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Data Structures and Algorithms in Java
Chapter 2
Complexity Analysis

Data Structures and Algorithms in Java 2
ObjectivesDiscuss the following topics: • Computational and Asymptotic Complexity• BigO Notation• Properties of BigO Notation• Ω and Θ Notations• Examples of Complexities• Finding Asymptotic Complexity: Examples• Amortized Complexity• The Best, Average, and Worst Cases• NPCompleteness

Data Structures and Algorithms in Java 3
Computational and Asymptotic Complexity
• Computational complexity measures the degree of difficulty of an algorithm
• Indicates how much effort is needed to apply an algorithm or how costly it is
• To evaluate an algorithm’s efficiency, use logical units that express a relationship such as:– The size n of a file or an array – The amount of time t required to process
the data

Data Structures and Algorithms in Java 4
Computational and Asymptotic Complexity (continued)
• This measure of efficiency is called asymptotic complexity
• It is used when disregarding certain terms of a function– To express the efficiency of an algorithm– When calculating a function is difficult or
impossible and only approximations can be found
f (n) = n2 + 100n + log10n + 1,000

Data Structures and Algorithms in Java 5
Computational and Asymptotic Complexity (continued)
Figure 21 The growth rate of all terms of function f (n) = n2 + 100n + log10n + 1,000

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BigO Notation
• Introduced in 1894, the bigO notation specifies asymptotic complexity, which estimates the rate of function growth
• Definition 1: f (n) is O(g(n)) if there exist positive numbers c and N such that f (n) ≤ cg(n) for all n ≥ N
Figure 22 Different values of c and N for function f (n) = 2n2 + 3n + 1 = O(n2) calculated according to the definition of bigO

Data Structures and Algorithms in Java 7
BigO Notation (continued)
Figure 23 Comparison of functions for different values of c and N from Figure 22

Data Structures and Algorithms in Java 8
Properties of BigO Notation
• Fact 1 (transitivity) If f (n) is O(g(n)) and g(n) is O(h(n)), then f(n) is O(h(n))
• Fact 2If f (n) is O(h(n)) and g(n) is O(h(n)), then f(n) + g(n) is O(h(n))
• Fact 3The function ank is O(nk)

Data Structures and Algorithms in Java 9
Properties of BigO Notation (continued)
• Fact 4The function nk is O(nk+j) for any positive j
• Fact 5If f(n) = cg(n), then f(n) is O(g(n))
• Fact 6The function loga n is O(logb n) for any positive numbers a and b ≠ 1
• Fact 7loga n is O(lg n) for any positive a ≠ 1, where lg n = log2 n

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Ω and Θ Notations
• BigO notation refers to the upper bounds of functions
• There is a symmetrical definition for a lower bound in the definition of bigΩ
• Definition 2: The function f(n) is Ω(g(n)) if there exist positive numbers c and N such that f(n) ≥ cg(n) for all n ≥ N

Data Structures and Algorithms in Java 11
Ω and Θ Notations (continued)
• The difference between this definition and the definition of bigO notation is the direction of the inequality
• One definition can be turned into the other by replacing “≥” with “≤”
• There is an interconnection between these two notations expressed by the equivalence
f (n) is Ω(g(n)) iff g(n) is O(f (n)) (prove?)

Data Structures and Algorithms in Java 12
Ω and Θ Notations (continued)
• Definition 3: f(n) is Θ(g(n)) if there exist positive numbers c1, c2, and N such that c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ N
• When applying any of these notations (bigO,Ω, and Θ), remember they are approximations that hide some detail that in many cases may be considered important

Data Structures and Algorithms in Java 13
Examples of Complexities
• Algorithms can be classified by their time or space complexities
• An algorithm is called constant if its execution time remains the same for any number of elements
• It is called quadratic if its execution time is O(n2)

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Examples of Complexities (continued)
Figure 24 Classes of algorithms and their execution times on a computer executing 1 million operations per second (1 sec = 106 μsec = 103 msec)

Data Structures and Algorithms in Java 15
Examples of Complexities (continued)
Figure 24 Classes of algorithms and their execution times on a computer executing 1 million operations per second (1 sec = 106 μsec = 103 msec)(continued)

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Examples of Complexities (continued)
Figure 25 Typical functions applied in bigO estimates

Data Structures and Algorithms in Java 17
Finding Asymptotic Complexity: Examples
• Asymptotic bounds are used to estimate the efficiency of algorithms by assessing the amount of time and memory needed to accomplish the task for which the algorithms were designed
for (i = sum = 0; i < n; i++)sum += a[i]
• Initialize two variables• Execute two assignments
– Update sum– Update iTotal 2+2n assignments for the complete executionAsymptotic complexity is O(n)

Data Structures and Algorithms in Java 18
Finding Asymptotic Complexity: Examples
• Printing sums of all the subarrays that begins with position 0
for (i = 0; i < n; i++) {for (j = 1, sum = a[0]; j

Data Structures and Algorithms in Java 19
Examples Continued• Printing sums of numbers in the last five cells of
the subarrays starting in position 0
for (i = 4; i < n; i++) {for (j = i3, sum = a[i4]; j

Data Structures and Algorithms in Java 20
Finding Asymptotic Complexity: Examples
• Finding the length of the longest subarray with the numbers in increasing order
• For example [1 2 5 ] in [1 8 1 2 5 0 11 12]
for (i = 0, length = 1; i < n1; i++) {for (i1 = i2 = k = i; k < n1 && a[k] < a[k+1];
k++, i2++);if (length < i2  i1 + 1)length = i2  i1 + 1;
System.out.println ("the length of the longestordered subarray is" + length);
}

Data Structures and Algorithms in Java 21
• If all numbers in the array are in decreasing order, the outer loop is executed n1 times
• But in each iteration, the inner loop executes just one time. The algorithm is O(n)
• If the numbers are in increasing order, the outer loop is executed n 1 times and the inner loop is executed n1i times for each i in {0,…, n2}. The algorithm is O(n2)

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Finding Asymptotic Complexity: Examples
int binarySearch(int[] arr, int key) {int lo = 0, mid, hi = arr.length1;while (lo

Data Structures and Algorithms in Java 23
The Best, Average, and Worst Cases
• The worst case is when an algorithm requires a maximum number of steps
• The best case is when the number of steps is the smallest
• The average case falls between these extremes
Cavg = Σip(inputi)steps(inputi)

Data Structures and Algorithms in Java 24
• The average complexity is established by considering possible inputs to an algorithm,
• determining the number of steps performed by the algorithm for each input,
• adding the number of steps for all the inputs, and dividing by the number of inputs
• This definition assumes that the probability of occurrence of each input is the same. It is not the case always.
• The average complexity is defined as the average over the numberof steps executed when processing each input weighted by the probability of occurrence of this input

Data Structures and Algorithms in Java 25
Consider searching sequentially an unordered arrayto find a number
• The best case is when the number is found in the first cell
• The worst case is when the number is in the last cell or not in the array at all
• The average case?

Data Structures and Algorithms in Java 26
• Assuming the probability distribution is uniform• The probability equals to 1/n for each position• To find a number in one try is 1/n • To find a number in two tries is 1/n• etc… • The average steps to find a number is
1 + 2 + … + nn =
n + 12

Data Structures and Algorithms in Java 27
• If the probabilities differ, the average case gives a different outcome
• If the probability of finding a number in the first cell is ½ , the probability in the second cell is ¼ and the probability is the same for remaining cells
=
• the average steps
1  ½  ¼n  2
14(n  2)
831
)2(86)1(1
)2(4...3
42
21 +
+=−−−
+=−
+++
nn
nnn
n

Data Structures and Algorithms in Java 28
Summation FormulasLet N > 0, let A, B, and C be constants, and let f and g be any functions. Then:
∑∑==
=N
k
N
kkfCkCf
11)()(
S1: factor out constant
∑∑∑===
±=±N
k
N
k
N
kkgkfkgkf
111)()())()((
S2: separate summed terms
NCCN
k=∑
=1
S3: sum of constant
2)1(
1
+=∑
=
NNkN
k
S4: sum of k
6)12)(1(
1
2 ++=∑=
NNNkN
k
S5: sum of k squared
122 10
−= +=∑ N
N
k
k
S6: sum of 2^k
12)1(21
1 +−=∑=
− NN
k
k Nk
S7: sum of k2^(k1)

Data Structures and Algorithms in Java 29
LogarithmsLet b be a real number, b > 0 and b ≠ 1. Then, for any real number x > 0, the logarithm of x to base b is the power to which b must be raised to yield x. That is:
x byx yb == ifonly and if )(log
For example:
642 because 6)64(log 62 ==
8/12 because 3)8/1(log 32 =−=−
12 because 0)1(log 02 ==
If the base is omitted, the standard convention in mathematics is that log base 10 is intended; in computer science the standard convention is that log base 2 is intended.

Data Structures and Algorithms in Java 30
Logarithms
Let a and b be real numbers, both positive and neither equal to 1. Let x > 0 and y > 0 be real numbers.
0)1(log =bL1:
1)(log =bbL2:
10 allfor 0)(log xbL4:
)(log)(log)(log yxxy bbb +=L7:
)(log)(loglog yxyx
bbb −=⎟⎟⎠
⎞⎜⎜⎝
⎛L8:
)(log)(log xyx by
b =L9:
yb yb =)(logL5:
xb xb =)(logL6:)(log)(log)(log
bxx
a
ab =
L10:

Data Structures and Algorithms in Java 31
Limit of a FunctionDefinition:
Let f(x) be a function with domain (a, b) and let a < c < b. The limit of f(x) as x approaches c is L if, for every positive real number ε, there is a positive real number δsuch that whenever xc < δ then f(x) – L < ε.
The definition being cumbersome, the following theorems on limits are useful. We assume f(x) is a function with domain as described above and that K is a constant.
KKcx
=→
limC1:
cxcx
=→
limC2: 0 allfor lim >=→
rcx rrcx
C3:

Data Structures and Algorithms in Java 32
Limit of a FunctionHere assume f(x) and g(x) are functions with domain as described above and that K is a constant, and that both the following limits exist (and are finite):
)(lim)(lim xfKxKfcxcx →→
=C4:
( ) )(lim)(lim)()(lim xgxfxgxfcxcxcx →→→
±=±C5:
Axfcx
=→
)(lim Bxgcx
=→
)(lim
Then:
( ) )(lim*)(lim)(*)(lim xgxfxgxfcxcxcx →→→
=C6:
( ) 0 provided )(lim/)(lim)(/)(lim ≠=→→→
Bxgxfxgxfcxcxcx
C7:

Data Structures and Algorithms in Java 33
Limit as x Approaches InfinityDefinition:
Let f(x) be a function with domain [0, ∞). The limit of f(x) as x approaches ∞ is L if, for every positive real number ε, there is a positive real number N such that whenever x > N then f(x) – L < ε.
The definition being cumbersome, the following theorems on limits are useful. We assume f(x) is a function with domain [0, ∞) and that K is a constant.
KKx
=∞→
limC8:
01lim =∞→ xx
C9: 0 allfor 01lim >=∞→
rxrx
C10:

Data Structures and Algorithms in Java 34
Limit of a Rational FunctionGiven a rational function the last two rules are sufficient if a little algebra is employed:
37
003007
5lim2lim3lim
10lim5lim7lim
523
1057lim
5231057lim
2
2
2
2
2
2
=
++++
=
++
++=
++
++=
++++
∞→∞→∞→
∞→∞→∞→
∞→∞→
xx
xx
xx
xxxxxx
xxx
xxx
xxDivide by highest power of x from the denominator.
Take limits term by term.
Apply theorem C3.

Data Structures and Algorithms in Java 35
Infinite LimitsIn some cases, the limit may be infinite. Mathematically, this means that the limit does not exist.
0 allfor lim >∞=∞→
rxrx
C11:
∞=+
++=
+
++=
+++
∞→∞→
∞→∞→∞→
∞→∞→
x
xx
x
xx
xxx
xx
xxx
xx
5lim2lim
10lim5lim7lim
52
1057lim
521057lim
2
( ) ∞=∞→
x
xelimC13:
Example:
( ) ∞=∞→
xbx loglimC12:

Data Structures and Algorithms in Java 36
l'Hôpital's RuleIn some cases, the reduction trick shown for rational functions does not apply:
In such cases, l'Hôpital's Rule is often useful. If f(x) and g(x) are differentiable functions such that
?? 52
10)log(57lim =+
++∞→ x
xxx
)()(lim
)()(lim
xgxf
xgxf
cxcx ′′
=→→
∞==→→
)(lim)(lim xgxfcxcx
then: This also applies if the limit is 0.

Data Structures and Algorithms in Java 37
l'Hôpital's Rule ExamplesApplying l'Hôpital's Rule:
27
2
57lim
5210)log(57lim =
+=
+++
∞→∞→
xx
xxxx
Another example:
06lim6lim3lim 10lim23
====+
∞→∞→∞→∞→ xxxxxxxx eex
ex
ex
Recall that: [ ] [ ])()()( xfDeeD xfxf =

Data Structures and Algorithms in Java 38
Mathematical Induction
Mathematical induction is a technique for proving that a statement is true for all integers in the range from N0 to ∞, where N0 is typically 0 or 1.
Let P(N) be a proposition regarding the integer N, and let S be the set of all integers k for which P(k) is true. If
1) N0 is in S, and
2) whenever N is in S then N+1 is also in S,
then S contains all integers in the range [N0, ∞).
First (or Weak) Principle of Mathematical Induction
To apply the PMI, we must first establish that a specific integer, N0, is in S (establishing the basis) and then we must establish that if a arbitrary integer, N ≥ N0, is in S then its successor, N+1, is also in S.

Data Structures and Algorithms in Java 39
Induction ExampleTheorem: For all integers n ≥ 1, n2+n is a multiple of 2.
proof: Let S be the set of all integers for which n2+n is a multiple of 2.
If n = 1, then n2+n = 2, which is obviously a multiple of 2. This establishes the basis, that 1 is in S.
Now suppose that some integer k ≥ 1 is an element of S. Then k2+k is a multiple of 2. We need to show that k+1 is an element of S; in other words, we must show that (k+1)2+(k+1) is a multiple of 2. Performing simple algebra:
(k+1)2+(k+1) = (k2 + 2k + 1) + (k + 1) = k2 + 3k + 2
Now we know k2+k is a multiple of 2, and the expression above can be grouped to show:
(k+1)2+(k+1) = (k2 + k) + (2k + 2) = (k2 + k) + 2(k + 1)
The last expression is the sum of two multiples of 2, so it's also a multiple of 2. Therefore, k+1 is an element of S.
Therefore, by PMI, S contains all integers [1, ∞).
QED

Data Structures and Algorithms in Java 40
Inadequacy of the First Form of InductionTheorem: Every integer greater than 3 can be written as a sum of 2's and 5's.
(That is, if N > 3, then there are nonnegative integers x and y such that N = 2x + 5y.)
This is not (easily) provable using the First Principle of Induction. The problem is that the way to write N+1 in terms of 2's and 5's has little to do with the way N is written in terms of 2's and 5's. For example, if we know that
N = 2x + 5y
we can say that
N + 1 = 2x + 5y + 1 = 2x + 5(y – 1) + 5 + 1 = 2(x + 3) + 5(y – 1)
but we have no reason to believe that y – 1 is nonnegative. (Suppose for example that N is 9.)

Data Structures and Algorithms in Java 41
"Strong" Form of Induction
Let P(N) be a proposition regarding the integer N, and let S be the set of all integers k for which P(k) is true. If
1) N0 is in S, and
2) whenever N0 through N are in S then N+1 is also in S,
then S contains all integers in the range [N0, ∞).
Second (or Strong) Principle of Mathematical Induction
Interestingly, the "strong" form of induction is logically equivalent to the "weak" form stated earlier; so in principle, anything that can be proved using the "strong" form can also be proved using the "weak" form.
There is a second statement of induction, sometimes called the "strong" form, that is adequate to prove the result on the preceding slide:

Data Structures and Algorithms in Java 42
Using the Second Form of InductionTheorem: Every integer greater than 3 can be written as a sum of 2's and 5's.
proof: Let S be the set of all integers n > 3 for which n = 2x + 5y for some nonnegative integers x and y.
If n = 4, then n = 2*2 + 5*0. If n = 5, then n = 2*0 + 5*1. This establishes the basis, that 4 and 5 are in S.
Now suppose that all integers from 4 through k are elements of S, where k ≥5. We need to show that k+1 is an element of S; in other words, we must show that k+1 = 2r + 5s for some nonnegative integers r and s.
Now k+1 ≥ 6, so k1 ≥ 4. Therefore by our assumption, k1 = 2x + 5y for some nonnegative integers x and y. Then, simple algebra yields that:
k+1 = k1 + 2 = 2x + 5y + 2 = 2(x+1) + 5y,
whence k+1 is an element of S.
Therefore, by the Second PMI, S contains all integers [4, ∞).
QED

Data Structures and Algorithms in Java 43
Amortized Complexity
• Amortized analysis:– Analyzes sequences of operations– Can be used to find the average complexity of a
worst case sequence of operations• By analyzing sequences of operations rather
than isolated operations, amortized analysis takes into account interdependence between operations and their results

Data Structures and Algorithms in Java 44
Amortized Complexity (continued)
Worst case:C(op1, op2, op3, . . .) = Cworst(op1) + Cworst(op2) + Cworst(op3) + . . .
Average case:C(op1, op2, op3, . . .) = Cavg(op1) + Cavg(op2) + Cavg(op3) + . . .
Amortized:C(op1, op2, op3, . . .) = C(op1) + C(op2) + C(op3) + . . .
Where C can be worst, average, or best case complexity

Data Structures and Algorithms in Java 45
Amortized Complexity (continued)
Figure 26 Estimating the amortized cost

Data Structures and Algorithms in Java 46
NPCompleteness
• A deterministic algorithm is a uniquely defined (determined) sequence of steps for a particular input– There is only one way to determine the next step
that the algorithm can make• A nondeterministic algorithm is an algorithm
that can use a special operation that makes a guess when a decision is to be made

Data Structures and Algorithms in Java 47
NPCompleteness (continued)
• A nondeterministic algorithm is considered polynomial: its running time in the worst case is O(nk) for some k
• Problems that can be solved with such algorithms are called tractable and the algorithms are considered efficient
• A problem is called NPcomplete if it is NP (it can be solved efficiently by a nondeterministic polynomial algorithm) and every NP problem can be polynomially reduced to this problem

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NPCompleteness (continued)
• The satisfiability problem concerns Boolean expressions in conjunctive normal form (CNF)

Data Structures and Algorithms in Java 49
Summary
• Computational complexity measures the degree of difficulty of an algorithm.
• This measure of efficiency is called asymptotic complexity.
• To evaluate an algorithm’s efficiency, use logical units that express a relationship.
• This measure of efficiency is called asymptotic complexity.

Data Structures and Algorithms in Java 50
Summary (continued)
• Introduced in 1894, the bigO notation specifies asymptotic complexity, which estimates the rate of function growth.
• An algorithm is called constant if its execution time remains the same for any number of elements.
• It is called quadratic if its execution time is O(n2).• Amortized analysis analyzes sequences of
operations.

Data Structures and Algorithms in Java 51
Summary (continued)
• A deterministic algorithm is a uniquely defined (determined) sequence of steps for a particular input.
• A nondeterministic algorithm is an algorithm that can use a special operation that makes a guess when a decision is to be made.
• A nondeterministic algorithm is considered polynomial.
Chapter 2��Complexity AnalysisObjectivesComputational and �Asymptotic ComplexityComputational and �Asymptotic Complexity (continued)Computational and �Asymptotic Complexity (continued)BigO NotationBigO Notation (continued)Properties of BigO NotationProperties of BigO Notation (continued)Ω and Θ NotationsΩ and Θ Notations (continued)Ω and Θ Notations (continued)Examples of ComplexitiesExamples of Complexities (continued)Examples of Complexities (continued)Examples of Complexities (continued)Finding Asymptotic Complexity: ExamplesFinding Asymptotic Complexity: ExamplesExamples ContinuedFinding Asymptotic Complexity: ExamplesFinding Asymptotic Complexity: ExamplesThe Best, Average, �and Worst CasesConsider searching sequentially an unordered array�to find a numberSummation FormulasLogarithmsLogarithmsLimit of a FunctionLimit of a FunctionLimit as x Approaches InfinityLimit of a Rational FunctionInfinite Limitsl'Hôpital's Rulel'Hôpital's Rule ExamplesMathematical InductionInduction ExampleInadequacy of the First Form of Induction"Strong" Form of InductionUsing the Second Form of InductionAmortized ComplexityAmortized Complexity (continued)Amortized Complexity (continued)NPCompletenessNPCompleteness (continued)NPCompleteness (continued)SummarySummary (continued)Summary (continued)