Datastructuren

DatastructurenData Structures

Fenia AivaloglouHendrik Jan Hoogeboom

Informatica – LIACSUniversiteit Leiden

najaar 2019

Datastructuren

Priority Queues

Contents

5 Priority QueuesADT Priority QueueBinary HeapLeftist heapsPairing Heap (niet)Double-ended Priority Queues

Datastructuren

Priority Queues

Abstract Data Structures

ADT – what, not how

Definition

An abstract data structure (ADT) is a specification of the valuesstored in the data structure as well as the description andsignatures of the operations that can be performed.

no representation or implementation in ADT“mathematical model”

Datastructuren

Priority Queues

Abstract Data Structures

stl container classes

helper: pair

sequences: contiguous: array (fixed length),vector (flexible length), deque (double ended),linked: forward list (single), list (double)

adaptors: based on one of the sequences:stack (lifo), queue (fifo),based on binary heap: priority queue

associative: based on balanced trees:set, map, multiset, multimap

unordered: based on hash table:unordered set, unordered map,unordered multiset,unordered multimap

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Priority Queues

Abstract Data Structures

STL priority queueclass Comp {

public:

int operator() ( const paar& p1, const paar& p2 ) {

return p1.second < p2.second;

}

};

int main() {

vector <paar> club // ’modern’ initialization

{ {"Jan", 1}, {"Piet", 6}, {"Katrien", 5}, {"Ramon", 2} };

using pqtype = priority_queue< paar, vector <paar>, Comp > ;

pqtype pq (club.begin(), club.end() );

// wow! converts into priority_queue

while ( !pq. empty() ) {

cout << pq.top().first << " (" << pq.top().second << ") ";

pq.pop();

}

return 0;

}

Piet (6) Katrien (5) Ramon (2) Jan (1)

Datastructuren

Priority Queues

ADT Priority Queue

dictionary vs. priority queue

Both store a set of (key,value) pairs

{ (’Detra’,17), (’Nova’,84), (’Charlie’,22), (’Henry’,75), (’Elsa’,29) }

both:Insert(’Roxanne’,29)

dictionary:Delete(’Detra’)Find(’Elsa’) returns 29Set(’Henry’,76)

priority queue:FindMax() returns (’Nova’,84)DeleteMax()

Datastructuren

Priority Queues

ADT Priority Queue

ADT dictionary / map / associative array

Stores a set of (key,value) pairs

Initialize, IsEmpty, Size

Insert: add (key,value) pair, provided key is not yet present

Delete: deletes (key,value) pair, given the key

Find: returns the value associated to a given key

Set: reassigns a new value to a (existing) given key

usually implemented as (balanced) binary serach tree,or hash table “unordered”

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Priority Queues

ADT Priority Queue

ADT priority queue

Initialize: construct an empty queue.

IsEmpty: check whether there are any elements in the queue.

Size: returns the number of elements.

Insert: given a data element with its priority, it is added tothe queue

DeleteMax: returns a data element with maximal priority,and deletes it.

GetMax: returns a data element with maximal priority.

IncreaseKey: given an element with its position in thequeue it is assigned a higher priority.

Meld, or Union: takes two priority queues and returns anew priority queue containing the data elements from both.

Datastructuren

Priority Queues

ADT Priority Queue

min & max queues

max-queue ≥

Initialize, IsEmpty, Size, Insert, DeleteMax, GetMax,IncreaseKey, Meld

min-queue ≤

Initialize, IsEmpty, Size, Insert, DeleteMin, GetMin,DecreaseKey, Meld

even opletten welke ordening

er staat vaak ook data (niet alleen prioriteit)

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Priority Queues

ADT Priority Queue

priority queue - use cases

sorting (heapsort)

graph algorithms (Dijkstra shortest path, Prim’s algorithm)

compression (Huffman)

operating systems: task queue, print job queue

discrete event simulation

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Priority Queues

ADT Priority Queue

implementations

Binary Leftist Pairing Fibonacci Brodal

GetMax Θ(1) Θ(1) Θ(1) Θ(1) Θ(1)Insert O(log n) Θ(log n) Θ(1) Θ(1) Θ(1)DeleteMax Θ(log n) Θ(log n) O(log n)† O(log n)† O(log n)

IncreaseKey Θ(log n) Θ(log n) O(log n)† Θ(1)† Θ(1)Meld Θ(n) Θ(log n) Θ(1) Θ(1) Θ(1)† amortized complexity

“. . . is based on heap ordered trees where [. . . ] nodes may violateheap order.” “The data structure presented is quite complicated.”

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Priority Queues

Binary Heap

binary search tree vs heap order

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Priority Queues

Binary Heap

representing binary tree with an array

root at index 1, left/right child i at index 2i/2i+1.

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10001001101010111100

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works well for complete binary treeswaste of space when ‘missing’ nodes

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Priority Queues

Binary Heap

binary heap: three levels

functioning: abstract (priority queue)

understanding: binary tree

implementation: array

internal operations (change key at position):bubble up, trickle down

“To add an element to a heap we must perform an up-heap operation(also known as bubble-up, percolate-up, sift-up, trickle-up, swim-up,heapify-up, or cascade-up), . . . ” What’s in a name? [Wikipedia]

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Priority Queues

Binary Heap

increasekey / bubble up

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BubbleUp : swap with parent until heap-ordered

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Priority Queues

Binary Heap

decreasekey / trickle down

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TrickleDown : swap with largest child until heap-ordered

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Priority Queues

Binary Heap

Insert to priority queue

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Insert: add as last, BubbleUp

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Priority Queues

Binary Heap

DeleteMax from priority queue

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DeleteMax: move last element to root, trickleDown

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Priority Queues

Binary Heap

heapify (1)

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TrickleDown new key: swap with parent until heap-ordered

Datastructuren

Priority Queues

Binary Heap

heapify (2)

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Priority Queues

Binary Heap

heapify (3)

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Priority Queues

Binary Heap

complexity heapify

Lemma∑hd=0 d2d = (h− 1)2h+1 + 2

n levels, N = 2n − 1 keys

top-down∑n−1`=0 2`` = (n− 2)2n = N lgN (ongeveer)

bottom-up∑n−1`=0 2`(n− 1− `) =

∑n−1`=0 2`(n− 1) +

∑n−1`=0 2`` = 2n − n− 1

which is O(N)

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Priority Queues

Leftist heaps

leftist heaps

“bladafstand”npl(x) nil path length, shortest distance to external leaf

Definition (Leftist tree)

An (extended) binary tree where for each internal node x,npl(left(x)) ≥ npl(right(x)).

Definition (Leftist heap)

A leftist tree where the priorities satisfy the heap order.

structure vs. node order

Datastructuren

Priority Queues

Leftist heaps

leftist tree (structure)

npl(left(x)) ≥ npl(right(x))

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Priority Queues

Leftist heaps

basic (internal) operation: ZIP

a b

T1 T2 T3 T4

︷ ︸︸ ︷Zipa

bT1

T2

T3 T4

︷ ︸︸ ︷Zipa ≥ b

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Priority Queues

Leftist heaps

example (step 1: recursive Zipping)

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Priority Queues

Leftist heaps

example (step 2: bottom-up swapping)

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Priority Queues

Leftist heaps

complexity

Lemma

Let T be a leftist tree with root v such that npl(v) = k, then(1) T contains at least 2k − 1 (internal) nodes, and(2) the rightmost path in T has exactly k (internal) nodes.

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2 2

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. . . . . .

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Priority Queues

Leftist heaps

priority queue operations: Insert

Zip︷ ︸︸ ︷38

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Priority Queues

Leftist heaps

priority queue operations: DeleteMax

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Priority Queues

Double-ended Priority Queues

dual structure min-max heap

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Pointer from min-heap item to same item in max-heap

Insertion: as in ordinary heap, but twice: once in each heap

Deletion: find item to delete in other heap using pointer,move last element to that position and do normal deletion

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Priority Queues

Double-ended Priority Queues

interval heap

2-92

8-80 11-75

17-69 42-70 44-73 14-39

24-33 23-65 55-60 44-50 54-57 61

[8,80] ⊆ [2,92]

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Priority Queues

Double-ended Priority Queues

interval heap: insert

2-92

11-75

44-73 14-39

54-57 6180

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44-73 14-39

54-57 61-80

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11-80

44-75 14-39

54-57 61-73

Datastructuren

Priority Queues

Double-ended Priority Queues

embedded min&max heap

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Priority Queues

Double-ended Priority Queues

Double ended priority queue - use case

wikipedia

One example application of the double-ended priority queue isexternal sorting. In an external sort, there are more elementsthan can be held in the computer’s memory.

Datastructuren

Priority Queues

Double-ended Priority Queues

end.