CS 473 -Algorithms I

1. CS473-Algorithms I Lecture X Ackermann’s Function and Its Inverse Lecture X

2. if i = 0 if i > 0 and lg(i-1)n > 0 if i > 0 and or lg(i-1)n is undefined undefined Repeated Exponentiation Functiong(i) i That is The parameter i gives the height of the stack of 2’s that make up the exponent 4 Repeated Logarithm Function lg(i)n The logarithm fn applied i time in succession starting with arg. n Lecture X

3. Iterated Logarithm Function lg(*)n lg* n = min { i 0 : lg(i)n 1 } lg* 2 = 1 lg* 4 = lg*(22) = 2 lg* 16 = lg*(24) = 3 lg* 65536 = lg*(216) = 4 lg* (265536) = 5 • Since the number of atoms in the observable universe • 1080 << 265536 • We rarely encounter a value of n such that lg*n > 5 Lecture X

4. P Iterated Logarithm Function Computation of lg* 65536 Therefore Lecture X

5. n n n-1 where n n-1 n-2 n-2 n-3 Iterated Logarithm Function The lg*n fn is essentially the inverse of repeated exponentiation Lecture X

6. . . . . . n-n=0 . Therefore n-i n Iterated Logarithm Function Lecture X

7. Ackermann’s Function • Defined for integers i, j > 1 by • A(1, j) = 2j for j 1 • A(i, 1) = A(i – 1, 2) for i 2 • A(i, j) = A(i – 1, A(i, j - 1)) for i, j 2 j = 1 j = 2 j = 3 j = 4 23 24 22 21 i = 1 i = 2 i = 3 Lecture X

8. Ackermann’s Function • Defined for integers i, j > 1 by • A(1, j) = 2j for j 1 • A(i, 1) = A(i – 1, 2) for i 2 • A(i, j) = A(i – 1, A(i, j - 1)) for i, j 2 j = 1 j = 2 j = 3 j = 4 23 24 22 21 i = 1 i = 2 i = 3 22 Lecture X

9. Ackermann’s Function • Defined for integers i, j > 1 by • A(1, j) = 2j for j 1 • A(i, 1) = A(i – 1, 2) for i 2 • A(i, j) = A(i – 1, A(i, j - 1)) for i, j 2 j = 1 j = 2 j = 3 j = 4 23 24 22 21 i = 1 i = 2 22 2 22 i = 3 Lecture X

10. Ackermann’s Function • Defined for integers i, j > 1 by • A(1, j) = 2j for j 1 • A(i, 1) = A(i – 1, 2) for i 2 • A(i, j) = A(i – 1, A(i, j - 1)) for i, j 2 j = 1 j = 2 j = 3 j = 4 23 24 22 21 i = 1 i = 2 22 2 2 22 22 i = 3 Lecture X

11. Ackermann’s Function • Defined for integers i, j > 1 by • A(1, j) = 2j for j 1 • A(i, 1) = A(i – 1, 2) for i 2 • A(i, j) = A(i – 1, A(i, j - 1)) for i, j 2 j = 1 j = 2 j = 3 j = 4 23 24 22 21 i = 1 4 3 i = 2 22 16 2 2 22 22 2 i = 3 Lecture X

12. . . Ackermann’s Function Second Row 1 2 3 4 . j Lecture X

13. Ackermann’s Function Third Row 16 16 16 16 16 ∙ ∙ ∙ ∙ Lecture X

14. ∙ ∙ 16 ∙ ∙ for j 2 j-1 of these 16 16 Ackermann’s Function Third Row • The first row, exponential in the column number j, is alreadyrapidly growing • The second row, consists of the widely spaced subset of columnsof the first row • The third row consists of even more widely spaced subset ofcolumns • , …, of the second row • which is even sparser subset of columns of the first row Lecture X

15. Explosive Growth of Ackermann’s Function Column Spacings S1 S2 S3 S4 1 2 S1 S2 Row 3 4 5 Lecture X

16. 16 16 Explosive Growth of Ackermann’s Function • The first row, exponential in the column number j, • is alreadyrapidly growing • The second row, consists of the widely spaced subset of columnsof the first row • The third row consists of even more widely spaced subset of columns • , …, of the second row • which is even sparser subset of columns of the first row Lecture X

17. α(m, n) = min{ i 1 : A( i, ) > lg n } Explosive Growth of Ackermann’s Function IN GENERAL The spacing between successive columns of row i-1 appearing in row i increases dramatically with both the column number and the rownumberi Inverse of Ackermann’s Function • Proceed in the -th column of the matrix starting from the first row • Until you encounter a value > lg n • Return the row index of that value Lecture X

18. Inverse of Ackermann’s Function • This fn is not the inverse of Ackermann’s fn in true math sense • It captures the spirit of the inverse in its growth • The mysterious lg n threshold appears in order to be able to prove the O(m α(m,n)) bound on the running time • Claim : for all practical purposes • since • Ackermann’s fn A(i, j) is strictly increasing with each argument • for all since Lecture X

19. , so for all practical purposes Inverse of Ackermann’s Function In particular, But we also have A(4,1) = A(3,2) = Hence, A(4,1) >> 1080 = Estimated no. of atoms in the universe Thus, for all practical purposes 16 O(mlg*n) bound is only slightly weaker than O(m α(m,n)) bound Lecture X

20. 1 1 2 ….. 1 2 3 4 5 . . . i2 A(i,c2) > A(i,c1) for all i where i1 A(i1,c1) = First value in col. c1 > lg n A(i2,c2) = First value in col. c2 > lg n An Important Property of Inverse Ackermann’s Function For a fixed value of n, as m increases the function α(m,n)is monotonically decreasing Proof :is monotonically increasing as m increases Thus, since n and hence lg n is fixed, the smallest row index i needed to bring above lg n is monotonicallydecreasing Lecture X

21. An Important Property of Inverse Ackermann’sFunction • This property corresponds to our intuition about • disjoint set forests with pathcompression • We would expect the average FIND-PATHlength to decrease • due to pathcompression • If we perform m operations in time O(m α(m,n)) • then, the average time per operation is O(α(m,n)) • which is monotonically decreasing as m increases Lecture X