tut1.pdf - SI 408 Data Structures Tutorial 1 1 Assume f g h...

SI 408: Data Structures
Tutorial 1
1. Assume f, g, h : N
+
→ R . Show the following:
(a) If f ( n ) = Θ( g ( n )), then g ( n ) = Θ( f ( n )).
(b) If f ( n ) = O ( g ( n )) and g ( n ) = O ( h ( n )), then f ( n ) = O ( h ( n )). Similarly for O replaced
by Ω or Θ .
(c) If f
1
, g
1
, f
2
, g
2
are functions such that f
1
∈ O ( f
2
) and g
1
∈ O ( g
2
), then show that
f
1
+ f
2
∈ O ( g
1
+ g
2
) and f
1
· f
2
∈ O ( g
1
· g
2
) . Show by example that the following can be
false : f
1
- f
2
∈ O ( g
1
- g
2
) .
2. Show that for any functions f, g : N → R . Let the function h : N → R be defined by
h ( n ) = max { f ( n ) , g ( n ) } . Show that h ( n ) = Θ( f ( n )+ g ( n )). Also let h
0
( n ) = min { f ( n ) , g ( n ) } .
Show that there are functions