2. optimal solution minimizing number of $3’s has at most one $3 ($3 + $3  $5 + $1) has at most two $1’s ($1+$1+$1  $3) has value at most $4 in $1 and $3  has the same number of $5 as greedy sol yes

3. no 8 = 4 + 4 8 = 6 + 1 + 1

4. yes optimal solution minimizing number of $4’s has at most one $4 ($4 + $4  $7 + $1) has at most three $1’s ($1+$1+$1+$1  $4) has value at most $6 in $1 and $4  has the same number of $7 as greedy sol

5. MAIN IDEA: if there is a counterexample then there is a small counterexample 1<A<B

6. LEMMA: optimal solution for the smallest counterexample doesn’t contain B E = O 1 + O A * A + O B * B E = G 1 + G A * A + G B * B E-B = O 1 + O A * A + (O B -1)*B E-B = G 1 + G A * A + (G B -1)*B

7. LEMMA: optimal solution for the smallest counterexample doesn’t contain B copies of A (A+A+...+A  B+B+...+B) A copies of 1 (1+1+...+1  A) E  (B-1)*A + (A-1) = A*B-1

8. THEOREM: if there is a counterexample then there is on with E  A*B-1 MAIN IDEA: if there is a counterexample then there is a small counterexample

9. for all C  AB-1 find optimum (dynamic programming) check if agrees with greedy

10. LEMMA: greedy solution for the smallest counterexample doesn’t contain A E = O 1 + O A * A E = G 1 + G A * A + G B * B E-A = O 1 + (O A -1)* A E-A = G 1 + (G A -1)* A + G B *B

11. LEMMA: optimal solution for the smallest counterexample doesn’t contain 1 E = O 1 + O A * A E = G 1 + G B * B E-1 = (O 1 -1)+ O A * A E-1 = (G 1 -1)+ + G B * B

12. LEMMA: If there exists a counterexample, then there exists a counterexample E = O A * A E = G 1 + G B * B with O A <B and G 1 <A

13. LEMMA: If there exists a counterexample, then there exists a counterexample E = O A * A E = G 1 + G B * B with O A <B and G 1 <A with G B = 1

14. polynomial-time solution  check if P=  B/A  A is a counterexample 

15.  check if P=  B/A  A is a counterexample 6 = 5 + 1 = 2*3 yes 8 = 6 + 1 + 1 = 2*4 no 8 = 7 +1 = 2*4 yes

16. each measurement has 3 outcomes 3 measurements  27 outcomes N=14  2*14 = 28 outcomes

17. SG S SGG 1/3 2/3

18. M[i,j] = min { P[j] + min M[k,j] + M[i-k,j] P[i] + min M[i,k] + M[i,j-k] 1  k  i-1 1  k  j-1

19. K[i,s]  K[i-1,s] if s  W[i] and K[i-1,s-W[i]]+V[i]>K[i,s] then K[i,s]  K[i-1,s-W[i]]+V[i]

20. M(m,n)=m * n - 1 proof: induction on m+n base case m+n=2  m=n=1 ok m * n  m 1 * n and m 2 * n, by IH (m 1 *n – 1) + (m 2 * n – 1 ) + 1 = m * n -1

21. binary search tree 5 47 2 13 depth  running time INSERT DELETE SEARCH

22. B-tree 5 47 2 1 3 branching factor > 2 * makes balancing easier * efficient for “burst memory” (HDD) uniform depth INSERT DELETE SEARCH

23. B-tree branching factor > 2 * makes balancing easier * efficient for “burst memory” (HDD)... 10 3 10 6 10 9

24. B-tree every node other than the root has  T-1 keys (i.e.,  T children) every node has  2T-1 keys (i.e.,  2T children) if T=2 the number of children is 2,3, or 4

25. B-tree - insert 102030 232429 2527 INSERT(T,26)

26. B-tree - insert 102030 232429 2527 INSERT(T,26)

27. B-tree - insert 102030 232429 2527 INSERT(T,26)

28. B-tree - insert 102030 232429 2526 INSERT(T,26) 27

29. B-tree - insert 102030 232429 2526 INSERT(T,28) 27 full leaf

30. B-tree - insert 102030 232429 2526 INSERT(T,28) 27 full leaf

31. B-tree - insert 102030 232429 2526 INSERT(T,28) 27 full leaf

32. B-tree - insert 10 20 30 232429 2526 INSERT(T,28) 27 full leaf

33. B-tree - insert 10 20 30 23 24 29 2526 INSERT(T,28) 27 full leaf

34. B-tree - insert 10 20 30 23 24 29 25 26 INSERT(T,28) 27 full leaf

35. B-tree - insert 10 20 30 23 24 29 25 26 INSERT(T,28) 27 full leaf 28

36. B-tree - insert 102030 232429 2527 INSERT(T,26) split proactively

37. B-tree - insert 10 20 30 232429 2527 INSERT(T,26) split proactively

38. B-tree - insert 10 20 30 232429 2527 INSERT(T,26) split proactively

39. B-tree - insert 10 20 30 23 24 29 2527 INSERT(T,26) split proactively

40. B-tree - insert 10 20 30 23 24 29 2527 INSERT(T,26) split proactively

41. B-tree - insert 10 20 30 23 24 29 25 INSERT(T,26) split proactively 2627

42. B-tree - insert 10 20 30 23 24 29 25 INSERT(T,28) split proactively 2627

43. B-tree - insert 10 20 30 23 24 29 25 INSERT(T,28) split proactively 26 2728

44. B-tree - insert 2T-1 keys  T-1 and T-1 keys every node other than the root has  T-1 keys (i.e.,  T children) every node has  2T-1 keys (i.e.,  2T children) uniform depth – the only operation increasing depth is splitting the root

45. B-tree - delete 10 20 30 23 24 29 25 26 2728 DELETE(T,28)

46. B-tree - delete 10 20 30 23 24 29 25 26 27 DELETE(T,28) leaf deletion

47. B-tree - delete 10 20 30 23 24 29 25 26 27 DELETE(T,27) leaf deletion ?