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CSCI 6212 Design and Analysis of Algorithms Which algorithm is better ? Dr. Juman Byun The George Washington University Please drop this course if you have not taken the following prerequisite. Sometimes enthusiasm alone is not enough. CSci 1311: Discrete Structures I (3) CSci 1112: Algorithms and Data Structures (3)
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Running Time Calculation
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Example: Running Time Analysis Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1
![Example: Running Time Analysis Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1](http://images.slideplayer.com./38/10813594/slides/slide_3.jpg "Example: Running Time Analysis Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1")
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LineCostTimes 1c1n 2c2n-1 3c3n-1 4c4n-1 5c5 6c6 7c7 8c8n-1
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Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
![Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key](http://images.slideplayer.com./38/10813594/slides/slide_5.jpg "Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key")
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Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
![Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key](http://images.slideplayer.com./38/10813594/slides/slide_6.jpg "Best Case: already sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key")
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Best Case: already sorted
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simply express it
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Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key
![Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key](http://images.slideplayer.com./38/10813594/slides/slide_9.jpg "Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key 6A[i +1] = A[i] 7i = i - 1 8A[i + 1] = key")
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Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key// for entire A[1..j-1] 6A[i +1] = A[i]// ∴ t j = (j - 1) + 1 7i = i - 1// = j 8A[i + 1] = key
![Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key// for entire A[1..j-1] 6A[i +1] = A[i]// ∴ t j = (j - 1) + 1 7i = i - 1// = j 8A[i + 1] = key](http://images.slideplayer.com./38/10813594/slides/slide_10.jpg "Worst Case: reverse sorted Insertion Sort (A) 1 for j = 2 to A.length 2 key = A[j] 3// Insert A[j] into the sorted sequence A[1..j-1] 4i = j - 1 5while i > 0 and A[i] > key// for entire A[1..j-1] 6A[i +1] = A[i]// ∴ t j = (j - 1) + 1 7i = i - 1// = j 8A[i + 1] = key")
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Worst Case: reverse sorted
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To abstract running time T(n) using constants Notation of "Worst-Case Running Time of Insertion Sort"
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Worst Case: reverse sorted Simply abstract it using constants
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To Denote Relative Algorithm Performance Algorithm of input size n with running time f(n) Asymptotic Notation
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O and o Notation "Big O" vs "little o"
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Θ "Theta"
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Ω and ω Notation "Omega" vs "little omega"
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Why is O popular ?
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