6 8 9 11 13 11 Heap: A Special Kind of Tree 15 26 16 25.

Heap: A Special Kind of Tree

Heap Order PropertyAt every node X, values inboth subtrees (if any)are greater than or equal tothe value in X.

Heap Structure PropertyA heap is also a complete tree. “Complete” means completely filled, except perhaps for the bottom row, which is filled from left to right.

Heap Structure PropertyA heap is also a completetree. “Complete” means completely filled, except perhaps for the bottom row, which is filled from left to right.

This is a binary heap.

This is not a binary heap.

Why not?

Heap orderviolation

Why not?

Bottom rownot filledleft-to-right

Why not?

This is not a binary heap.Interior

row notfilled.

This is a binary heap.

Binary Search Tree PropertyAt every node X, values inleft subtree are smaller thanthe value in X, and values in right subtree are largerthan the value in X.

versus

This is a heap.

versus

This is a heap. It is not abinary searchtree.

versus

This is nota heap.

It is abinary searchtree.

Leftist Heaps: Easily Mergeable Trees

Leftist Structure PropertyFor every node X, the nullpath length of the left childis at least as large as that of the right child.

The null path length,npl(X), of any node X isthe length of the shortestpath from X to a node withzero or one child.

npl(X) = min(npl(left(X)), npl(right(X))) + 1npl() = 1

Leftist Structure Property

npl(left(X)) npl(right(X))

2This is a leftist heap.

Note that itis also abinary heap.

2This is a leftist heap.Suppose we

remove thisnode Note that it

is also abinary heap.

Is the tree stillleftist?

Is it stilla binaryheap?

No,because. . .

No,becauseit is notcomplete.

Complete trees have good balance,but . . .

Complete trees have good balance,but leftist trees can be very unbalanced.

Although leftist heaps can be very unbalanced, they are easy to merge.

As a trivial case,consider two entirelyleft-leaning leftist heaps:

To merge them,. . . 7

To merge them,first make the onewith the largerroot node valuethe rightchildof theother.

In the case shown,the result is alreadya leftist heap,and the mergeis finished.

60If we had started with

. . . then after the firststep the tree is not leftist, since the left child of the root hasnull path length–1, which is less thanthat of theright subtree.

In this case, the mergealgorithm swapsthe left and rightsubtrees of theroot.

In either case, the mergeis done inconstant time!

Remember, though, thesewere special cases(entirely left-leaningtrees).

In general, the mergecan’t be done in constant time,but . . .

In general, the mergecan’t be done in constant time,but it can be donein O(log N) time,where N isthe number ofnodes in thelarger of the twoheaps.

T (H , H ) =

O(max(r , r ))

r = rpl(H ) + 1ii

rThe right path length, rpl(X), of any node X is the length of the pathfollowing right links to a leaf.

rpl(X) = rpl(right(X))) + 1rpl() = 1

Hrpl(H) = 2

Theorem. (Theorem 6.2 in textbook, restated.)If H is a leftist tree, then size(H) 2 – 1, where r = rpl(H) + 1.

Corollary.The number ofnodes on the rightpath of a leftist tree with N nodesis at most log(N + 1).

N = size(H)

N = size(H)Proof of the Corollary:By the theorem, N 2 – 1, thus . . .

N = size(H)Proof of the Corollary:By the theorem, N 2 – 1, thus N + 1 2 and . . .

N = size(H)Proof of the Corollary:By the theorem, N 2 – 1, thus N + 1 2 and log(N + 1) r,

N = size(H)Proof of the Corollary:By the theorem, N 2 – 1, thus N + 1 2 and log(N + 1) r,so log(N + 1) r.

N = size(H)Proof of the Corollary:By the theorem, N 2 – 1, thus N + 1 2 and log(N + 1) r,so log(N + 1) r. QED

Proof of the theorem: By induction on r.

Basis case. r = 1. In this case the tree has at least one node; i.e., size(H) 1 = 2 – 1. 1

Inductive step. Consider a leftist tree H' withr + 1 nodes on its right path.

Inductive step. Consider a leftist tree H' withr + 1 nodes on its right path. (We will showthat size(H') 2 – 1.)r+1

Inductive step. Consider a leftist tree H' withr + 1 nodes on its right path. Then the roothas a right subtree with r nodes on the rightpath, and a left subtree with at least r nodes on its right path — why?

Inductive step. Consider a leftist tree H' withr + 1 nodes on its right path. Then the roothas a right subtree with r nodes on the rightpath, and a left subtree with at least r nodes on its right path (otherwise H' would not be leftist).

Applying the inductive hypothesis, and using thenotation in the diagram, . . .

Applying the inductive hypothesis, and using thenotation in the diagram, size(H ) 2 – 1, size(H) 2 – 1,

Applying the inductive hypothesis, and using thenotation in the diagram, size(H ) 2 – 1, size(H) 2 – 1,and therefore size(H' ) 2 – 1.

Applying the inductive hypothesis, and using thenotation in the diagram, size(H ) 2 – 1, size(H) 2 – 1,and therefore size(H' ) 2 – 1. QED

Merge algorithm, applied to:

Merge algorithm: If either of the two heaps is empty, we can return the other.

Merge algorithm: In this case, neither is empty.

Merge algorithm: In this case, neither is empty, so we compare their roots. . .

Merge algorithm: . . . and recursively merge the heap with the largerroot . . .

Merge algorithm: . . . with the right subheap of the other heap.

Merge algorithm: . . . producing:

Merge algorithm: . . . The resulting tree satisfies the heap order property . . .

Merge algorithm: . . . but is it leftist? . . .

Merge algorithm: . . . but is it leftist? No, the left subtree has a null path length of 1 while the right subtree has a null path length of 2.

Merge algorithm: . . . so the algorithm swaps the left subtree with the right . . .

Merge algorithm: . . . producing:

Merge algorithm: . . . The result is then a leftist heap, and the algorithm terminates. 3

T (H , H ) =

O(max(r , r ))

r = rpl(H ) + 1ii

Again,

This can be proved by induction on r = max(r , r )1 2

T (H , H ) =

O(log (N + 1))

By the theorem and its corollary,

N = max(N , N )

What about insert and delete_min?

insert( , ) = ?, v

insert( , v ) =

merge( , ) v

A one element heap

insert( , v ) =

merge( , ) v

A one element heap

T (H, v) = O(log(N + 1)), where N = size(H).insert

delete_min( ) = ?

delete_min( ) = merge(

T (H) = ? delete_min

6 8 9 11 13 11 Heap: A Special Kind of Tree 15 26 16 25.

Documents

Transcript of 6 8 9 11 13 11 Heap: A Special Kind of Tree 15 26 16 25.

Franklin University 1 COMP 620 Algorithm Analysis Module 3: Advanced Data Structures Trees, Heap, Binomial Heap, Fibonacci Heap.

CS20L-Lecture Set 11 - Binary Heap

CSE 4080 PROJECT: SKEW HEAP, LEFTIST HEAP AND ALTERNATIVESaaw/Pourhashemi/Paper/Paper.pdf · 2007-05-02 · CSE 4080 PROJECT: SKEW HEAP, LEFTIST HEAP AND ALTERNATIVES Winter Term

Heap Spray

Practical Windows XP SP3 / 2003 Heap Exploitationillmatics.com › XP2003_Exploitation_Slides.pdf · Practical Windows XP SP3 / 2003 Heap Exploitation ... heap – / } }

ad-heap: an Efficient Heap Data Structure for Asymmetric Multicore ...

CMSC 341 Binomial Queues and Fibonacci Heaps. Basic Heap Operations OpBinary Heap Leftist Heap Binomial Queue Fibonacci Heap insertO(lgN) deleteMinO(lgN)

binary heap, d-ary heap, binomial heap, amortized analysis ...

Heap Leaching

Procedures on Heap - WordPress.com … · 28/06/2016 · Heap Sort Algorithm The heap sort algorithm starts by using procedure BUILD-HEAP to build a heap on the input array A[1 .

On heap cache vs off-heap cache

skew heap, leftist heap - Department of Computer Science and

11. Heap Sort

HEAP Chart

Windows 8 Heap Internals final - Illmatics.comillmatics.com/Windows 8 Heap Internals (Slides).pdf · Windows 8 Heap Internals Why • Learn how the Heap Manager and Kernel Pool Allocator

Oracle Enterprise Manager 10g · The link Show Heap leads you to the Heap summary Page which breaks down the JVM heap usage by heap space and object type. The link Dump Heap to File

Volatile Use of Persistent Memory - pmem.io · Creating a Fixed Size Heap ... Creating a Variable Size Heap When PMEM_MAX_SIZE is set to zero, ... retrieve the kind of memory pool

Imogen Heap

Windows Heap Overflows - orkspace.net · Windows Heap Overflows Introduction This presentation will examine how to exploit heap based buffer overflows on the Windows platform. Heap

1 Chapter 6 Heapsort. 2 About this lecture Introduce Heap Shape Property and Heap Property Heap Operations Heapsort: Use Heap to Sort Fixing heap.