1. Applications of Recursion
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2. Applications of Recursion
Binary search
Hanoi tower
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3. Binary Search Given an array A, search for a specific element
Linear search
for(int i=0; i<size; i++) {
if(A[i]….) …. }
Time complexity: O(n)
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4. Binary Search (continued)
Linear search is used for unsorted array
For sorted array, we can use binary search
Given array A[0..size]
A[0] ≤ A[1] ≤ A[2] ≤ A[3] ≤ … ≤ A[size]
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5. Binary Search Algorithm
To search A[lo] through A[hi] do the following
found = false;
mid = approximate midpoint between lo and hi
If(key==a[mid]) found = true;
else if(key<A[mid]) {
search A[lo] through A[mid-1]; }
else // key > A[mid]
Search A[mid+1] through A[hi]
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6. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

7. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

8. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

9. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33.
a[lo]  value  a[hi]
Ex. Binary search for 33.

10. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

11. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

12. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

13. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

14. Binary Search Demo (1) Algorithm maintains Ex. Binary search for 33
a[lo]  value  a[hi]
Ex. Binary search for 33

15. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

16. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

17. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

18. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

19. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

20. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

21. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

22. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40

23. Binary Search Demo (2) Algorithm maintains Ex. Binary search for 40
a[lo]  value  a[hi]
Ex. Binary search for 40
terminate search (lo>hi)

24. C++ Solution (Iterative)

25. C++ Solution (recursive)

26. Complexity of Binary Search
O(Log2(n)) n
log(n) 10 3
100 7
1,000
1,000,000 20
1,000,000,000 30
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27. The Towers of Hanoi A Stack-based Application
GIVEN: three poles and a set of discs on the first pole, discs of different sizes, the smallest discs at the top
GOAL: move all the discs from the left pole to the right one.
CONDITIONS
Only one disc may be moved at a time.
A disc can be placed either on an empty pole or on top of a larger disc.

28. Towers of Hanoi

29. Towers of Hanoi

30. Towers of Hanoi

31. Towers of Hanoi

32. Towers of Hanoi

33. Towers of Hanoi

34. Towers of Hanoi

35. Towers of Hanoi

36. Data Structure: Stack void Push(element) element Pop()
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37. Stack: Animation
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38. Recursive Solution
void hanoi (int N, Stack left, Stack middle, Stack right) {
int value;
if( N == 1)
value = left.pop();
right.push(value); }
else
hanoi(N-1, left, right, middle);
hanoi(N-1,middle, left, right);

39. Is the End of the World Approaching?
Problem complexity 2n
64 gold discs
Given 1 move a second
 600,000,000,000 years until the end of the world 

40. Online-Animation