Tower of Hanoi

1
Divide Conquer

• Themes
• Reasoning about code (correctness and cost)
• iterative code, loop invariants, and sums
• recursion, induction, and recurrence relations
• Divide and Conquer
• Examples
• sorting (insertion sort merge sort)
• computing powers
• Euclidean algorithm (computing gcds)

2
Identities for Sums
3
Some Common Sums
4
Arithmetic Series

• Sum of consecutive integers (written slightly
  differently)

See http//www.cs.drexel.edu/kschmidt/CS520/Lectu
res/2/arithmetic.swf
5
Arithmetic Series (Proof)

6
Geometric Series
1 x x2
xn-1
See http//www.cs.drexel.edu/kschmidt/CS520/Lectu
res/2/geometric.swf
7
Geometric Series (Proof)

8
Floor and Ceiling

• Let x be a real number
• The floor of x, ?x?, is the largest integer less
  than or equal to x
• If an integer k satisfies k ? x lt k1, k ?x?
• E.G. ?3.7? 3, ?3? 3
• The ceiling of x, ?x?, is the smallest integer
  greater than or equal to x
• If an integer k satisfies k-1 lt x ? k, k ?x?
• E.G. ?3.7? 4, ?3? 3

9
logarithm

• y logb(x) ? by x
• Two important cases
• ln(x) loge(x)
• lg(x) log2(x) frequently occurs in CS
• Properties
• log(cd) log(c) log(d)
• log(cx) xlog(c)
• logb(bx) x blogb(x)
• d ln(x)/dx 1/x

10
logarithm

• 2k ?? x lt 2k1 ? k ?lg(x)?
• E.G. 16 ? 25 lt 32 ? 4 ? lg(25) lt 5
• lg(25) ? 4.64
• Change of base
• logc(x) logb(x) / logb(c)
• Proof. y logc(x) ? cy x ? ylogb(c) logb(x)
  ? y logb(x) / logb(c)

11
Insertion Sort

• To sort x0,,xn-1,
• recursively sort x0,,xn-2
• insert xn-1 into x0,,xn-2
• (see code for details)
• Loop invariant (just before test, iltn)
• x0,, xi-1 sorted
• initialize t xi

12
Insertion Sort (Example)

• (7,6,5,4,3,2,1,0)
• after recursive call (1,2,3,4,5,6,7,0)
• Insert 0 into sorted subarray
• Let 0 fall as far as it can
• Number of comparisons to sort inputs that are in
  reverse sorted order (worst case)
• C(n) C(n-1) (n-1)
• C(1) 0

See (http//www.cs.drexel.edu/kschmidt/CS520/Prog
rams/sorts.cc)
13
Merge Sort

• To sort x0,,xn-1,
• recursively sort x0,,xa-1 and xa,,xn-1, where a
  n/2
• merge two sorted lists
• Insertion sort is a special case where a1
• loop invariant for merge similar to insert
  (depends on implementation)

14
Merge Sort (Example)

• (7,6,5,4,3,2,1,0)
• after recursive calls (4,5,6,7) and (0,1,2,3)
• Number of comparisons needed to sort, worst case
  (the merged lists are perfectly interleaved)
• M(n) 2M(n/2) (2n-2)
• M(1) 0
• What is the best case (all in one list gt other
  list)?
• M(n) 2M(n/2) n/2

15
Comparison of Insertion and Merge Sort

• Count the number of comparisons for different
  n2k (see and run sort.cpp)
• M(n)/C(n) ?0 as n increases
• C(n) is of higher order. I.e., C(n) bounds M(n)
  from above, but not tightly
• C(2n)/C(n) ? 4
• So, apparently quadratic.
• C(n) T(n2)
• M(2n)/M(n) ? 2 as n increases

n M(n) C(n) M(n)/C(n)
2 1 1 1
4 4 6 0.667
8 12 28 0.429
16 32 120 0.267
32 80 496 0.161
64 192 2016 0.095
128 4481 8128 0.055
16
Solve Recurrence for C(n)
See (http//www.cs.drexel.edu/kschmidt/CS520/Lect
ures/2/insertionSortSwaps.swf)
17
Solve Recurrence for M(n)
See http//www.cs.drexel.edu/kschmidt/CS520/Lectu
res/2/mergeSortSwaps.swf
18
Computing Powers

• Recursive definition, multiplying as we learned
  in 4th grade (whatever)
• an a ? an-1, n gt 0
• a0 1
• Number of multiplications
• M(n) M(n-1) 1, M(0) 0
• M(n) n

19
Binary Powering (Recursive)

• Binary powering
• (see http//www.cs.drexel.edu/kschmidt/CS520/Prog
  rams/power.cc)
• x16 (((x2)2)2)2, 16 (10000)2
• x23 (((x2)2x)2x)2x, 23 (10111)2
• Recursive (right-left)
• xn (x?n/2?)2 ? x(n 2)
• M(n) M(?n/2?) n 2

20
Binary Powering (Iterative)
(See http//www.cs.drexel.edu/kschmidt/CS520/Prog
rams/power.cc

• Example
• N y z
• 1 x
• 11 x x2
• 5 x3 x4
• 2 x7 x8
• 1 x7 x16
• 0 x23 x32

• Loop invariant
• xn y ? zN
• N n y 1 z x
• while (N ! 0)
• if (N 2 1)
• y zy
• z zz N N/2

21
Binary Powering

• Number of multiplications
• Let ?(n) number of 1bits in binary
  representation of n
• num bits ?lg(n)? 1
• M(n) ?lg(n)? ?(n)
• ?(n) lg(n) gt
• M(n) 2lg(n)

22
Greatest Common Divisors

• g gcd(a,b) ? ga and gb
• if ea and eb ? eg
• gcd(a,0) a
• gcd(a,b) gcd(b,ab)
• since if ga and gb then gab and if gb and
  gab then ga

23
Euclidean Algorithm (Iterative)

• (see http//www.cs.drexel.edu/kschmidt/CS520/Prog
  rams/gcd.cc)
• a0 a, a1 b
• ai qi ai1 ai2 , 0 ? ai2 lt ai1
• an qn an1
• g an1

24
Number of Divisions

• ai qi ai1 ai2 , 0 ? ai2 lt ai1 gt
• ai ? qi ai2 ai2 (qi 1)ai2 ? 2ai2
• gt an ? 2an2 gt an / an2 ? 2
• a1 / a3 ? a2 / a4 ? ? an-1 / an1 ? an / an2 ?
  2n
• n ? lg(a1a2 /g2)
• if a,b ? N, n ? 2lg(N)

25
(blank, for notes)