cs201: part 1
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CS201: PART 1. Data Structures & Algorithms S. Kondakcı. Analysis of Algorithms. Algorithm. Input. Output. An algorithm is a step-by-step procedure for solving a problem in a finite amount of time. Theoretical Analysis of Algorithms: - PowerPoint PPT PresentationTRANSCRIPT
CS201: PART 1
Data Structures & AlgorithmsS. Kondakcı
Analysis of Algorithms 2
Analysis of Algorithms
AlgorithmInput Output
An algorithm is a step-by-step procedure for
solving a problem in a finite amount of time.
Theoretical Analysis of Algorithms:
•Uses a high-level description of the algorithm instead of an implementation
•Characterizes running time as a function of the input size n.
•Takes into account all possible inputs Allows us to evaluate the speed of any design independent of its implementation.
Analysis of Algorithms 3
Program Efficiency
Program efficiency: is a measure of the amount of resources required to produce desired results.
Efficiency Aspects:
1) What are the important resources we should try to optimize?
2) Where are the important efficiency gains to be made?3) How important is efficiency in the first place?
Analysis of Algorithms 4
Efficiency Today
User Efficiency. The amount of time and effort users will spend to learn how o use the program, how to prepare the data, how to configure and customize the program, and how to interpret and use the output.Maintenance Efficiency. The amount of time and effort maintenance group will spend reading a program and its technical documentation in order to understand it well enough to make any necessary modifications. Algorithmic Complexity. The inherent efficiency of the method itself, regardless of which machine we run it on or how we code it. Coding Efficiency. This is the traditional efficiency measure. Here we are concerned with how much processor time and memory space a computer program requires to produce desired results. Coding efficiency is the key step towards optimal usage of machine resources.
Analysis of Algorithms 5
Programmer’s Duty
Programmers should should keep these in mind:
1. Correct, robust, and reliable.2. Easy to use for its intended end-user
group.3. Easy to understand and easy to modify. 4. Portable.5. Consistency in Input/Output behavior.6. User documentation.
Analysis of Algorithms 6
Optimization
Optimization on CPU-Time: Consider a network security assessment tool as a real-time application. The application works like a security scanner protocol designed to audit, monitor, and correct all aspects of network security. Real-time processing of the intercepted network packets containing inspection information requires faster data processing. Besides, such a process should generate some auditing information.Optimization on Memory: Developing programs that do not fit into the memory space available on your systems is often quite a bit demanding. Kernel level processing of the network packets requires kernel memory optimization and a powerful and failsafe memory management capability.Providing Run-time Continuity: Extensive machine-level optimization is a major requirement for continuously running programs, such as the security scanner daemons.Reliability and Correctness: One of the inevitable efficiency requirements is the absolute reliability. The second important efficiency factor is correctness. That is, your program should do exactly what it is supposed to do. Choosing and implementing a reliable inspection methodology should be done with precision.Optimization on Programmer’s Time: How efficient a programmer works depends on the choice of team policy and developmen tool selection.
Analysis of Algorithms 7
Coding Efficiency: Unstructured Code
8: read(x,y,z); if (x >= y) goto 1; if (y >= z) goto 5; goto 4;1: if (x >= z) goto 2; goto 4;5: big = y; goto 3;2: big = x; goto 3;4: big = z;3: if (!eof) goto 8
Figure 5.2: Unstructured coding, see Figure 5.3 /Efficient Programming/S. Kondakci-1999
Analysis of Algorithms 8
Coding Efficiency: Structured Code
while (!eof) {read(x,y,z);if ((x >= y) && (x >= z)) big = x;else if ((y >= x) && (y >= z)) big = yelse big = z;
}
Figure 5.3: Structured coding versus unstructured coding /Efficient Programming/S. Kondakci-1999
Analysis of Algorithms 9
Protecting Against Run-time Errors
Illegal pointer operations.Array subscript out of bound.Endless loops may cause stacks grow into the heap area.Presentational errors, such as network byte order, number conversions, division by zero, undefined results, e.g., tan(90) = undefined.Trying to write over the kernel’s text area, or the data area.Referencing objects declared as prototype but not defined. Performing operations on a pointer pointing at NULL.Operating system weaknesses.
Analysis of Algorithms 10
Assertions A general pitfall: making assumptions that turn out not to be justified.
Most of the mistakes arise from simply misunderstanding the interaction between various pieces of code
The assertion rule states that you should always express yourself boldly or forcefully of the fact that there are some other things that you have not covered clear enough yet. Any assumptions you make in writing your programs should be documented somewhere in the code itself, particularly if you know or expect the assumption to be false in other environments.
Analysis of Algorithms 11
Does the Machine Understand Your Assumptions? Remember those assumptions are yours: They should be presented to the machine by any means that you are supposed to provide in your code. The machine will not be able to check your assumptions. This is simply a matter of including explicit checks in your code, even for things that “cannot happen”.
if (p == NULL)panic(“Driver routine: p is NULL\n”);
if (p->p_flags & BUSY); /* Safe to continue */…<etcetera> ASSERT(p !=NULL); If (p->p_flags & BUSY); /* Safe to continue */…<etcetera> …
Analysis of Algorithms 12
Guidelines for the implementation
1. Protect input parameters using call-by-value.2. Avoid global variables and functions with side effects.3. Make all temporary variables local to functions where they
are used.4. Never halt or sleep in a function. Spawn a dedicated function
if necessary.5. Avoid producing output within a function unless the sole
purpose of the function is output.6. Where appropriate use return values to return the status of
function calls.7. Avoid confusing programming tricks.8. Always strive for simplicity and clarity. Never sacrifice clarity
of expression for cleverness of expression.9. If any keep your assertions local to your code.10. Never sacrifice clarity of expression for minor reductions in
execution time.
Analysis of Algorithms 13
Debugging and TracingMaking use of the preprocessor can allow you to incorporate many debugging aids in your module, for instance, the driver module. Later, in the production version these debugging aids can be removed.
#ifdef DEBUG#define TRACE_OPEN (debugging && 0x01)#define TRACE_CLOSE (debugging && 0x02)#define TRACE_READ (debugging && 0x04)#define TRACE_WRITE (debugging && 0x08)
int debugging = -1; /* enable all traces output */#else#define TRACE_OPEN 0#define TRACE_CLOSE 0#define TRACE_READ 0#define TRACE_WRITE 0#endif...
Analysis of Algorithms 14
Tracing: Later in the Program
if (TRACE_READ)printf(‘’Device driver read, Packet number (%d) \n’’,pack_no);… <etcetera>…
Later, in the code the output of the trace information can be done by a manner similar to this:
Analysis of Algorithms 15
Checking Programs With lint (Unix)The lint utility is intended to verify some facets of a C program, such as its potential portability. lint derives from the idea of picking the “fluff” out of a C program. It does this, by advising on C constructs (including functions) and usage which might turn out to be ‘bugs’, portability problems, inconsistent declarations, bad function and argument types, or dead code. See the manual section lint(1) for further explanations.
$ cat mytest.cmain() /* nonsense */{
char y = ‘1’, z;int x = ‘p’;extern float duble();
goto LABEL1;while (1 !=0) {
LABEL1: x+= (long) y;printf(“Fasten your seat belts! %d\n”, x);}
return (y);}
Analysis of Algorithms 16
Now, Lint’ing$ lint –hxa mytest.c
(8) warning: loop not entered at top
(8) warning: constant in conditional context
variable unused in function
(3) z in main
implicitly declared to return int
(10) printf
declaration unused in block
(5) duble
function returns value, which is always ignored
printf
$ cat mytest.cmain() /* nonsense */{
char y = ‘1’, z;int x = ‘p’;extern float duble();
goto LABEL1;while (1 !=0) {
LABEL1: x+= (long) y;printf(“Fasten your seat belts! %d\n”, x);}
return (y);}
Analysis of Algorithms 17
Test Coverage Analysis Yet another tool born for execution tracing and analysis of programs called tcov,it can be used to trace and analyze a source code to report a coverage test. tcov does this by analysing the source code step-by-step. The extra code is generated by giving the –xa option to the compiler command, i.e.,
$ gcc -xa -o src src.c
The –xa option invokes a runtime recording mechanism that creates a .d file for every .c file. The .d file accumulates execution data for the corresponding source file.
The tcov utility can then be run on the source file to generate statistics about the program. The following example source file, getmygid.c, is analysed as:
$ cc -xa -o getmygid getmygid.c
$ tcov -a getmygid.c
$ ls –l getmy???*
-rwxr-xr-x 1 staff 25120 Feb 11 12:07 getmygid
-rw------- 1 staff 519 Sep 9 1994 getmygid.c
-rw-r--r-- 1 staff 9 Feb 11 12:07 getmygid.d
-rw-r--r-- 1 staff 1025 Feb 11 12:08 getmygid.tcov
Analysis of Algorithms 18
Example: getmygid.c $ cat getmygid.c #include <stdio.h>char *msg = "I am sorry I cannot tell you everything" ;int gid,egid;int uid,euid, pid ,ppid, i;int main(){ gid = getgid(); if (gid >= 0) printf("1- My GID is: %d\n", gid); egid = getegid(); if (egid >=0 ) printf("2- My EGID is: %d\n", egid); uid = getuid(); if ( uid >=0) printf("3- My uid is: %d\n", uid); euid = geteuid(); if (euid >= 0) printf("4- My Euid is: %d\n", euid); pid = getpid(); if ( pid >=0 ) printf("5- My pid is: %d\n", pid); ppid = getppid(); if ( ppid >= 0) printf("6- My ppid is: %d\n", ppid); prt_msg("We came to end!!!"); return 0; prt_msg(msg);}prt_msg(char *mesg){ printf("%s \n", mesg);}
Analysis of Algorithms 19
Tcov’ing getmygid.c $ cat getmygid.tcov ##### -> #include <stdio.h> ##### -> char *msg = "I am sorry I cannot tell you everything" ; ##### -> ##### -> int gid,egid; ##### -> int uid,euid, pid ,ppid, i; ##### -> int main() ##### -> { 2 -> gid = getgid(); 2 -> if (gid >= 0) printf("1- My GID is: %d\n", gid); 2 -> egid = getegid(); 2 -> if (egid >=0 ) printf("2- My EGID is: %d\n", egid); 2 -> uid = getuid(); 2 -> if ( uid >=0) printf("3- My uid is: %d\n", uid); 2 -> euid = geteuid(); 2 -> if (euid >= 0) printf("4- My Euid is: %d\n", euid); 2 -> pid = getpid(); 2 -> if ( pid >=0 ) printf("5- My pid is: %d\n", pid); 2 -> ppid = getppid(); 2 -> if ( ppid >= 0) printf("6- My ppid is: %d\n", ppid); 2 -> prt_msg("We came to end!!!"); 2 -> return 0; 2 -> prt_msg(msg);
2 -> } 2 -> prt_msg(mesg) 2 -> char *mesg; 2 -> { 2 -> printf("%s \n", mesg); 2 -> }
Analysis of Algorithms 20
Tcov’ing getmygid.c
Top 10 Blocks Line Count 9 2 11 2 13 2 15 2 17 2 19 2 21 2
29 2 8 Basic blocks in this file 8 Basic blocks executed 100.00 Percent of the file executed 16 Total basic block executions 2.00 Average executions per basic block
As shown, tcov(1) generates an annotated listing of the source file (getmygid.tcov), where each line is prefixed with a number indicating the count of execution of each statement on the line. Finally per line and per block statistics are shown.
Analysis of Algorithms 21
Have nice break!
Analysis of Algorithms
AlgorithmInput Output
An algorithm is a step-by-step procedure forsolving a problem in a finite amount of time.
Analysis of Algorithms 23
Running Time
Most algorithms transform input objects into output objects.The running time of an algorithm typically grows with the input size.Average case time is often difficult to determine.We focus on the worst case running time.
Easier to analyze Crucial to applications
such as games, finance and robotics
0
20
40
60
80
100
120
Runnin
g T
ime
1000 2000 3000 4000
Input Size
best caseaverage caseworst case
Analysis of Algorithms 24
Experimental Studies
Write a program implementing the algorithmRun the program with inputs of varying size and compositionUse a function, like the built-in clock() function, to get an accurate measure of the actual running timePlot the results
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 50 100
Input Size
Tim
e (
ms)
Analysis of Algorithms 25
Limitations of Experiments
It is necessary to implement the algorithm, which may be difficultResults may not be indicative of the running time on other inputs not included in the experiment. In order to compare two algorithms, the same hardware and software environments must be used
Analysis of Algorithms 26
Theoretical Analysis
Uses a high-level description of the algorithm instead of an implementationCharacterizes running time as a function of the input size, n.Takes into account all possible inputsAllows us to evaluate the speed of an algorithm independent of the hardware/software environment
Analysis of Algorithms 27
Pseudocode
High-level description of an algorithmMore structured than English proseLess detailed than a programPreferred notation for describing algorithmsHides program design issues
Algorithm arrayMax(A, n)Input array A of n integersOutput maximum element of A
currentMax A[0]for i 1 to n 1 do
if A[i] currentMax thencurrentMax A[i]
return currentMax
Example: find max element of an array
Analysis of Algorithms 28
Pseudocode Details
Control flow if … then … [else …] while … do … repeat … until … for … do … Indentation replaces
braces Method declarationAlgorithm method (arg [, arg…])
Input …
Output …
Method/Function callmethod (arg [, arg…])
Return valuereturn expression
Expressions Assignment
(like in C++) Equality testing
(like in C++)n2 Superscripts and
other mathematical formatting allowed
Analysis of Algorithms 29
The Random Access Machine (RAM) Model
A CPU
A potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character
01
2
Memory cells are numbered and accessing any cell in memory takes unit time.
Analysis of Algorithms 30
Primitive Operations
Basic computations performed by an algorithmIdentifiable in pseudocodeLargely independent from the programming languageExact definition not importantAssumed to take a constant amount of time in the RAM model
Examples: Evaluating an
expression Assigning a
value to a variable
Indexing into an array
Calling a method Returning from a
method
Analysis of Algorithms 31
Counting Primitive Operations
By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size
Algorithm arrayMax(A, n)
# operations
currentMax A[0] 2for i 1 to n 1 do 2 n
if A[i] currentMax then 2(n 1)currentMax A[i] 2(n 1)
{ increment counter i } 2(n 1)return currentMax 1
Total 7n 1
Analysis of Algorithms 32
Estimating Running Time
Algorithm arrayMax executes 7n 1 primitive operations in the worst case. Define:a = Time taken by the fastest primitive operationb = Time taken by the slowest primitive operation
Let T(n) be worst-case time of arrayMax. Thena a (7(7nn 1) 1) TT((nn)) bb(7(7nn 1) 1)
Hence, the running time T(n) is bounded by two linear functions
Analysis of Algorithms 33
Growth Rate of Running Time
Changing the hardware/ software environment Affects T(n) by a constant factor, but Does not alter the growth rate of T(n)
The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax
Analysis of Algorithms 34
Growth Rates
Growth rates of functions: Linear n Quadratic n2
Cubic n3
In a log-log chart, the slope of the line corresponds to the growth rate of the function
1E+01E+21E+41E+61E+8
1E+101E+121E+141E+161E+181E+201E+221E+241E+261E+281E+30
1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n
T(n
)
Cubic
Quadratic
Linear
Analysis of Algorithms 35
Constant Factors
The growth rate is not affected by constant factors
or lower-order terms
Examples 102n 105 is a
linear function 105n2 108n is a
quadratic function
1E+01E+21E+41E+61E+8
1E+101E+121E+141E+161E+181E+201E+221E+241E+26
1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n
T(n
)
Quadratic
Quadratic
Linear
Linear
Analysis of Algorithms 36
Big-Oh Notation
Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constantsc and n0 such that
f(n) cg(n) for n n0
Example: 2n 10 is O(n) 2n 10 cn (c 2) n 10 n 10(c 2) Pick c 3 and n0 10
1
10
100
1,000
10,000
1 10 100 1,000n
3n
2n+10
n
Analysis of Algorithms 37
Big-Oh Example
Example: the function n2 is not O(n)
n2 cn n c The above
inequality cannot be satisfied since c must be a constant
1
10
100
1,000
10,000
100,000
1,000,000
1 10 100 1,000n
n^2
100n
10n
n
Analysis of Algorithms 38
More Big-Oh Examples
7n-2
7n-2 is O(n)need c > 0 and n0 1 such that 7n-2 c•n for n n0
this is true for c = 7 and n0 = 1
3n3 + 20n2 + 53n3 + 20n2 + 5 is O(n3)need c > 0 and n0 1 such that 3n3 + 20n2 + 5 c•n3 for n
n0
this is true for c = 4 and n0 = 21 3 log n + log log n3 log n + log log n is O(log n)need c > 0 and n0 1 such that 3 log n + log log n c•log n
for n n0
this is true for c = 4 and n0 = 2
Analysis of Algorithms 39
Big-Oh and Growth Rate
The big-Oh notation gives an upper bound on the growth rate of a functionThe statement “f(n) is O(g(n))” means that the growth rate of f(n) is no more than the growth rate of g(n)
We can use the big-Oh notation to rank functions according to their growth rate
f(n) is O(g(n)) g(n) is O(f(n))
g(n) grows more Yes No
f(n) grows more No Yes
Same growth Yes Yes
Analysis of Algorithms 40
Big-Oh Rules
If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e.,
1. Drop lower-order terms2. Drop constant factors
Use the smallest possible class of functions
Say “2n is O(n)” instead of “2n is O(n2)”Use the simplest expression of the class
Say “3n 5 is O(n)” instead of “3n 5 is O(3n)”
Analysis of Algorithms 41
Asymptotic Algorithm Analysis
The asymptotic analysis of an algorithm determines the running time in big-Oh notationTo perform the asymptotic analysis
We find the worst-case number of primitive operations executed as a function of the input size
We express this function with big-Oh notationExample:
We determine that algorithm arrayMax executes at most 7n 1 primitive operations
We say that algorithm arrayMax “runs in O(n) time”Since constant factors and lower-order terms are eventually dropped anyhow, we can disregard them when counting primitive operations
Analysis of Algorithms 42
Computing Prefix Averages
We further illustrate asymptotic analysis with two algorithms for prefix averagesThe i-th prefix average of an array X is average of the first (i 1) elements of X:
AA[[ii]] XX[0] [0] XX[1] [1] … … XX[[ii])/(])/(ii+1)+1)0
5
10
15
20
25
30
35
1 2 3 4 5 6 7
X
A
Analysis of Algorithms 43
Prefix Averages (Quadratic)The following algorithm computes prefix averages in quadratic time by applying the definition
Algorithm prefixAverages1(X, n)Input array X of n integersOutput array A of prefix averages of X #operations A new array of n integers nfor i 0 to n 1 do n
s X[0] nfor j 1 to i do 1 2 … (n 1)
s s X[j] 1 2 … (n 1)A[i] s (i 1) n
return A 1
Analysis of Algorithms 44
Arithmetic Progression
The running time of prefixAverages1 isO(1 2 …n)
The sum of the first n integers is n(n 1) 2 There is a simple
visual proof of this fact
Thus, algorithm prefixAverages1 runs in O(n2) time
0
1
2
3
4
5
6
7
1 2 3 4 5 6
Analysis of Algorithms 45
Prefix Averages (Linear)The following algorithm computes prefix averages in linear time by keeping a running sum
Algorithm prefixAverages2(X, n)Input array X of n integersOutput array A of prefix averages of X #operationsA new array of n integers ns 0 1for i 0 to n 1 do n
s s X[i] nA[i] s (i 1) n
return A 1Algorithm prefixAverages2 runs in O(n) time
Analysis of Algorithms 46
Computing Spans
We show how to use a stack as an auxiliary data structure in an algorithmGiven an an array X, the span S[i] of X[i] is the maximum number of consecutive elements X[j] immediately preceding X[i] and such that X[j] X[i] Spans have applications to financial analysis
E.g., stock at 52-week high
6 3 4 5 2
1 1 2 3 1
X
S
01234567
0 1 2 3 4
Analysis of Algorithms 47
Quadratic Algorithm
Algorithm spans1(X, n)Input array X of n integersOutput array S of spans of X #S new array of n integers nfor i 0 to n 1 do n
s 1 nwhile s i X[i s] X[i] 1 2 … (n 1)
s s 1 1 2 … (n 1)S[i] s n
return S 1
Algorithm spans1 runs in O(n2) time
Analysis of Algorithms 48
Have nice break!
Analysis of Algorithms 49
RecursionRecursion = a function calls itself as a function for unknown times. We call this recursive call
1
1
n
i
sum i
for (i = 1 ; i <= n-1; i++)
sum = sum +1;
int sum(int n) {
if (n <= 1)
return 1
else
return (n + sum(n-1));
}
Analysis of Algorithms 50
Recursive function
int f( int x )
{
if( x == 0 )
return 0;
else
return 2 * f( x - 1 ) + x * x;
}
22 ( 1)f f x x
Analysis of Algorithms 51
RecursionCalculate factorial (n!) of a positive integer:
n! = n(n-1)(n-2)...(n-n-1), 0! = 1! = 1
0! 1, ! (( 1)!) ( 0)n n n n int factorial(int n) {
if (n <= 1)
return 1;
else
return (n * factorial(n-1));
}
Analysis of Algorithms 52
Fibonacci numbers, Bad algorith for n>40 !
long fib(int n) {
if (n <= 1)
return 1;
else
return fib(n-1) + fib(n-2);
}
0 1 2 3 4 1 2
1 1
1
1, 1, 2, 3, 5,...,
( 1) (5 / 3)
i i i
k k k
k
F F F F F F F F
F F F
T k
( ) (3 / 2)Nfib N
Analysis of Algorithms 53
Algorithm IterativeLinearSum(A,n)
Algorithm IterativeLinearSum(A,n):
Input: An integer array A and an integer n (size)
Output: The sum of the first n integers
if n = 1 then
return A[0]
else
while n 0 do
sum = sum + A[n]
n n - 1
return sum
Analysis of Algorithms 54
Algorithm LinearSum(A,n)
Algorithm LinearSum(A,n):
Input: An integer array A and an integer n (size)
Output: The sum of the first n integers
if n = 1 then
return A[0]
else
return LinearSum(A,n-1) + A[n-1]
Analysis of Algorithms 55
Iterative Approach: Algorithm IterativeReverseArray(A,i,n)
Algorithm IterativeReverseArray(A,i,n):
Input: An integer array A and an integers i and n
Output: The reversal of n integers in A starting at index i
while n > 1 do
swap A[i] and A[i+n-1]
i i +1
n n-2
return
Analysis of Algorithms 56
Algorithm ReverseArray(A,i,n)
Algorithm ReverseArray(A,i,n):
Input: An integer array A and an integers i and n
Output: The reversal of n integers in A starting at index i
if n > 1 then
swap A[i] and A[i+n-1]
call ReverseArray(A, i+1, n-2)
return
Analysis of Algorithms 57
Higher-Order Recursion
Algorithm BinarySum(A,i,n):
Input: An integer array A and an integers i and n
Output: The sum of n integers in A starting at index i
if n = 1 then
return A[i]
return BinarySum(A,i,[n/2])+BinarySum(A,i+[n/2],[n/2])
Making recursive calls more than a single call at a time.
Analysis of Algorithms 58
Kth Fibonacci Numbers0 1 1 2
0
1
2 1 0
3 2 1
4 3 2
5 4 3
6 5 4
7 6 5
8 7 6
0, 1, ,
1
1
1
1 1 1 1 3
1 3 1 1 5
1 5 3 1 9
1 9 5 1 15
1 15 9 1 25
1 25 15 1 41
1 41 25 1 67
i i iF F and F F F
For i
n
n
n n n
n n n
n n n
n n n
n n n
n n n
n n n
Analysis of Algorithms 59
Algorithm BinaryFib(k):
Input: An integer k
Output: A pair ( ) such that is the kth Fibonacci number and is the (k-1)st Fibonacci number
if (k <= 1) then
return (k,0)
else
(i,j) LinearFibonacci(k-1)
return (i+j,i)
kth Fibonacci NumbersLinear recursion
, 1k kF F kF
1kF
Analysis of Algorithms 60
kth Fibonacci Numbers
Algorithm BinaryFib(k):
Input: An integer k
Output: The kth Fibonacci number
if (k <= 1) then
return k
else
return BinaryFib(k-1)+BinaryFib(k-2)
Binary recursion
Analysis of Algorithms 61
properties of logarithms:
logb(xy) = logbx + logby
logb (x/y) = logbx - logby
logbxa = alogbx
logba = logxa/logxbproperties of exponentials:a(b+c) = aba c
abc = (ab)c
ab /ac = a(b-c)
b = a logab
bc = a c*logab
Summations Logarithms and Exponents
Proof techniquesBasic probability
Math you need to Review
Analysis of Algorithms 62
Relatives of Big-Oh
big-Omega f(n) is (g(n)) if there is a constant c > 0
and an integer constant n0 1 such that
f(n) c•g(n) for n n0
big-Theta f(n) is (g(n)) if there are constants c’ > 0 and c’’ > 0 and
an integer constant n0 1 such that c’•g(n) f(n) c’’•g(n) for n n0
little-oh f(n) is o(g(n)) if, for any constant c > 0, there is an integer
constant n0 0 such that f(n) c•g(n) for n n0
little-omega f(n) is (g(n)) if, for any constant c > 0, there is an integer
constant n0 0 such that f(n) c•g(n) for n n0
Analysis of Algorithms 63
Intuition for Asymptotic Notation
Big-Oh f(n) is O(g(n)) if f(n) is asymptotically less than or equal to
g(n)big-Omega
f(n) is (g(n)) if f(n) is asymptotically greater than or equal to g(n)
big-Theta f(n) is (g(n)) if f(n) is asymptotically equal to g(n)
little-oh f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n)
little-omega f(n) is (g(n)) if is asymptotically strictly greater than g(n)
Analysis of Algorithms 64
Example Uses of the Relatives of Big-Oh
f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n0 0 such that f(n) c•g(n) for n n0
need 5n02 c•n0 given c, the n0 that satisfies this is n0 c/5 0
5n2 is (n)
f(n) is (g(n)) if there is a constant c > 0 and an integer constant n0 1 such that f(n) c•g(n) for n n0
let c = 1 and n0 = 1
5n2 is (n)
f(n) is (g(n)) if there is a constant c > 0 and an integer constant n0 1 such that f(n) c•g(n) for n n0
let c = 5 and n0 = 1
5n2 is (n2)