Chapter 1blk/algorithms-undergrad/... · Here is an instrumented recursiv e solution to the gcd...

Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

Chapter 1

Introduction

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 1 1-1

What is an algorithm?

An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.

“computer”

problem

algorithm

input output


Historical Perspective

Euclid’s algorithm for finding the greatest common divisor

Muhammad ibn Musa al-Khwarizmi – 9th century mathematician www.lib.virginia.edu/science/parshall/khwariz.html


Example of computational problem: sorting

Statement of problem:• Input: A sequence of n numbers <a1, a2, …, an>

• Output: A reordering of the input sequence <a 1, a 2, …, a n> so that a i≤ a j whenever i < j

Instance: The sequence <5, 3, 2, 8, 3>

Algorithms:• Selection sort• Insertion sort• Merge sort• (many others)


Selection Sort

Input: array a[1],…,a[n]

Output: array a sorted in non-decreasing order

Algorithm:

for i=1 to nswap a[i] with smallest of a[i],…a[n]

• see also pseudocode, section 3.1


Some Well-known Computational Problems

SortingSearchingShortest paths in a graphMinimum spanning treePrimality testingTraveling salesman problemKnapsack problemChessTowers of HanoiProgram termination


Basic Issues Related to Algorithms

How to design algorithms

How to express algorithms

Proving correctness

Efficiency• Theoretical analysis

• Empirical analysis

Optimality


Analysis of Algorithms

How good is the algorithm?• Correctness• Time efficiency• Space efficiency

Does there exist a better algorithm?• Lower bounds• Optimality


What is an algorithm?

Recipe, process, method, technique, procedure, routine,… with following requirements:

1. Finitenessterminates after a finite number of steps

2. Definitenessrigorously and unambiguously specified

3. Inputvalid inputs are clearly specified

4. Outputcan be proved to produce the correct output given a valid input

5. Effectivenesssteps are sufficiently simple and basic


Why study algorithms?

Theoretical importance

• the core of computer science

Practical importance

• A practitioner’s toolkit of known algorithms

• Framework for designing and analyzing algorithms for new problems


Euclid’s Algorithm

Problem: Find gcd(m,n), the greatest common divisor of two nonnegative, not both zero integers m and n

Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ?

Euclid’s algorithm is based on repeated application of equalitygcd(m,n) = gcd(n, m mod n)

until the second number becomes 0, which makes the problemtrivial.

Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12


Two descriptions of Euclid’s algorithm

Step 1 If n = 0, return m and stop; otherwise go to Step 2Step 2 Divide m by n and assign the value of the remainder to rStep 3 Assign the value of n to m and the value of r to n. Go to

Step 1.

while n ≠ 0 dor ← m mod nm← n n ← r

return m


Other methods for computing gcd(m,n)

Consecutive integer checking algorithmStep 1 Assign the value of min{m,n} to tStep 2 Divide m by t. If the remainder is 0, go to Step 3;

otherwise, go to Step 4Step 3 Divide n by t. If the remainder is 0, return t and stop;

otherwise, go to Step 4Step 4 Decrease t by 1 and go to Step 2


Other methods for gcd(m,n) [cont.]

Middle-school procedureStep 1 Find the prime factorization of mStep 2 Find the prime factorization of nStep 3 Find all the common prime factorsStep 4 Compute the product of all the common prime factors

and return it as gcd(m,n)

Is this an algorithm?


Sieve of EratosthenesInput: Integer n ≥ 2Output: List of primes less than or equal to nfor p ← 2 to n do A[p] ← p

for p ← 2 to ⎣n⎦ doif A[p] ≠ 0 //p hasn’t been previously eliminated from the list

j ← p* pwhile j ≤ n do

A[j] ← 0 //mark element as eliminated

j ← j + p

Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Two main issues related to algorithms

How to design algorithms

How to analyze algorithm efficiency


Algorithm design techniques/strategies

Brute force

Divide and conquer

Decrease and conquer

Transform and conquer

Space and time tradeoffs

Greedy approach

Dynamic programming

Iterative improvement

Backtracking

Branch and bound


Important problem types

sorting

searching

string processing

graph problems

combinatorial problems

geometric problems

numerical problems


Fundamental data structures

list

• array

• linked list

• string

stack

queue

priority queue

graph

tree

set and dictionary


Instrumenting an Algorithm

We want to count the number of operations performed when executing an algorithm

We will increment the counter for each check of a condition in an if statement or a while statement and we will increment the counter for each assignment statement or return

We do not include the counting operations themselvescount++ // count the first test to enter the while loopwhile n ≠ 0 do

r ← m mod nm← n n ← rcount ← count + 4 // three assignments and next loop entry

count++ // count the return that appears nextreturn m

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 1

Refining our Instrumentation

1-20

Operation Count Comment Example

Assignment 1 Assign single value or one operation

x = y + z;

Assignment 2 Assign a complex expression

x = y * y + z * z;

Condition 1 Only a single operation number > 0

Condition 2 Complex expression (number > 0) && (number < 10)

Method call 10 For each method call sort(array)

Return 1 or 2 Depends on complexity of expression (see assign)

return number (Note: increment before the instruction)

Each array access

1 or 2 Depends on complexity of subscript (see assign)

Arr[i] * Arr[j] (Note: three total, two accesses and one *)

Each field access

1 Assumes fields are local, not using getter

data. number

Instrumentation itself

0 Never include the instrumentation

Count++; // instrumentation// don’t count it


Instrumenting a Recursive Algorithm

Here is a recursive solution to the gcd problemgcd(m, n)

if n == 0 return melse return gcd(n, m % n)

Here is an instrumented recursive solution to the gcd problemgcd(m, n)

if n == 0 {count += 2; // one for condition, one for returnreturn m

} else {count += 13; // 1 for cond., 1 for %, 10 for method, 1 for retreturn gcd(n, m % n)

}

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 1

MicroLab01 and Program01

Both of these activities will involve different versions of implementing Euclid’s algorithm to calculate the GCDMicroLab01 (ML01)• Implement and test GCD01 – iterative version using mod (%)• Implement and test GCD02 – recursive version using mod (%)• Instrument both GCD01 and GCD02 and collect data

Program01 (P01)• Implement and test GCD03 – iterative version using consecutive

integer checking (see slide 12)• Implement and test GCD04 – iterative version using subtraction, as

described on assignment sheet• Instrument both GCD03 and GCD04 and collect data• Write a brief report of your results

1-22

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