to measure its complexity. It is intended for students studying computer science and discrete mathematics.
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Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry
Fall 2007
Computer Science & Engineering 235 Introduction to Discrete Mathematics 1 / 18^ Section 3.1 of Rosen
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Brief Introduction
Real World Computing World Objects Data Structures, ADTs, Classes Relations Relations and functions Actions Operations
Problems are specified by (1) a formulation and (2) a query. Formulation is a set of objects and a set of relations between them Query is the information one is trying to extract from the formulation, the question to answer. Algorithms^1 are methods or procedures that solve instances of a problem 2 / 18^1 ”Algorithm” is a distortion of^ al-Khwarizmi, a Persian mathematician
Algorithms: A Brief Introduction CSE
Introduction Algorithms PseudocodeDesign
Examples Greedy algorithm
General Techniques
There are many broad categories of Algorithms: Randomized algorithms, Monte-Carlo algorithms, Approximation algorithms, Parallel algorithms, et al. Usually, algorithms are studied corresponding to relevant data structures. Some general styles of algorithms include
(^1) Brute Force (enumerative techniques, exhaustive search) (^2) Divide & Conquer (^3) Transform & Conquer (reformulation) (^4) Greedy Techniques
Algorithms: A Brief Introduction CSE
Introduction Algorithms PseudocodeDesign
Examples Greedy algorithm
Algorithms are usually presented using some form of pseudo-code. Good pseudo-code is a balance between clarity and detail. Bad pseudo-code gives too many details or is too implementation specific (i.e. actual C++ or Java code or giving every step of a sub-process). Good pseudo-code abstracts the algorithm, makes good use of mathematical notation and is easy to read.
Algorithms: A Brief Introduction CSE
Introduction Algorithms PseudocodeDesign
Examples Greedy algorithm
A general approach to designing algorithms is as follows. (^1) Understand the problem, assess its difficulty (^2) Choose an approach (e.g., exact/approximate, deterministic/probabilistic) (^3) (Choose appropriate data structures) (^4) Choose a strategy (^5) Prove termination (^6) Prove correctness (^7) Prove completeness (^8) Evaluate complexity (^9) Implement and test it. (^10) Compare to other known approaches and algorithms.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
When designing an algorithm, we usually give a formal statement about the problem we wish to solve. Problem Given a set A = {a 1 , a 2 ,... , an} integers. Output the index i of the maximum integer ai.
A straightforward idea is to simply store an initial maximum, say a 1 then compare it to every other integer, and update the stored maximum if a new maximum is ever found.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Analysis
This is a simple enough algorithm that you should be able to:
Prove it correct Verify that it has the properties of an algorithm. Have some intuition as to its cost.
That is, how many “steps” would it take for this algorithm to complete its run? What constitutes a step? How do we measure the complexity of the step? These questions will be answered in the next few lectures, for now let us just take a look at a couple more examples.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Check Bubble Sort and Insertion Sort in your textbooks, which you have seen ad nauseum, in CSE155, CSE156, and will see again in CSE310. I will be glad to discuss them with any of you if you have not seen them yet.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Change-Making Problem
For anyone who’s had to work a service job, this is a familiar problem: we want to give change to a customer, but we want to minimize the number of total coins we give them. Problem Given An integer n and a set of coin denominations (c 1 , c 2 ,... , cr) with c 1 > c 2 > · · · > cr Output A set of coins d 1 , d 2 , · · · , dk such that
∑k i=1 di^ =^ n^ and k is minimized.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Change-Making Algorithm
Change Input : An integer n and a set of coin denominations (c 1 , c 2 ,... , cr ) with c 1 > c 2 > · · · > cr. Output : A set of coins d 1 , d 2 , · · · , dk such that Pki=1 di = n and k is minimized. 1 C ← ∅ 2 for i = 1,... , r do 3 while n ≥ ci do 4 C ← C ∪ {ci} 5 n ← n − ci 6 end 7 end 8 output C
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Analysis
Will this algorithm always produce an optimal answer? Consider a coinage system:
where c 1 = 20, c 2 = 15, c 3 = 7, c 4 = 1 and we want to give 22 “cents” in change.
What will this algorithm produce? Is it optimal?
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Analysis
Will this algorithm always produce an optimal answer? Consider a coinage system:
where c 1 = 20, c 2 = 15, c 3 = 7, c 4 = 1 and we want to give 22 “cents” in change.
What will this algorithm produce? Is it optimal? It is not optimal since it would give us one c 4 and two c 1 , for three coins, while the optimal is one c 2 and one c 3 for two coins.
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Proving optimality
Algorithms: A Brief Introduction CSE
Introduction Algorithms Examples Greedy algorithm
Proving optimality
Let C = {d 1 , d 2 ,... , dk} be the solution given by the greedy algorithm for some integer n. By way of contradiction, assume there is another solution C′^ = {d′ 1 , d′ 2 ,... , d′ l} with l < k.
