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Introduction
Mathematical Induction II
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Mathematical Induction II
212
Principle of Mathematical Induction
Principle of mathematical induction
:
To prove that a statement that assert that some property
P(n)
is
true for all positive integers
n, we complete two steps
Basis step
:
We verify that
P(1)
is true.
Inductive step
:
We show that the conditional statement
P(k)
→
P(k + 1)
is true for all positive integers
k
To prove the conditional statement, we assume that
P(k)
is true
(it is called
inductive hypothesis
) and show that under this
assumption
P(k + 1)
is also true
Discrete Mathematics – Mathematical Induction II
213
Analysis of Algorithms
Consider the following problem
There is a group of proposed talks to be given. We want to
schedule as many talks as possible in the main lecture room. Let
be the talks,
talk
begins at time
and ends at
time
.
(No two lectures can proceed at the same time, but a
lecture can begin at the same time another one ends.)
We assume
that
.
m
t
t
t
,
,
,
2
1
K
j
t
j
b
j
e
m
e
e
e
≤
≤
≤
K
2
1
9:00
10:00
11:00
12:00
1
t
2
t
3
t
5
t
4
t
6
t
7
t
8
t
Greedy algorithm:
At every step choose a talk with the earliest ending time among
all those talks that begin after all talks already scheduled end.
We prove that the greedy algorithm is optimal in the sense that it
always schedules the most talks possible in the main lecture hall.
9:00
10:00
11:00
12:00
1
t
2
t
3
t
5
t
4
t
6
t
7
t
8
t
Discrete Mathematics – Mathematical Induction II
214
Greedy Algorithm
Discrete Mathematics – Mathematical Induction II
215
Greedy Algorithm (cntd)
Let
P(n)
be the proposition that if the greedy algorithm schedules
n
talks, then it is not possible to schedule more than
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 Spring '10
 AndreiBulatov

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