lecture 2 sorting
DESCRIPTION
Lecture 2 Sorting. Sorting Problem. e.g.,. Insertion Sort, Merge Sort. Efficiency. Running time from receiving the input to producing the output. Running time. Insertion Sort. Merge Sort. Heapsort. Heap, a data structure Max-Heapify procedure Building a heap Heapsort. - PowerPoint PPT PresentationTRANSCRIPT
Lecture 2 Sorting
Sorting Problem
.'''such that sequence
input of }',...,','{n permutatioA :Output
}.,...,,{ numbers of sequenceA :Input
21
21
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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.
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6
6 5 3 2 4 1
1
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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”.
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Running time
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e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.
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Analysis
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Running time
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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
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1
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1, 14, 10, 8, 7, 9, 3, 2, 4, 16.
14
8
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14, 8, 10, 4, 7, 9, 3, 2, 1, 16.
1
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8
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1, 8, 10, 4, 7, 9, 3, 2, 14, 16.
10
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4 1
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10, 8, 9, 4, 7, 1, 3, 2, 14, 16.
2
8
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4 1
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9
2
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9
2, 8, 9, 4, 7, 1, 3, 10, 14, 16.
9
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4 1
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3
9
8
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4 1
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3
9, 8, 3, 4, 7, 1, 2, 10, 14, 16.
2
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3
8
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Running Time
for-end
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O(lg n)
O(n)
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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 ][][
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such that ]..1[ and ]1..[
subarrays into ]..[array Partition
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Example
9. 8, 7, 6, ,5̂ 4, 3, 2, 1,
6. 7, 9, 8, ,5̂ 1, 2, 4, 3,
Quicksort
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qpA
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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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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
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Expected Running Time
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Randomized Quicksort
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Randomized Quicksort
);,1,(Quichsort-Ramdomized
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),,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?