MIT OpenCourseWare http ocw mit edu 6 006 Introduction to Algorithms Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Lecture 9 Sorting II Heaps Lecture 9 Sorting II Heaps Lecture Overview Review Heaps and MAX HEAPIFY Building a Heap Heap Sort Priority Queues Recitation Readings CLRS 6 1 6 4 Review Heaps Parent i i 2 Left i 2i Right i 2i 1 Max heap property A Parent i A i MAX HEAPIFY A 2 heap size A 10 A 2 A 4 MAX HEAPIFY A 4 A 4 A 9 1 6 006 Spring 2008 Lecture 9 Sorting II Heaps 6 006 Spring 2008 1 16 2 4 8 10 4 6 14 9 2 3 5 8 7 1 7 3 9 10 Violation 1 2 3 4 5 6 16 4 10 14 7 9 7 8 9 10 3 2 8 1 etc O lg n time Figure 1 Review from last lecture Building a Heap A 1 n converted to a max heap Observation Elements A n 2 1 n are all leaves of the tree and can t have children BUILD MAX HEAP A heap size A length A O n times for i length A 2 downto 1 O lg n time do MAX HEAPIFY A i O n lg n overall See Figure 2 for an example 2 Lecture 9 Sorting II Heaps 1 A 4 2 4 8 2 3 1 5 9 14 3 8 6 16 7 7 10 1 MAX HEAPIFY A 3 Swap A 3 and A 7 4 2 4 8 14 9 2 3 3 1 5 8 6 16 7 7 10 9 10 1 MAX HEAPIFY A 2 Swap A 2 and A 5 Swap A 5 and A 10 4 2 4 8 14 9 2 3 10 1 5 8 6 16 7 4 1 3 2 16 9 10 14 8 7 MAX HEAPIFY A 5 no change MAX HEAPIFY A 4 Swap A 4 and A 8 10 9 7 3 9 10 1 4 3 2 8 2 14 9 5 8 MAX HEAPIFY A 1 Swap A 1 with A 2 Swap A 2 with A 4 Swap A 4 with A 9 10 16 4 7 1 6 006 Spring 2008 6 7 9 3 10 Figure 2 Example Building Heaps 3 Lecture 9 Sorting II Heaps 6 006 Spring 2008 Sorting Strategy Build max heap from unordered array Find maximum element A 1 Put it in correct position A n A n goes to A 1 New root could violate max heap property but children remain max heaps Discard node n from heap decrement heapsize Heap Sort Algorithm O n lg n BUILD MAX HEAP A n times for i length A downto 2 do exchange A 1 A i heap size A heap size A 1 O lg n MAX HEAPIFY A 1 O n lg n overall See Figure 3 for an illustration 4 Lecture 9 Sorting II Heaps 6 006 Spring 2008 1 1 16 3 2 10 14 4 8 6 8 5 9 2 4 7 1 3 10 14 3 4 8 10 1 2 7 9 heap size 9 MAX HEAPIFY A 1 6 8 5 9 2 4 7 7 3 9 10 16 not part of heap Note cannot run MAX HEAPIFY with heapsize of 10 1 1 14 2 4 8 4 2 10 8 5 9 1 3 6 7 2 7 3 9 4 8 1 3 10 8 6 4 5 7 9 2 14 7 3 9 10 not part of heap 16 MAX HEAPIFY A 1 1 1 10 2 4 8 4 2 3 9 8 5 7 6 1 2 7 3 4 4 2 3 5 8 10 9 8 7 6 9 10 14 16 1 MAX HEAPIFY A 1 and so on Figure 3 Illustration Heap Sort Algorithm 5 7 3 not part of heap Lecture 9 Sorting II Heaps 6 006 Spring 2008 Priority Queues This is an abstract datatype as it can be implemented in di erent ways INSERT S X MAXIMUM S EXTRACT MAX S INCREASE KEY S x k inserts X into set S returns element of S with largest key removes and returns element with largest key increases the value of element x s key to new value k assumed to be as large as current value 6
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