CMSC 451 Design and Analysis of Computer Algorithms1 David M. Mount Department of Computer Science University of Maryland Fall 2003 1 Copyright, David M. of Computer Science, University of Maryland, College Park, MD, 20742. These lecture notes were prepared by David Mount for the course CMSC 451, Design and Analysis of Computer Algorithms, at the University of Maryland.
Permission to use, copy, modify, and distribute these notes for educational purposes and without fee is hereby granted, provided that this copyright notice appear in all copies. Lecture Notes 1 CMSC 451 Lecture 1: Course Introduction Read: (All readings are from Cormen, Leiserson, Rivest and Stein, Introduction to Algorithms, 2nd Edition). What is an algorithm? Our text defines an algorithm to be any well-defined computational procedure that takes some values as input and produces some values as output. Like a cooking recipe, an algorithm provides a step-by-step method for solving a computational problem.
Unlike programs, algorithms are not dependent on a particular programming language, machine, system, or compiler. They are mathematical entities, which can be thought of as running on some sort of idealized computer with an infinite random access memory and an unlimited word size. Algorithm design is all about the mathematical theory behind the design of good programs. Why study algorithm design? Programming is a very complex task, and there are a number of aspects of program- ming that make it so complex.
The first is that most programming projects are very large, requiring the coor- dinated efforts of many people. (This is the topic a course like software engineering.) The next is that many programming projects involve storing and accessing large quantities of data efficiently. (This is the topic of courses on data structures and databases.) The last is that many programming projects involve solving complex computational problems, for which simplistic or naive solutions may not be efficient enough. The complex problems may involve numerical data (the subject of courses on numerical analysis), but often they involve discrete data.
This is where the topic of algorithm design and analysis is important. Although the algorithms discussed in this course will often represent only a tiny fraction of the code that is generated in a large software system, this small fraction may be very important for the success of the overall project. An unfortunately common approach to this problem is to first design an inefficient algorithm and data structure to solve the problem, and then take this poor design and attempt to fine-tune its performance. The problem is that if the underlying design is bad, then often no amount of fine-tuning is going to make a substantial difference.
The focus of this course is on how to design good algorithms, and how to analyze their efficiency. This is among the most basic aspects of good programming. Course Overview: This course will consist of a number of major sections. The first will be a short review of some preliminary material, including asymptotics, summations, and recurrences and sorting.
These have been covered in earlier courses, and so we will breeze through them pretty quickly. We will then discuss approaches to designing optimization algorithms, including dynamic programming and greedy algorithms. The next major focus will be on graph algorithms. This will include a review of breadth-first and depth-first search and their application in various problems related to connectivity in graphs.
Next we will discuss minimum spanning trees, shortest paths, and network flows. We will briefly discuss algorithmic problems arising from geometric settings, that is, computational geometry. Most of the emphasis of the first portion of the course will be on problems that can be solved efficiently, in the latter portion we will discuss intractability and NP-hard problems. These are problems for which no efficient solution is known.
Finally, we will discuss methods to approximate NP-hard problems, and how to prove how close these approximations are to the optimal solutions. Issues in Algorithm Design: Algorithms are mathematical objects (in contrast to the must more concrete notion of a computer program implemented in some programming language and executing on some machine). As such, we can reason about the properties of algorithms mathematically. When designing an algorithm there are two fundamental issues to be considered: correctness and efficiency.
It is important to justify an algorithm’s correctness mathematically. For very complex algorithms, this typically requires a careful mathematical proof, which may require the proof of many lemmas and properties of the solution, upon which the algorithm relies. For simple algorithms (BubbleSort, for example) a short intuitive explanation of the algorithm’s basic invariants is sufficient. (For example, in BubbleSort, the principal invariant is that on completion of the ith iteration, the last i elements are in their proper sorted positions.) Lecture Notes 2 CMSC 451 Establishing efficiency is a much more complex endeavor.
Intuitively, an algorithm’s efficiency is a function of the amount of computational resources it requires, measured typically as execution time and the amount of space, or memory, that the algorithm uses. The amount of computational resources can be a complex function of the size and structure of the input set. In order to reduce matters to their simplest form, it is common to consider efficiency as a function of input size. Among all inputs of the same size, we consider the maximum possible running time.
This is called worst-case analysis. It is also possible, and often more meaningful, to measure average-case analysis. Average-case analyses tend to be more complex, and may require that some probability distribution be defined on the set of inputs. To keep matters simple, we will usually focus on worst-case analysis in this course.
Throughout out this course, when you are asked to present an algorithm, this means that you need to do three things: • Present a clear, simple and unambiguous description of the algorithm (in pseudo-code, for example). They key here is “keep it simple.” Uninteresting details should be kept to a minimum, so that the key compu- tational issues stand out. (For example, it is not necessary to declare variables whose purpose is obvious, and it is often simpler and clearer to simply say, “Add X to the end of list L” than to present code to do this or use some arcane syntax, such as “L.”) • Present a justification or proof of the algorithm’s correctness. Your justification should assume that the reader is someone of similar background as yourself, say another student in this class, and should be con- vincing enough make a skeptic believe that your algorithm does indeed solve the problem correctly.
Avoid rambling about obvious or trivial elements. A good proof provides an overview of what the algorithm does, and then focuses on any tricky elements that may not be obvious. • Present a worst-case analysis of the algorithms efficiency, typically it running time (but also its space, if space is an issue). Sometimes this is straightforward, but if not, concentrate on the parts of the analysis that are not obvious.
Note that the presentation does not need to be in this order. Often it is good to begin with an explanation of how you derived the algorithm, emphasizing particular elements of the design that establish its correctness and efficiency. Then, once this groundwork has been laid down, present the algorithm itself. If this seems to be a bit abstract now, don’t worry.
We will see many examples of this process throughout the semester. Lecture 2: Mathematical Background Read: Review Chapters 1–5 in CLRS. Algorithm Analysis: Today we will review some of the basic elements of algorithm analysis, which were covered in previous courses. These include asymptotics, summations, and recurrences.
Asymptotics: Asymptotics involves O-notation (“big-Oh”) and its many relatives, Ω, Θ, o (“little-Oh”), ω. Asymp- totic notation provides us with a way to simplify the functions that arise in analyzing algorithm running times by ignoring constant factors and concentrating on the trends for large values of n. For example, it allows us to reason that for three algorithms with the respective running times n3 log n + 4n2 + 52n log n ∈ Θ(n3 log n) 15n2 + 7n log3 n ∈ Θ(n2 ) 3n + 4 log5 n + 19n2 ∈ Θ(n2 ). Thus, the first algorithm is significantly slower for large n, while the other two are comparable, up to a constant factor.
Since asymptotics were covered in earlier courses, I will assume that this is familiar to you. Nonetheless, here are a few facts to remember about asymptotic notation: Lecture Notes 3 CMSC 451 Ignore constant factors: Multiplicative constant factors are ignored. For example, 347n is Θ(n). Constant factors appearing exponents cannot be ignored.
For example, 23n is not O(2n ). Focus on large n: Asymptotic analysis means that we consider trends for large values of n. Thus, the fastest growing function of n is the only one that needs to be considered. For example, 3n2 log n + 25n log n + (log n)7 is Θ(n2 log n).
Polylog, polynomial, and exponential: These are the most common functions that arise in analyzing algo- rithms: Polylogarithmic: Powers of log n, such as (log n)7. We will usually write this as log7 n. √ Polynomial: Powers of n, such as n4 and n = n1/2. Exponential: A constant (not 1) raised to the power n, such as 3n.
An important fact is that polylogarithmic functions are strictly asymptotically smaller than polynomial function, which are strictly asymptotically smaller than exponential functions (assuming the base of the exponent is bigger than 1). For example, if we let ≺ mean “asymptotically smaller” then loga n ≺ nb ≺ cn for any a, b, and c, provided that b > 0 and c > 1. Logarithm Simplification: It is a good idea to first simplify terms involving logarithms. For example, the following formulas are useful.
Here a, b, c are constants: loga n logb n = = Θ(loga n) loga b loga (nc ) = c loga n = Θ(loga n) bloga n = nloga b. Avoid using log n in exponents. The last rule above can be used to achieve this. For example, rather than saying 3log2 n , express this as nlog2 3 ≈ n1.
Following the conventional sloppiness, I will often say O(n2 ), when in fact the stronger statement Θ(n2 ) holds. (This is just because it is easier to say “oh” than “theta”.) Summations: Summations naturally arise in the analysis of iterative algorithms. Also, more complex forms of analy- sis, such as recurrences, are often solved by reducing them to summations. Solving a summation means reducing it to a closed form formula, that is, one having no summations, recurrences, integrals, or other complex operators.
In algorithm design it is often not necessary to solve a summation exactly, since an asymptotic approximation or close upper bound is usually good enough. Here are some common summations and some tips to use in solving summations. Constant Series: For integers a and b, b X 1 = max(b − a + 1, 0). i=a Notice that when b = a − 1, there are no terms in the summation (since the index is assumed to count upwards only), and the result is 0.
Be careful to check that b ≥ a − 1 before applying this formula blindly. Arithmetic Series: For n ≥ 0, n X n(n + 1) i = 1 + 2 + ··· + n =. (The starting bound could have just as easily been set to 1 as 0.) Lecture Notes 4 CMSC 451 Geometric Series: Let x 6= 1 be any constant (independent of n), then for n ≥ 0, n X xn+1 − 1 xi = 1 + x + x2 + · · · + xn =. i=0 x−1 If 0 < x < 1 then this is Θ(1).
If x > 1, then this is Θ(xn ), that is, the entire sum is proportional to the last element of the series.
