Discrete Mathematics
Advanced Counting Technique
(Recurrence)
M. Affan Aleem
1
Recursive Definition
To define a function with the set of
nonnegative integers as its domain,
1. Specify the value of the function at zero
2. Give a rule for finding its value as an
integer from its values at smaller
integers
Such a definition is called recursive or
inductive definition
2
Recursive Definition
Example:
f(0) = 3
f(n+1) = 2*f(n) + 3
Find f(1), f(2), f(3)
3
Recursive Definition cont…
Example:
Give an inductive definition of the factorial
function F(n) = n!
F(0) = F(1) = 1
F(n+1) = (n+1)F(n)
When F(0) is the only value of the function
that occurs, no more reductions
4
Definition
A recurrence relation for the sequence {an}
is an equation that expresses an in terms of
one or more of the previous terms of the
sequence, namely, a0, a1, …,an-1, for all
integers n with n  n0 is a nonnegative
integer
5
Definition
A sequence is called a solution of a
recurrence relation if its terms satisfy the
recurrence relation.
6
Example#1
Let {an} be a sequence that satisfies the
recurrence relation an = an-1 – an-2 for
n = 2,3,4… and suppose that a0 = 3 and
a1 = 5 what are the a2 and a3.
7
Initial Condition
The initial conditions for a sequence specify
the terms that precede the first term where
the recurrence relation takes effect.
8
Modeling with Recurrence Relations
The recurrence relations to model a wide
variety of problems, such as finding
compound interest, counting rabbits on a
island, determining the number of moves in
the Tower of Hanoi puzzle, and counting bit
strings with certain properties
9
10
Examples
Compound interest suppose that a person
deposits $10,000 in a saving account at a
bank yielding 11% per year with interest
compounded annually. How much will be in
the account after 30 years.
11
Solution
• To solve this problem, let Pn denote the
amount in the account after n years.
• Since the amount in the account after n
years equal the amount in the account after
n-1 year plus interest for the nth year, we
see that the sequence {Pn} satisfies the
recurrence relation.
12
Solution contd…
pn  pn 1  0.11 pn 1  (1.11) pn 1
• The initial condition is p0 = 10,000
• We can use an iterative approach to find a
formula for Pn note that
13
Tower of Hanoi
(Problem Statement)
14
Tower of Hanoi-Problem Statement
• There three pegs, some disks are in first
peg in descending order from bottom to up
• Transfer the all disk from first peg to third
peg with following conditions:
• Only one disk can be moved at a time,
from one peg to another, such that it never
is placed on a disk of smaller diameter.
15
Tower of Hanoi
(Step by Step Solution)
16
Tower of Hanoi- one disk
1-moves
17
Tower of Hanoi- two disk
Note: that we perform the solution to the one-disk
Tower of Hanoi twice (once in the top row, and
once in the bottom row). Between rows, we move
the bottom disk. Thus, we require
2· 1+1 = 3 Moves
18
Tower of Hanoi- three disk
Note that we perform the solution to the two-disk
Tower of Hanoi twice (once in the top row, and
once in the bottom row). Between rows we move
the bottom disk. Thus, we require
2(2 · 1+ 1) +1= 7 Moves
19
Tower of Hanoi: What we know so far
Let M(n) denote the minimum number of
legal moves required to complete a
tower of Hanoi puzzle that has n disks
n
1
2
M(n)
1
3
3
7
20
Tower of Hanoi: What we know so far
Following the pattern, for n = 4 we need to
solve the three-disk puzzle twice, plus one
more operation to move the largest disk.
Thus,
M(4) =2 · M(3) +1 = 2· 7+1= 15.
21
Tower of Hanoi: What we know so far
Similarly, for n = 5 disks, we expect that we
will need to perform
M(5) = 2 • M(4) +1 = 2• 15+1= 31
22
Tower of Hanoi: n-disk
Let M(n) denote the minimum number of
legal moves required to complete a
tower of Hanoi puzzle that has n disks.
We thus obtain the recursion relation,
M(n) = 2M(n - 1) + 1
23
Tower of Hanoi: Solution
With the solution for a single disk
M(1) = 1
the recursion relation
M(n) = 2M(n - 1) + 1
n = 2, 3,4 … (number of disk)
24
Tower of Hanoi: Solution
Make a series of different values of n
1, 3, 7,15, 31…nth term
Each term can be generated by Tn = 2n-1
25
Tower of Hanoi: practical Consequence
The practical difficulty with exponential
algorithms is that they can quickly grow
out of hand. (With each additional disk, the
minimum number of operations essentially
doubles.) N.B., 1 century ~ 4.5 × 109
seconds. (The age of the universe is
4 × 1017 seconds.)
26
n
2n - 1
n
2n - 1
n
2n - 1
8
255
19
524287
30
1073741823
9
511
20
1048575
31
2147483647
10
1027
21
2097151
32
4294967295
11
2047
22
4194303
33
8589934591
12
4095
23
8388607
34
17179869183
13
8191
24
16777215
35
34359738367
14
16383
25
33554431
36
68719476735
15
32767
26
67108863
37
137438953471
16
65635
27
134217727
38
274877906943
17
131071
28
268435455
39
549755813887
18
262143
29
536870911
64
18446744073709551615
27