Selected Solutions for Chapter 21:
Data Structures for Disjoint Sets
Solution to Exercise 21.23
We want to show that we can assign
O.1/
charges to M
AKE
S
ET
and F
IND
S
ET
and an
O.
lg
n/
charge to U
NION
such that the charges for a sequence of these
operations are enough to cover the cost of the sequence—
O.m
C
n
lg
n/
, according
to the theorem. When talking about the charge for each kind of operation, it is
helpful to also be able to talk about the number of each kind of operation.
Consider the usual sequence of
m
M
AKE
S
ET
, U
NION
, and F
IND
S
ET
operations,
n
of which are M
AKE
S
ET
operations, and let
l < n
be the number of U
NION
operations. (Recall the discussion in Section 21.1 about there being at most
n
NUL
1
U
NION
operations.) Then there are
n
M
AKE
S
ET
operations,
l
U
NION
operations,
and
m
NUL
n
NUL
l
F
IND
S
ET
operations.
The theorem didn’t separately name the number
l
of U
NION
s; rather, it bounded
the number by
n
. If you go through the proof of the theorem with
l
U
NION
s, you
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 Spring '10
 .
 LG, Basic concepts in set theory, lg l/

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