# 05 - QuickSort and Others Data Structures and Algorithms...

This preview shows pages 1–7. Sign up to view the full content.

QuickSort and Others Data Structures and Algorithms Andrei Bulatov

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Algorithms – Quicksort 5-2 Heap Property A heap is a nearly complete binary tree, satisfying an extra condition Let Parent(i) denote the parent of the vertex i Max-Heap Property : Key(Parent(i)) Key(i) for all i Min-Heap Property : Key(Parent(i)) Key(i) for all i 1 4 2 3 9 7 8 14 10 16
Algorithms – Quicksort 5-3 Insertion Insert(H,key) set n:=length(n), set H[n+1]:=key HeapifyUp(H,n+1) HeapifyUp(H,i) H is almost heap with H[i] too big if decreasing H[i] by a certain amount turns H into a heap if i>1 then set j:=parent(i)= i/2 if Key[H[i]]>Key[H[j]] then swap array entries H[i] and H[j] HeapifyUp(H,j) endif endif

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Algorithms – Quicksort 5-4 Deletion Delete(H,i) set n:=length(n), set H[i]:=H[n] if Key[H[i]]>Key[H[parent(i)]] then HeapifyUp(H,i) endif if Key[H[i]]<Key[H[leftChild(i)]] or Key[H[i]]<Key[H[rightChild(i)]] then HeapifyDown(H,i) endif H is almost heap with H[i] too small if increasing H[i] by a certain amount turns H into a heap
Algorithms – Quicksort 5-5 Deletion (cntd) HeapifyDown(H,i) set n:=length(H) if 2i>n then Terminate with H unchanged else if 2i<n then do set left:=2i, right:=2i+1 let j be the index that minimizes Key[H[left]] and Key[H[right]] else if 2i=n then set j:=2i endif if Key[H[j]]>Key[H[i]] then swap array entries H[i] and H[j] HeapifyDown(H,j) endif

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Algorithms – Quicksort 5-6 HeapifyDown: Soundness Theorem The procedure HeapifyDown(H,i) fixes the heap property in O(log i) time, assuming that the array H is almost a heap with the key of H[i] too small.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern