Chapter 6Heaps

Introduction

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some systems applications require that items be processedin specialized ways

printingmay not be best to place on a queuesome jobs may be more ________________small 1-page jobs should be printed before a 100-pagejob

operating system _________________processes run only for a slice of timequeue: FIFO

some jobs may take too longother jobs are more important and should not wait

Heap Model

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specialized queue requiredheap (priority queue)provides at least

_______________deleteMin: finds, returns, and removes min (or max)other operations common

used in other applicationsexternal sorting

algorithmsdiscrete event simulation

Heap Implementation

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heaps can be implemented with, with insertions at head

insertdeleteMin

ordered linked list (worse due to number of )insertdeleteMin

binary search treeinsertdeleteMinsince only min is deleted, tree will be _______________overkill since other included operations not required

Binary Heap

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binary heap commonsimply termed heap

two properties________________heap _____________

operations can one of the propertiesoperation must continue until heap properties have beenrestored, which is typically simple

Binary Heap

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structure propertycompletely filled

except level, which is filled from left to rightcomplete binary tree has between and nodesheight:

can be represented with an array (no necessary)array position

left child inright child inparent in

operations simplemaximum heap size must be ______________________

Binary Heap

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Binary Heap

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heap-order propertyallows operations to be performed quicklywant to be able to find minimum quickly

smallest element at ___________any should also be a heap

any node should be than all its descendants

Heap Operations

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two main heap operationsinsertdeleteMin

insertneed to maintain structure propertycreate in next available location (at bottom of tree)place new value there if possibleotherwise, slide into hole and bubble up holecontinue until new value can be insertedtermed strategy

Binary Heap

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example: insert 14

Heap Operations

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insert (cont.)could have used repeated swaps, but a swap requires_____________ assignmentsif element percolated up levels

swap method: assignmentsnon-swap method: assignments

if new element is smaller than all others in heap, hole willpercolate to the __________hole will be at index 1 and we will break out of the loop

if extra check for 1 in loop, adds unnecessary timecould place a copy of new value in position ____

Heap Operations

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insert (cont.)could require as much ason average, percolation terminates ___________

on average 2.607 comparisons are required

Heap Operations

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deleteMinfinding minimum easy

minimum more difficulthole is created at root

element in complete binary tree must moveif last element can be placed in hole, done

child into holehole slides down one levelrepeat until last element can be placed in hole

worst case:average case:

Binary Heap

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example: deleteMin (13)

Binary Heap

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example: deleteMin (13) (cont.)

Heap Operations

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other heap operationsfinding minimum ___________finding maximum not possible without scanthrough entire heap

maximum in one of the leavescould use separate data structure, such as _____________

Heap Operations

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other heap operations (cont.)decreaseKey

lowers value at position by given amountpercolate up

example: change process for more run timeincreaseKey

increases value at position by given amountpercolate down

example: drop process priority if taking ______________remove

decreaseKey to root, then deleteMinexample: process is by user

Heap Operations

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other heap operations (cont.)buildHeap

place items into the tree in any ____________maintains propertyuse percolateDown( ) from node

Binary Heap

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example: start with random placementstart with percolateDown(7)each dashed line represents 2 comparisons

Binary Heap

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example: (cont.)

Heap Applications

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selection problemfrom a list of elements, find th largestoriginal algorithm

list and index th elementwith simple sort,

alternative algorithmread ____ elements into array and sort themsmallest is in th positionother elements processed one by one, placing them intothe arrayrunning time

if ,

Heap Applications

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selection problem (cont.)if heap is used

build heap of elementsperform deleteMin operations

element extracted is th smallest elementif ,

another algorithm using heapas in previous algorithm, but put elements in heap

is in th positionother elements processed one by one, placing them intothe arrayrunning time

Heap Applications

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event simulationbank where customers arrive and wait until one of tellersis available

customer and basedon probability distribution functioncompute statistics on the length of time a customer mustwait, or the length of the lineneed to consider event that will occur in the least amountof time

heap can be used to order events

-Heaps

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-heapsame as heap, but all nodes have childrentends to be more than binary heaprunning time reduced todeleteMin more expensive since more _______________requireduseful when heap is too large to fit entirely into mainmemory4-heaps may outperform 2-heaps (binary heaps)

-Heaps

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example: -heap with

Leftist Heaps

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one weakness of heaps so far is two heapsis difficultthree data structures that can help

heapsheaps

binomial queues

Leftist Heaps

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leftist heapscan be difficult to design a data structure for ____________that uses an array, but runs efficiently

linked data structure therefore requiredleftist heap

structure and ordering properties of binary heapsdifference is that heap is not ______________________

very unbalanced is desired

Leftist Heaps

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leftist heaps (cont.)null path length (npl): length of shortest path from currentnode to a node without children

npl of a node with zero or one child is 0npl of a null pointer is -1

npl of each node is 1 more than the minimum of the nullpath lengths of its ________________leftist heap property

npl of the left child is at least as large as that of the rightchildbiases tree to left subtree

Leftist Heaps

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example: leftist heap

Leftist Heaps

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leftist heaps operationsall work should be done on the right path, which isguaranteed to be short

and may destroy the leftistheap property

not difficult to fix

Leftist Heaps

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leftist heaps operations (cont.)merging

insertion special case of merging a 1-node heap with alarger heapif either of the two heaps is , return the otherheapotherwise, compare __________

merge heap with the larger root withthe right subheap of the heap with the smaller root

Leftist Heaps

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leftist heaps operations (cont.)merging example

Leftist Heaps

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leftist heaps operations (cont.)merging example

Leftist Heaps

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leftist heaps operations (cont.)merging example

result is not leftist: left npl = 1, right npl = 2

Leftist Heaps

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leftist heaps operations (cont.)merging example

fix by swapping children

Skew Heaps

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self-adjusting version of leftist heaprelationship of skew heap to leftist heap is analogous to splaytrees and treesskew heaps

binary trees with heap orderbut no structural constraint

no information kept about null path lengthright path can be arbitrarily _________

worst case of all operations:for operations, total worst case: , or__________________

Skew Heaps

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skew heaps (cont.)fundamental operation is mergingafter merging, for leftists heaps, check both children forstructure and swap children if neededin skew heaps, swap children

except nodes on right paths do not swapchildrenno extra required to maintain path lengthsno tests required to determine when to swap children

Skew Heaps

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skew heaps examplemerge two skew heaps

tree with larger root will merge onto tree with smaller root

Skew Heaps

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skew heaps example (cont.)recursively merge with the subheap of rooted at 8

heap happens to be ______________

Skew Heaps

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skew heaps example (cont.)make this heap the new left child of and the old left childof becomes the new right child

Binomial Queues

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binomial queueskeep a collection of heap-ordered trees, known as a____________at most one binomial tree of every ____________

heap order imposed on each binomial treecan represent any priority queueexample: a priority queue of size 13 can be representedby the forest or 1101

worst case of all operations:

Binomial Queues

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binomial queues example

Binomial Queues

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binomial queue of size 6 example

Binomial Queues

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binomial queue operationsminimum element found by scanning of all trees

found incan keep ongoing information to reduce to

merging two queuesmerge takes

Binomial Queues

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merge example

merge of two trees

Binomial Queues

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merge example (cont.)now we have three binomial trees of height 3; keep oneand merge the other two

Binomial Queues

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merge values 1 7 exampleinsert 1

insert 2

insert 3

insert 4

Binomial Queues

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merge values 1 7 example (cont.)insert 5

insert 6

insert 7

Binomial Queues

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deleteMin example

separate tree with minimum root from rest of tree

remaining trees after removing min

Binomial Queues

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deleteMin example (cont.)after merge