. . Januray 25th, 2011 Biostatistics 615/815 - Lecture 6 Hyun Min Kang Januray 25th, 2011 Hyun Min Kang and Elementary Data Structures Linear Sorting Algorithms Biostatistics 615/815 Lecture 6: . . . . . . . Summary Array Radix sort Introduction . . . . . . . . . . 1 / 32 . . . . . . . . . . . . . . . . . . . . . . . . .
. . Homework #2 will be announced in the next lecture . 815 projects . . . . . . . . 5-6 team pairs in total Team assignment will be made during this week Each team should set up a meeting with the instructor to kick-start the project Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . . . . . . . . . . . . Introduction Radix sort Array Summary Announcements . A good and bad news . . . . 2 / 32 . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 815 projects . . . . . . . . 5-6 team pairs in total Team assignment will be made during this week Each team should set up a meeting with the instructor to kick-start the project Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . . . . . . . . . . . . Introduction Radix sort Array Summary Announcements . A good and bad news . . . 2 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . • Homework #2 will be announced in the next lecture
. . . . . 815 projects . . . . . . . . the project Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . Array . . . . . . . . . . Introduction . Radix sort Summary Announcements . A good and bad news . . 2 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . • Homework #2 will be announced in the next lecture • 5-6 team pairs in total • Team assignment will be made during this week • Each team should set up a meeting with the instructor to kick-start
. . . . Insertion sort is loop invariant . . . . . . . . Time complexity of Insertion sort . . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . Algorithm description . . . . . . . . . . Introduction Radix sort Array Summary Recap on sorting algorithms : Insertion sort . 3 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 For each j ∈ [2 · · · n ] , iterate element at indices j − 1 , j − 2 , · · · 1 . 2 If A [ i ] > A [ j ] , swap A [ i ] and A [ j ] 3 If A [ i ] < = A [ j ] , increase j and go to step 1 At the start of each iteration, A [1 · · · j − 1] is loop invariant iff: • A [1 · · · j − 1] consist of elements originally in A [1 · · · j − 1] . • A [1 · · · j − 1] is in sorted order. • Worst and average case time-complexity is Θ( n 2 )
. . . Data : array A and indices p and r Mergesort ( A , p , q ); Merge ( A , p , q , r ); end . Time complexity of Mergesort . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . Summary . . . . . . . . . . Introduction Radix sort Array . 4 / 32 Recap on sorting algorithms : Mergesort . Algorithm Mergesort . . . . . . . . . . . . . . . . . . . . . . . . . . . . Result : A [ p .. r ] is sorted if p < r then q = ⌊ ( p + r )/2 ⌋ ; Mergesort ( A , q + 1 , r ); • Worst and average case time-complexity is Θ( n log n )
. . . Data : array A and indices p and r q = Partition ( A , p , r ); end . Time complexity of Quicksort . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . Summary . . . . . . . . . . Introduction Radix sort . Array Recap on sorting algorithms : Quicksort Algorithm Quicksort . . . . 5 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . Result : A [ p .. r ] is sorted if p < r then Quicksort ( A , p , q − 1 ); Quicksort ( A , q + 1 , r ); • Average case time-complexity is Θ( n log n ) • Worst case time-complexity is Θ( n 2 ) , but practically faster than other Θ( n log n ) algorithms.
. . Januray 25th, 2011 Biostatistics 615/815 - Lecture 6 Hyun Min Kang How Partition algorithm works Summary Array Radix sort Introduction . . . . . . . . . . 6 / 32 . . . . . . . . . . . . . . . . . . . . . . . . .
. . Januray 25th, 2011 Biostatistics 615/815 - Lecture 6 Hyun Min Kang . . . . . . . . Running example with 100,000 elements (in UNIX or MacOS) . Performance of sorting algorithms in practice . . . . . . . . . Summary . 7 / 32 Introduction Radix sort Array . . . . . . . . . . . . . . . . . . . . . . . . . user@host: ˜ /> time cat src/sample.input.txt | src/stdSort > /dev/null real 0m0.430s user 0m0.281s sys 0m0.130s user@host: ˜ /> time cat src/sample.input.txt | src/insertionSort > /dev/null real 1m8.795s user 1m8.181s sys 0m0.206s user@host: ˜ /> time cat src/sample.input.txt | src/mergeSort > /dev/null real 0m0.898s user 0m0.755s sys 0m0.131s user@host: ˜ /> time cat src/sample.input.txt | src/quickSort > /dev/null real 0m0.427s user 0m0.285s sys 0m0.129s
. a sequence. . . . . . . . Any comparison sort algorithm can be represented as a binary decision tree, where each node represents a comparison. Each path from the root to leaf represents possible series of comparisons to sort Each leaf of the decision tree represents one of n possible An informal proof permutations of input sequences We have n l h , where l is the number of leaf nodes, and h is the height of the tree, equivalent to the # of comparisons. Then it implies h log n n log n Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . . . the worst case . . . . . . . . . . Introduction Radix sort Array Summary Lower bounds for comparison sorting . CLRS Theorem 8.1 . . . . . . . . 8 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . Any comparison-based sort algorithm requires Ω( n log n ) comparisons in
. a sequence. . . . . . . . . decision tree, where each node represents a comparison. Each path from the root to leaf represents possible series of comparisons to sort Each leaf of the decision tree represents one of n possible . permutations of input sequences We have n l h , where l is the number of leaf nodes, and h is the height of the tree, equivalent to the # of comparisons. Then it implies h log n n log n Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 An informal proof . the worst case . . . . . . . . . . . Introduction Radix sort Array Summary Lower bounds for comparison sorting CLRS Theorem 8.1 . . . . . . . . 8 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . Any comparison-based sort algorithm requires Ω( n log n ) comparisons in • Any comparison sort algorithm can be represented as a binary
. a sequence. . . . . . . . . decision tree, where each node represents a comparison. Each path from the root to leaf represents possible series of comparisons to sort permutations of input sequences . We have n l h , where l is the number of leaf nodes, and h is the height of the tree, equivalent to the # of comparisons. Then it implies h log n n log n Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 An informal proof . the worst case . . . . . . . . . . . Introduction Radix sort Array Lower bounds for comparison sorting Summary CLRS Theorem 8.1 . . . . . . . . 8 / 32 . . . . . . . . . . . . . . . . . . . . . . . . . Any comparison-based sort algorithm requires Ω( n log n ) comparisons in • Any comparison sort algorithm can be represented as a binary • Each leaf of the decision tree represents one of n ! possible
. decision tree, where each node represents a comparison. Each path An informal proof . . . . . . . . from the root to leaf represents possible series of comparisons to sort . a sequence. permutations of input sequences the height of the tree, equivalent to the # of comparisons. Then it implies h log n n log n Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . the worst case . Summary . . . . . . . . . . Introduction Radix sort Array 8 / 32 Lower bounds for comparison sorting . . . CLRS Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Any comparison-based sort algorithm requires Ω( n log n ) comparisons in • Any comparison sort algorithm can be represented as a binary • Each leaf of the decision tree represents one of n ! possible • We have n ! ≤ l ≤ 2 h , where l is the number of leaf nodes, and h is
. . the worst case . An informal proof . . . . . . . . decision tree, where each node represents a comparison. Each path from the root to leaf represents possible series of comparisons to sort a sequence. permutations of input sequences the height of the tree, equivalent to the # of comparisons. Hyun Min Kang Biostatistics 615/815 - Lecture 6 Januray 25th, 2011 . 8 / 32 . . . Introduction Radix sort . . Array . Summary Lower bounds for comparison sorting . CLRS Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Any comparison-based sort algorithm requires Ω( n log n ) comparisons in • Any comparison sort algorithm can be represented as a binary • Each leaf of the decision tree represents one of n ! possible • We have n ! ≤ l ≤ 2 h , where l is the number of leaf nodes, and h is • Then it implies h ≥ log ( n !) = Θ( n log n )
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