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The Coin Changing problem
The Coin Changing problem
Suppose we need to make change for 67
. We want to do
nickels, dimes and quarters are available.this using the fewest number of coins possible. Pennies,
Optimal solution for 67
has six coins: two quarters, one
dime, a nickel, and two pennies.
We can use a
greedy algorithm
to solve this problem:
remaining sum, until the desired sum is obtained.repeatedly choose the largest coin less than or equal to the
This is how millions of people make change every day (*).
CS404/
Computer Science
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Design and Analysis of Algorithms: Lecture 19
The Coin-Changing problem:
formal
description
The Coin-Changing problem:
formal
description
Let
D
d
1 , d
2
, ..., d
k
}
be a finite set of distinct coin
denominations. Example:
d
1
d
2
d
3
, and
d
4
We can assume each
d
i
is an integer and
d
1
d
2
... > d
k .
Each denomination is available in unlimited quantity.
The Coin-Changing problem:
Make change for
n
cents, using a minimum total number
of coins.
Assume that
d k
= 1 so that there is always a solution.
CS404/
Computer Science
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Design and Analysis of Algorithms: Lecture 19
A Dynamic Programming Solution:
Step (i)
A Dynamic Programming Solution:
Step (i)
Step (i):
Characterize the structure of a coin-change solution
Define
C
[
j ] to be the minimum number of coins we need to
make change for
j
cents.
making change forIf we knew that an optimal solution for the problem of
j
cents used a coin of denomination
d
i ,
we would have:
C
[
j ] = 1 +
C [ j − d i
].
CS404/
Computer Science
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Design and Analysis of Algorithms: Lecture 19
A Dynamic Programming Solution:
Step (ii)
A Dynamic Programming Solution:
Step (ii)
Step (ii):
Recursively define the value of an optimal solution
C
[
j ] =
if
j <
if
j
min
1 ≤ i ≤ k { C [ j − d i ] }
if
j
CS404/
Computer Science
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Design and Analysis of Algorithms: Lecture 19
An example
An example
C
[11] = min
C
[
50]
C
[
25]
C
[
10]
C
[
1]
C C[20] = 2; ..., C[29] = 5;
[30] = min
C
[
50]
C
[
25]
C
[5] = 6
C
[
10]
C
[20] = 3;
C
[
1]
C
[29] = 6;
% $
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Computer Science
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Design and Analysis of Algorithms: Lecture 19
A Dynamic Programming Solution:
Step (iii)
A Dynamic Programming Solution:
Step (iii)
Step (iii):
Compute values in a bottom-up fashion.
Avoid examining
C
[
j ] for
j <
0 by ensuring that
j
d
i
before
looking up
C [ j − d i
].
C
OMPUTE-
C
HANGE(
n, d, k
C
[0] := 0
for j := 1 to n do
C
[
j ] :=
for i := 1 to k do
if
j
d
i
and 1 +
C [ j − d i ]
< C
[
j ] then
C
[
j ] := 1 +
C [ j − d i ]
return
c
Complexity: Θ(
nk
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Computer Science
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Design and Analysis of Algorithms: Lecture 19
Step (iv):
Print optimal solution
Step (iv):
Print optimal solution
P
RINT-COINS(
denom, j
if
j >
P
RINT-COINS(
denom, j
denom
[
j ])
print
denom
[
j ]
Initial call is PRINT-COINS(
denom, n
Example: CS404/
Computer Science
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Design and Analysis of Algorithms: Lecture 19
Time complexity of DP algorithms
Time complexity of DP algorithms
Usually the complexity of a DP algorithm is:
of sub-problems
×
choices for each sub-problem
For example:
0/1 Knapsack Problem:
C[
i,̟
] = max( C[
i
−
], C[
i
−
w
i ] +
p
i ).
n
×
M
sub-problems, each needs to check
choices.
nM
C Matrix Chain Multiplication:
[
i, j
] = min
i ≤
k<j
C
[
i, k
] +
C
[
k
, j
] +
rows
[
A
i ]
∗
col
[
A
k
]
∗
col
[
A
j ] }
n
×
n
sub-problems, each needs to check
O
n
) choices
— O ( n 3 ) Coin Changing Problem:
size of
C
n
,
k
possible coin types
for each
C
[
j ]. — Θ(
nk
% $
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Computer Science
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Design and Analysis of Algorithms: Lecture 19
Another Dynamic Programming Solution
Another Dynamic Programming Solution
Step (i):
Characterize the structure of a coin-change solution
Making change for
j
cents with coins
d 1
, d
2
, ..., d
i
can be
- Don’t use coindone in two ways:
d i
(even if it’s possible):
C
[
i, j
] =
C
[
i
−
, j
]
- Use coin
d
i
and reduce the amount:
C
[
i, j
] = 1 +
C
[
i, j
d i ].
We will pick the solution with least number of coins:
C
[
i, j
] = min(
C
[
i
−
, j
]
C
[
i, j
d
i ])
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Computer Science
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Design and Analysis of Algorithms: Lecture 19
Another Dynamic Programming Solution
Another Dynamic Programming Solution
Step (ii):
Recursively define the value of an optimal solution
C
[
i, j
] =
if
j <
if
j
j
if
i
= 0
min
{ C [ i − 1
, j
]
C
[
i, j
d
i ] }
if
j
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Computer Science
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Design and Analysis of Algorithms: Lecture 19
Example:
Bottom-up computation
Example:
Bottom-up computation
Suppose we have coin set
d
1 , d
2
, d
3 }
c,
c,
c }
and
n
c .
d3 = 6 | 0 1 2 3 1 2 1 2 2d2 = 4 | 0 1 2 3 1 2 3 4 2d1 = 1 | 0 1 2 3 4 5 6 7 8----------------------------- C[i,j] | 0 1 2 3 4 5 6 7 8
C
[
8] = min(
C
[
8]
C
[
d
3
]) = min(
Evidently, the optimal solution does NOT use
d
3
.
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Design and Analysis of Algorithms: Lecture 19
Another Dynamic Programming Solution
Another Dynamic Programming Solution
Step (iv):
Construct an optimal solution.
Two strategies:
Online: use an additional matrix
S
[
..k,
..n
], where
S
[
i, j
]
indicates which of the values
C
[
i
−
, j
] and
C
[
i, j
d
i ] was
used to compute
C
[
i, j
] (use two symbols:
and
Compute
S
in parallel with
C
optimal solution by starting backwards fromBatch: recover the denominations of the coins used in the
C
[
k, n
], after
computing the entire matrix
C
space complexity. HW exercise: write the pseudocode for each, analyze time & CS404/
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Design and Analysis of Algorithms: Lecture 19
