Lecture 2 Sorting. Sorting Problem Insertion Sort, Merge Sort e.g.,
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Transcript of Lecture 2 Sorting. Sorting Problem Insertion Sort, Merge Sort e.g.,
Lecture 2 Sorting
Sorting Problem
.'''such that sequence
input of }',...,','{n permutatioA :Output
}.,...,,{ numbers of sequenceA :Input
21
21
21
n
n
n
aaa
aaa
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Insertion Sort, Merge Sort
6. 5, 4, 3, 2, 1, :Output
3. 1, 6, 4, 2, 5, :Input
e.g.,
Efficiency
• Running time from receiving the input to producing the output.
)log(
)( 2
nnO
nOInsertion Sort
Merge Sort
Running time
Is array a data structure?
Is array a data structure?
• No!
• A data structure is a standard part in construction of algorithms.
• What data structures do you know on array?
Is array a data structure?
• No!
• A data structure is a standard part in construction of algorithms.
• What data structures do you know on array?
• Stack, queue, list, …, heap.
Heapsort
• Heap, a data structure
• Max-Heapify procedure
• Building a heap
• Heapsort
A Data Structure Heap
• A heap is an array object that can be viewed as a nearly complete binary tree.
35
2 4 1
6
6 5 3 2 4 1
1
2 3
4 5 6
;12return
)(Right
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Max-Heap
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Min-Heap
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Max-Heapify
• Max-Heapify(A,i) is a subroutine.
• When it is called, two subtrees rooted at Left(i) and Right(i) are max-heaps, but A[i] may not satisfy the max-heap property.
• Max-Heapify(A,i) makes the subtree rooted at A[i] become a max-heap by letting A[i] “float down”.
4
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2 8 1
8 7
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if-end
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];g[][ exchangebegin then
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estlarA
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Running time
).(lg in time runsHeapify -Max Hence,
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Alengthi
AlengthAsizeheap
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e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.
4
31
16 10
7
2 9
14 8
.2/ is child a has holocation wlast The n
Proof. .on induction by is proof This n
n
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n 1n
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.12/1)/2(n odd. is 2. Case nn
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.12/1)/2(n odd. is 2. Case nn
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Building a Max-Heap
);,(Heapify-Max do
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Alengthi
AlengthAsizeheap
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e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.
4
31
16 10
7
2 9
14 8
4
31
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4
31
16 10
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14 9
2 8
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2 8
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1
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2 8
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2 8
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1
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2 8
10
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1
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2 8
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1
4 9
2 8
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1
8 9
2 4
10
Analysis
).( and levelat
nodes ofnumber theis )(# where
)()(# timerunning0
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)(- wherelg
16
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1
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2 4
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Running time
).(2
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2
because
)()2
)(2()(2)(2
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2/1
2/11
2/1
2/11
1/2
2
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Heapsort
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16
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7 3
1
8 9
2 4
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Input: 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.Build a max-heap
16, 14, 10, 8, 7, 9, 3, 2, 4, 1.
1
14
7 3
16
8 9
2 4
10
1
14
7 3
16
8 9
2 4
10
1, 14, 10, 8, 7, 9, 3, 2, 4, 16.
14
8
7 3
16
4 9
2 1
10
14
8
7 3
16
4 9
2 1
10
14, 8, 10, 4, 7, 9, 3, 2, 1, 16.
1
8
7 3
16
4 9
2 14
10
1
8
7 3
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4 9
2 14
10
1, 8, 10, 4, 7, 9, 3, 2, 14, 16.
10
8
7 3
16
4 1
2 14
9
10
8
7 3
16
4 1
2 14
9
10, 8, 9, 4, 7, 1, 3, 2, 14, 16.
2
8
7 3
16
4 1
10 14
9
2
8
7 3
16
4 1
10 14
9
2, 8, 9, 4, 7, 1, 3, 10, 14, 16.
9
8
7 2
16
4 1
10 14
3
9
8
7 2
16
4 1
10 14
3
9, 8, 3, 4, 7, 1, 2, 10, 14, 16.
2
8
7 9
16
4 1
10 14
3
8
7
2 9
16
4 1
10 14
3
1
7
2 9
16
4 8
10 14
3
7
4
2 9
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1 8
10 14
3
2
4
7 9
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4
2
7 9
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1 8
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3
1
2
7 9
16
4 8
10 14
3
3
2
7 9
16
4 8
10 14
1
1
2
7 9
16
4 8
10 14
3
2
1
7 9
16
4 8
10 14
3
1
2
7 9
16
4 8
10 14
3
Running Time
for-end
;)1(Heapify-Max
;1][-][-
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begin do
2 downto ][for
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)(Heapsort
A,
AsizeheapAsizeheap
iAA
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A
A
O(lg n)
O(n)
)lg( nnO
Quicksort
• Worst-case running time
• Expected running time
• The best practical choice (why?)
)( 2n
)lg( nnO
Divide and Conquer
• Divide the problem into subproblems.
• Conquer the subproblems by solving them recursively.
• Combine the solutions to subproblems into the solution for original problem.
Idea of Quicksort
them.combine toneeded is work no
],..1[ and ]1..[sort after Then
.1for ][][
1for ][][
such that ]..1[ and ]1..[
subarrays into ]..[array Partition
rqAqpA
riqiAqA
qipqAiA
rqAqpA
rpA
Example
9. 8, 7, 6, ,5̂ 4, 3, 2, 1,
6. 7, 9, 8, ,5̂ 1, 2, 4, 3,
Quicksort
then-end
);,1,(Quicksort
);1,,(Quicksort
);,,(Partition
beginn the
if
),,(Quicksort
rqA
qpA
rpAq
rp
rpA
How to find such a partition?
• Such a partition may not exist.
e.g., 5, 4, 3, 2, 1.
• Hence, we may need to make such a partition.
• Take a A[i].
• Classify other A[j] by comparing it with A[i].
8 6, 5, 7, ,4̂ 3, 1, 2,
4̂ 6, 5, 7, 8, ,3 1, 2,
4̂ ,6 5, 7, 8, ,3 1, 2,
4̂ 6, ,5 7, 8, ,3 1, 2,
4̂ 6, 5, ,3 8, 7, ,1 2,
4̂ 6, 5, 3, ,1 7, 8, ,2
4̂ 6, 5, 3, 1, 7, ,8 ,2
4̂ 6, 5, 3, 1, 7, 8, ,2
4̂ 6, 5, 3, 1, 7, 8, ,2 ,
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7̂ 6, 5, 8, ,4 ,1 3, 2,
4̂ 6, 5, 8, ,7 ,1 3, 2,
4̂ 6, 5, 8, ,7 ,1 3, 2,
4̂ 6, 5, 8, ,1 ,7 3, 2,
4̂ 6, 5, ,8 1, 7, ,3 2,
4̂ 6, 5, ,3 1, 7, ,8 2,
4̂ ,6 5, 3, 1, 7, 8, ,2
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Expected Partition
partition. balanced ain iswhich
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expected theThus, ./1y probabilit thehas
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n
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Expected Running Time
1
11
1
1
1
1
lg2
))]0([E)]1([E(1
))]2([E)]1([E(1
))]1([E)]0([E(1
)](E[
:Induction
log)](E[
Quicksort of timerunning expected the:Guess
partition of timerunning the:Assume
n
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11
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cc
nccncnn
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Randomized Quicksort
);,,(Partition
];[][ exchange
);,(Random
),,(Partition-Randomized
rpA
rAiA
rpi
rpA
Randomized Quicksort
);,1,(Quichsort-Ramdomized
);1,,(Quichsort-Ramdomized
);,,(Partition-Randomized
),,Quicksort(-Randomized
rqA
qpA
rpAq
rpA
What we learnt in this lecture?
• What is heap, max-heap and min-heap?
• Heapsort and Quicksort.
• What is expected running time?
