lecture 4

CS 332: Algorithms Solving Recurrences Continued The Master Theorem Introduction to heapsort

Review: Merge Sort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Analysis of Merge Sort ,[object Object],[object Object],[object Object],MergeSort(A, left, right) { T(n) if (left < right) {  (1) mid = floor((left + right) / 2);  (1) MergeSort(A, left, mid); T(n/2) MergeSort(A, mid+1, right); T(n/2) Merge(A, left, mid, right);  (n) } }

Review: Solving Recurrences ,[object Object],[object Object],[object Object]

Review: Solving Recurrences ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Solving Recurrences ,[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Master Theorem ,[object Object],[object Object],[object Object],[object Object]

The Master Theorem ,[object Object]

Using The Master Method ,[object Object],[object Object],[object Object],[object Object],[object Object]

Sorting Revisited ,[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],Heaps 16 14 10 8 7 9 3 2 4 1

[object Object],[object Object],Heaps 16 14 10 8 7 9 3 2 4 1 1 1 1 1 1

Heaps ,[object Object],16 14 10 8 7 9 3 2 4 1 A = = 16 14 10 8 7 9 3 2 4 1

Heaps ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],16 14 10 8 7 9 3 2 4 1 A = = 16 14 10 8 7 9 3 2 4 1

Referencing Heap Elements ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Heap Property ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Heap Height ,[object Object],[object Object]

Heap Operations: Heapify() ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Heap Operations: Heapify() ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Heapify() Example 16 4 10 14 7 9 3 2 8 1 16 4 10 14 7 9 3 2 8 1 A =

Heapify() Example 16 4 10 14 7 9 3 2 8 1 16 10 14 7 9 3 2 8 1 A = 4

Heapify() Example 16 4 10 14 7 9 3 2 8 1 16 10 7 9 3 2 8 1 A = 4 14

Heapify() Example 16 14 10 4 7 9 3 2 8 1 16 14 10 7 9 3 2 1 A = 4 8

Analyzing Heapify(): Informal ,[object Object],[object Object],[object Object]

Analyzing Heapify(): Formal ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Analyzing Heapify(): Formal ,[object Object],[object Object],[object Object],[object Object],[object Object]

Heap Operations: BuildHeap() ,[object Object],[object Object],[object Object],[object Object],[object Object]

BuildHeap() ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

BuildHeap() Example ,[object Object],4 1 3 2 16 9 10 14 8 7

Analyzing BuildHeap() ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Analyzing BuildHeap(): Tight ,[object Object],[object Object],[object Object],[object Object],[object Object]

Heapsort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Heapsort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Analyzing Heapsort ,[object Object],[object Object],[object Object]

Priority Queues ,[object Object],[object Object],[object Object],[object Object],[object Object]

Priority Queue Operations ,[object Object],[object Object],[object Object],[object Object]

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