The document presents algorithms related to data structures, focusing on dynamic programming, optimal binary search trees, string editing, and the 0/1 knapsack problem. It describes optimal binary search trees, detailing the average number of comparisons in successful searches and providing examples with specific probabilities. Additionally, it covers string editing and the 0/1 knapsack problem with recursive formulas and multiple examples.

1. Data structure algorithm
presented by
NIBIYA.G
I-MSC(INFORMATION TECHNOLOGY)
NADAR SARASWATHI COLLEGE OF ARTS
AND SCIENCE
VADAPUTHUPATTI, THENI – 625531

2. DYNAMIC PROGRAMMING
 Optimal binary search Tree
 string editing
 0/1 knapsack problem

3. OPTIMAL BINARY SEARCH
TREE
 A binary search tree is one of the most important data
structures which contain a set of element with the operation
of searching, insertion, and deletion.

4.  An optimal binary search tree is the one for which the
average number of comparisons in a search is the smallest
possible, if the probability of searching each element is given.
 Consider four keys A,B,C, and D to be searched for with
probabilities 0.1, 0.2, 0.4 and 0.3 respectively.

5.  The below figure depicts two out of 14 possible binary search
trees containing these keys.

6.  The average number of comparisons in a successful search in
the first of these tree
0.1, 1+0.2, 2+0.4, 3+0.3, 4 = 2.9.
and for the second one it is
0.1, 2+0.2, 1+0.4, 2+0.3, 3 = 2.1.
 Neither of these two trees is, in fact, optimal.

7.  The total number of binary search trees with n keys is equal
to the nth catalan number.
 Let a1,………, aa be distinct keys ordered from smallest to the
largest and let p1,…….pa be the probabilities of searching for
them.

8.  Let C( I , j ) be the smallest average number of comparisons made
in a successful search in a binary search tree T1j.
 Suppose the root contains key ak, the left subtree Tik-1 contains keys
ai,…….ak-1 optimally arranged, and the right sutree tjk+1 contain keys
ak+1….aj also optimally arranged

10.  The recurrence relation
 Here C(i,i-1) = 0 for 1

14. Figure 5.12 two possible binary search tree

15. Figure 5.13 binary search tree of figure 5.12 with external
nodes added

16. Optimal binary search tree(*)

17. Optimal binary search tree(*)

18. Figure 5.16 computation of c(0,4), w(0,4), and
r(0,4)

20. string editing

21. string editing

22. string editing FORMULA

23. string editing
(example)

25. 0/1 knapsack problem

26. 0/1 knapsack problem

27. 0/1 knapsack problem
(recursive formula sub-program)

28. 0/1 knapsack algorithm

29. Example -1

30. Example -2

31. Example -3

32. Example -4

33. Example -5

34. Example -6

35. Example -7

36. Example -8

37. Example -9

38. Example -10

39. Example -11

40. Example -12

41. Example -13

42. Example -14

43. Example -15

44. Example -16

45. Example -17

46. Example -18