1. RECURSION

2. What is recursion? a function calls itself
direct recursion
a function calls its invoker
indirect recursion f1 f f2

3. Recursion is an alternative to iteration (loop)
often more "elegant" and concise than a loop
sometimes very inefficient
recursion should be considered when
a problem can be defined in terms of successive smaller problems of the same type, i.e. recursively
eventually the problem gets small enough that the answer is known (base case)
reducing the problem size leads to the base case
sometimes the "work" of the algorithm is done after the base case is reached

4. Outline of a Recursive Function
if (answer is known)
provide the answer
else
make a recursive call to
solve a smaller version
of the same problem
base
case
recursive
case - “leap
of faith”

5. Factorial (n) - recursive
Factorial (n) = n * Factorial (n-1)
for n > 0
Factorial (0) = 1
int RecFact (int n) {
if (n > 0)
return n * RecFact (n-1);
else
return 1; }
base case

6. How Recursion Works
a recursive function call is handled like any other function call
each recursive call has an activation record on the stack
stores values of parameters and local variables
when base case is reached return is made to previous call
the recursion “unwinds”

7. Recursive Call Tree RecFact (4) 24 int RecFact (int n) 4 {3 2 1 6 24
int RecFact (int n) {
if (n > 0)
return n * RecFact (n-1);
else
return 1; }

8. Factorial (n) - iterative
Factorial (n) = n * (n-1) * (n-2) * ... * 1
for n > 0
Factorial (0) = 1
int IterFact (int n) {
int fact =1;
for (int i = 1; i <= n; i++)
fact = fact * i;
return fact; }

9. Another Recursive Definition
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, ….
for n == 1
fib(n) = for n == 2
fib(n-2) + fib(n-1) for n>2

10. Tracing fib(6) 5 6 4 4 3 3 2 3 2 2 1 2 1 2 1

11. When to use recursion?
if recursive and iterative algorithms are of similar efficiency --> iteration
if the problem is inherently recursive and a recursive algorithm is less complex to write than an iterative algorithm --> recursion
recursion often simpler than iteration when it is not tail recursion or there are multiple recursive calls
recursion very inefficient when values are recomputed

12. Euclid'sAlgorithm
Finds the greatest common divisor of two nonnegative integers that are not both 0
Recursive definition of gcd algorithm
gcd (a, b) = a (if b is 0)
gcd (a, b) = gcd (b, a % b) (if b != 0)
Write a recursive gcd function
prototype?
definition?

13. Ackermann's function
A(0, n) = n + 1
A(m+1, 0) = A(m, 1)
A(m+1, n+1) = A(m, A(m+1, n))

14. Iterative gcd int gcd (int a, int b) { int temp; while (b != 0)
temp = b;
b = a % b;
a = temp; }
return a;

15. iterative binary search
bool binarySearch (const int a[ ], int numItems, int item) {
bool found = false;
int first = 0;
int last = numItems – 1;
while (first <= last && ! found) {
mid = (first + last) / 2;
if (item == a[mid])
found = true;
else if (item > a[mid])
first = mid +1;
else last = mid – 1; }
return found;

16. bool binarySearch(const int a[ ], int first, int last, int item){
if (first > last)
return false; // base case 1 (unsuccessful search)
else
int mid = (first + last) / 2;
if (item == a[mid])
return true; // base case 2 (successful search)
else if (item > a[mid])
return binarySearch(a, mid+1, last, item); // X
return binarySearch(a, first, mid-1, item); // Y }

17. a = [ ]
binarySearch(a, 0, 7, 6)
binarySearch(a, 0, 7, 9)
2, 1,
0, 7, 3
0, 2, 1
2, 2, 2 X Y
false
0, 7, 3
0, 2, 1
2, 2, 2 X Y
true
base case 2
base case 1

18. What Does a Compiler Do? Lexical analysis Parsing Generate object code
divide a stream of characters into a stream of tokens
total = cost * cost;
if ( ( cond1 && ! cond2 ) )
Parsing
do the tokens form a valid program, i.e. follow the syntax rules?
Generate object code

19. BNF (Backus-Naur form)
a language used to define the syntax rules of a programming language
consists of
productions – rules for forming some construct of the language
meta-terms – symbols of BNF
terminals – appear as shown
non-terminals – syntax defined by another production

20. Syntax Rules for a simplified boolean expression meta-terms are in red
bexpr -> bterm || bterm | bterm
bterm -> bfactor && bfactor | bfactor
bfactor -> !bfactor|(bexpr)|true|false|ident
ident -> alpha {alpha|digit|_}
alpha -> a .. z | A .. Z
digit ->
meta-terms are in red
terminals are in blue
non-terminals are in black

21. bexpr -> bterm || bterm | bterm
does a sequence of tokens form a valid bexpr?
if (there is a valid bterm) if (nextToken is ||) if (there is a valid bterm) return true else return false else return true else return false

22. Comparing Algorithms and ADT Data Structures
big Oh Notation
Comparing Algorithms and ADT Data Structures

23. Algorithm Efficiency
a measure of the amount of resources consumed in solving a problem of size n
time
space
benchmarking – code the algorithm, run it with some specific input and measure time taken
better for measuring and comparing the performance of processors than for measuring and comparing the performance of algorithms
Big Oh (asymptotic analysis) provides a formula that associates n, the problem size, with t, the processing time required to solve the problem