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ICS 353Design and Analysis of Algorithms

Spring Semester 2006 - 2007 (062)

King Fahd University of Petroleum & Minerals

Information & Computer Science Department

Spring Semester 2006 - 2007 (062)

King Fahd University of Petroleum & Minerals

Information & Computer Science Department

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Basic Concepts in Algorithmic Analysis

• Topics– Introduction– Time Complexity– Space Complexity– Optimal Algorithms– How to estimate the running time of an algorithm– Worst Case Analysis and Average Case Analysis– Amortized Analysis– Input Size and Problem Instance

• Reading Assignment– All Chapter 1 from the textbook

• In particular, sections 1-3,6,8-14 will be discussed in class.

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What is an algorithm?

• An algorithm is defined as a finite set of steps, each of which may require one or more operations and if carried out on a set of inputs, will produce one or more outputs after a finite amount of time.

• Examples of Algorithms• Examples of computations that are not

algorithms

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Properties of Algorithms

• Definiteness: It must be clear what should be done.• Effectiveness: Each step must be such that it can, at

least in principle, be carried out by a person using pencil and paper in a finite amount of time. E.g. integer arithmetic.

• An algorithm produces one or more outputs and may have zero or more externally supplied inputs.

• Finiteness: Algorithms should terminate after a finite number of operations.

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Our Objective

• Find the most efficient algorithm for solving a particular problem.

• In order to achieve the objective, we need to determine:– How can we find such algorithm?– What does it mean to be an efficient algorithm?– How can one tell that it is more efficient than

other algorithms?

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In the First Chapter

• We will answer the following two questions– What does it mean to be an efficient algorithm?– How can one tell that it is more efficient than

other algorithms?

based on some easy-to-understand searching and sorting algorithms that we may have seen earlier.

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Searching Problem

• Assume A is an array with n elements A[1], A[2], … A[n]. For a given element x, we must determine whether there is an index j; 1 ≤ j ≤ n, such that x = A[j]

• Two algorithms, among others, address this problem– Linear Search– Binary Search

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Linear Search Algorithm

Algorithm: LINEARSEARCH

Input: array A[1..n] of n elements and an element x.

Output: j if x = A[j], 1 ≤ j ≤ n, and 0 otherwise.1. j 12. while (j < n) and (x A[j])3. j j + 14. end while

5. if x = A[j] then return j else return 0

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Analyzing Linear Search

• One way to measure efficiency is to count how many statements get executed before the algorithm terminates

• One should keep an eye, though, on statements that are executed “repeatedly”.

• What will be the number of “element” comparisons if x– First appears in the first element of A– First appears in the middle element of A– First appears in the last element of A– Doesn’t appear in A.

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Binary Search

• We can do “better” than linear search if we knew that the elements of A are sorted, say in non-decreasing order.

• The idea is that you can compare x to the middle element of A, say A[middle].– If x < A[middle] then you know that x cannot be an

element from A[middle+1], A[middle+2], …A[n]. Why?

– If x > A[middle] then you know that x cannot be an element from A[1], A[2], …A[middle-1]. Why?

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Binary Search AlgorithmAlgorithm: BINARYSEARCH

Input: An array A[1..n] of n elements sorted in nondecreasing order and an element x.

Output: j if x = A[j], 1 ≤ j ≤ n, and 0 otherwise.1. low 1; high n; j 02. while (low ≤ high) and (j = 0)

3. mid (low + high)/24. if x = A[mid] then j mid5. else if x < A[mid] then high mid - 1

6. else low mid + 17. end while

8. return j

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Worst Case Analysis of Binary Search• What to do: Find the maximum number of element

comparisons• How to do:

– The number of “element” comparisons is equal to the number of iterations of the while loop in steps 2-7. HOW?

– How many elements of the input do we have in the• First iteration

• Second iteration

• Third iteration

• …

• ith iteration

– The last iteration occurs when the size of input we have =

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Theorem

• The number of comparisons performed by Algorithm BINARYSEARCH on a sorted array of size n is at most 1log n

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Insertion Sort

Algorithm: INSERTIONSORTInput: An array A[1..n] of n elements.Output: A[1..n] sorted in nondecreasing order.1. for i 2 to n2. x A[i]3. j i - 14. while (j > 0) and (A[j] > x)5. A[j + 1] A[j]6. j j - 17. end while8. A[j + 1] x9. end for

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Insertion Sort Example

42 85 9

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Insertion Sort Example

42 85 9x=2

45 8 2 9x=9

45 8 2 9x=8

45 9 2 8x=4

94 8 2 5

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Analyzing Insertion Sort

• The minimum number of element comparisons is which occurs when

• The maximum number of element comparisons is which occurs when

• The number of element assignments is

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Time Complexity

• One way of measuring the performance of an algorithm is how fast it executes. The question is how to measure this “time”?– Is having a digital stop watch suitable?

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Order of Growth• As measuring time is subjective to many factors, we

look for a more “objective” measure, i.e. the number of operations

• Since counting the exact number of operations is cumbersome, sometimes impossible, we can always focus our attention to asymptotic analysis, where constants and lower-order terms are ignored.– E.g. n3, 1000n3, and 10n3+10000n2+5n-1 are all “the same”

– The reason we can do this is that we are always interested in comparing different algorithms for arbitrary large number of inputs.

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Example

Growth rate for some function

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Example

Growth rate for same previous functions showing larger input sizes

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Running Times for Different Sizes of Inputs of Different Functions

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Asymptotic Analysis: Big-oh (O())

• Definition: For T(n) a non-negatively valued function, T(n) is in the set O(f(n)) if there exist two positive constants c and n0 such that T(n) cf(n) for all n > n0.

• Usage: The algorithm is in O(n2) in [best, average, worst] case.

• Meaning: For all data sets big enough (i.e., n>n0), the algorithm always executes in less than or equal to cf(n) steps in [best, average, worst] case.

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Big O()

• O() notation indicates an upper bound.

• Usually, we look for the tightest upper bound:– while T(n) = 3n2 is in O(n3), we prefer O(n2).

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Big O() Examples

• Example 1: Find c and n0 to show that

T(n) = (n+2)/2 is in O(n)

• Example 2: Find c and n0 to show that T(n)=c1n2+c2n is in O(n2)

• Example 3: T(n) = c. We say this is in O(1).

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Asymptotic Analysis: Big-Omega (())

• Definition: For T(n) a non-negatively valued function, T(n) is in the set (g(n)) if there exist two positive constants c and n0 such that T(n) >= cg(n) for all n > n0.

• Meaning: For all data sets big enough (i.e., n > n0), the algorithm always executes in more than or equal to cg(n) steps.

() notation indicates a lower bound.

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() Example

• Find c and n0 to show that T(n) = c1n2 + c2n is in (n2) .

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Asymptotic Analysis: Big Theta (())

• When O() and () meet, we indicate this by using () (big-Theta) notation.

• Definition: An algorithm is said to be (h(n)) if it is in O(h(n)) and it is in (h(n)).

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Example

• Show that log(n!) is in (n log n).

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Complexity Classes and small-oh (o())

• Using () notation, one can divide the functions into different equivalence classes, where f(n) and g(n) belong to the same equivalence class if f(n) = (g(n))

• To show that two functions belong to different equivalence classes, the small-oh notation has been introduced

• Definition: Let f(n) and g(n) be two functions from the set of natural numbers to the set of non-negative real numbers. f(n) is said to be in o(g(n)) if for every constant c > 0, there is a positive integer n0 such that f(n) < cg(n) for all n n0.

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Simplifying Rules• If f(n) is in O(g(n)) and g(n) is in O(h(n)), then

f(n) is in O(h(n))

• If f(n) is in O(kg(n)) for any constant k > 0, then f(n) is in ………

• If f1(n) is in O(g1(n)) and f2(n) is in O(g2(n)), then (f1 + f2)(n) is in ………

• If f1(n) is in O(g1(n)) and f2(n) is in O(g2(n)) then f1(n)f2(n) is in ………

• You can safely “globally” replace O with or in the above, where the above rules will still hold.

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Very Useful Simplifying Rule

• Let f(n) and g(n) be be two functions from the set of natural numbers to the set of non-negative real numbers such that:

Thenif L < then f(n) is in

if L > 0 then f(n) is in if 0 < L < then f(n) is in if L = 0 then f(n) is in

)(

)(lim0

ng

nfL

n

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Space Complexity

• Space complexity refers to the number of memory cells needed to carry out the computational steps required in an algorithm excluding memory cells needed to hold the input.

• Compare additional space needed to carry out SELECTIONSORT to that of BOTTOMUPSORT if we have an array with 2 million elements!

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Examples

• What is the space complexity for– Linear search– Binary search– Selection sort– Insertion sort– Merge (that merges two sorted lists)– Bottom up merge sort

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Optimal Algorithms

• If one can show that there is no algorithm that solves a certain problem in asymptotically less than that of a certain algorithm A, we call A an optimal algorithm.

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Optimal Algorithms: Example

A decision tree for sorting three elementsFigure 12.1 page 338 from the textbook

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Optimal Algorithms: Example

• Consider the sorting problem of n distinct elements using element comparison-based sorting– Using the decision tree model, the number of possible

solutions (leaf nodes) in the binary tree is equal to ................

– You have learnt earlier that a binary tree with n nodes has height of at least log n (Observation 3.3 page 111 of the textbook)

– Hence, the length of the longest path in a decision tree for sorting n distinct elements is at least .............

– Therefore, • Insertion sort is• Selection sort is• Merge sort is• Quick sort is

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Estimating the Running Time of an Algorithm

• As mentioned earlier, we need to focus on counting those operations which represent, in general, the behavior of the algorithm

• This is achieved by– Counting the frequency of basic operations.

• Basic operation is an operation with highest frequency to within a constant factor among all other elementary operations

– Recurrence Relations

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Counting the Frequency of Basic Operations

• Sometimes, it is easier to compute the frequency of an operation that is a good representative of the overall time complexity of the algorithm– For example, Algorithm MERGE.

• Counting the number of iterations – The number of iterations in a while loop and/or a for

loop is a good indication of the total number of operations

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Example 1

sum = 0;

for (j=1; j<=n; j++)

for (i=1; i<=j; i++)

sum++;

for (k=0; k<n; k++)

A[k] = k;

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Example 2

for j := 1 to n do sum[j] := 0;

for i := 1 to j2 do

sum[j] := sum[j] + i;

end for;

end for;

return sum[1..n];

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Example 3

count := 0;

for i := 1 to n do

m := n/i for j := 1 to m do

count := count + 1 ;

end for;

end for;

return count;

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Example 4

count := 0; while n >= 1 do for j := 1 to n do execute_algorithm_x; count := count + 1; end for n := n / 2; end while return count;

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Examples 5 & 6

sum1 = 0;for (k=1; k<=n; k*=2) for (j=1; j<=n; j++) sum1++;

sum2 = 0;for (k=1; k<=n; k*=2) for (j=1; j<=k; j++) sum2++;

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Example 7

count := 0; for i := 1 to n do j := 2; while j <= n do j := j2; count := count + 1; end while end for; return count;

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Recurrence Relations

• The number of operations can be represented as a recurrence relation.

• There are very well known techniques, other than expanding the recurrence relation, which we will study in order to solve these recurrences

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Example• Recursive Merge Sort

MergeSort(A,p,r) if p < r then

q := (p+r)/2; MergeSort(A,p,q);

MergeSort(A,q+1,r);

Merge(A,p,q,r);

end if;

– What is the call to sort an array with n elements?

– Let us assume that the overall cost of sorting n elements is T(n), assuming that n is a power of two.

• If n = 1, do we know T(n)?

• What is the cost of MergeSort(A,p,q)?

• What is the cost of MergeSort(A,q+1,r)?

• What is the cost of Merge(A,p,q,r)?

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Worst Case Analysis

• In worst case analysis of time complexity we select the maximum cost among all possible inputs of size n.– One can do that for the() notation as well as

the O() notation.– However, it is better use it with the ()

notation.• Why?

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Average Case Analysis

• Probabilities of all inputs is an important piece of prior knowledge in order to compute the number of operations on average

• Usually, average case analysis is lengthy and complicated, even with simplifying assumptions.

k

i 1

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Computing the Average Running Time

• The running time in this case is taken to be the average time over all inputs of size n.– Assume we have k inputs, where each input

costs Ci operations, and each input can occur with probability Pi, 1 i k, the average running time is given by

k

iiiCP

1

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Average Case Analysis of Linear Search

• Assume that the probability that key x appears in any position in the array (1, 2, …, n) or does not appear in the array is equally likely

– This means that we have a total of ……… different inputs, each with probability ………

– What is the number of comparisons for each input?

– Therefore, the average running time of linear search = ………

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Average Case Analysis of Insertion Sort

• Assume that array A contains the numbers from 1..n ( i.e. elements are distinct)

• Assume that all n! permutations of the input are equally likely.

• What is the number of comparisons for inserting A[i] in its proper position in A[1..i]? What about on average?

• Therefore, the total number of comparisons on average is