GAP (datastructures) - Chapter 3: Heaps

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3 Heaps
3.1 Introduction
3.2 API
3.2-1 IsHeap
3.2-2 Heap
3.2-3 NewHeap
3.2-4 Push
3.2-5 Peek
3.2-6 Pop
3.2-7 Merge
3.3 Binary Heaps
3.3-1 BinaryHeap
3.4 Pairing Heaps
3.4-1 PairingHeap
3.5 Declarations
3.5-1 IsBinaryHeapFlatRep
3.6 Implementation
3.6-1 IsPairingHeapFlatRep

### 3 Heaps

#### 3.1 Introduction

A heap is a tree datastructure such that for any child C of a node N it holds that C \leq N, according to some ordering relation \leq.

The fundamental operations for heaps are Construction, `Push`ing data onto the heap, `Peek`ing at the topmost item, and `Pop`ping the topmost item off of the heap.

For a good heap implementation these basic operations should not exceed O(\log n) in runtime where n is the number of items on the heap.

We currently provide two types of heaps: Binary Heaps 3.3 and Pairing Heaps 3.4.

The following code shows how to use a binary heap.

    gap> h := BinaryHeap();
    <binary heap with 0 entries>
    gap> Push(h, 5);
    gap> Push(h, -10);
    gap> Peek(h);
    5
    gap> Pop(h);
    5
    gap> Peek(h);
    -10

The following code shows how to use a pairing heap.

    gap> h := PairingHeap( {x,y} -> x.rank > y.rank );
    <pairing heap with 0 entries>
    gap> Push(h, rec( rank  := 5 ));
    gap> Push(h, rec( rank  := 7 ));
    gap> Push(h, rec( rank  := -15 ));
    gap> h;
    <pairing heap with 3 entries>
    gap> Peek(h);
    rec( rank := -15 )
    gap> Pop(h);
    rec( rank := -15 )

#### 3.2 API

For the purposes of the datastructures, we provide a category `IsHeap` (???) . Every implementation of a heap in the category `IsHeap` (???) must follow the API described in this section.

##### 3.2-1 IsHeap

`‣ IsHeap`( arg ) | ( filter )

Returns: `true` or `false`

The category of heaps. Every object in this category promises to support the API described in this section.

##### 3.2-2 Heap

`‣ Heap`( arg ) | ( function )

Wrapper function around constructors

##### 3.2-3 NewHeap

`‣ NewHeap`( [filter, func, data] ) | ( constructor )

Returns: a heap

Construct a new heap

##### 3.2-4 Push

`‣ Push`( heap, object ) | ( operation )

Puts the object object a new object onto heap.

##### 3.2-5 Peek

`‣ Peek`( heap ) | ( operation )

Inspect the item at the top of heap.

##### 3.2-6 Pop

`‣ Pop`( heap ) | ( operation )

Returns: an object

Remove the top item from heap and return it.

##### 3.2-7 Merge

`‣ Merge`( heap1, heap2 ) | ( operation )

Merge two heaps (of the same type)

Heaps also support `IsEmpty` (Reference: IsEmpty) and `Size` (Reference: Size)

#### 3.3 Binary Heaps

A binary heap employs a binary tree as its underlying tree datastructure. The implemenataion of binary heaps in datastructures stores this tree in a flat array which makes it a very good and fast default choice for general purpose use. In particular, even though other heap implementations have better theoretical runtime bounds, well-tuned binary heaps outperform them in many applications.

For some reference see http://stackoverflow.com/questions/6531543

##### 3.3-1 BinaryHeap

`‣ BinaryHeap`( [isLess[, data]] ) | ( function )

Returns: A binary heap

Constructor for binary heaps. The optinal argument isLess must be a binary function that performs comparison between two elements on the heap, and returns `true` if the first argument is less than the second, and `false` otherwise. Using the optional argument data the user can give a collection of initial values that are pushed on the stack after construction.

#### 3.4 Pairing Heaps

A pairing heap is a heap datastructure with a very simple implementation in terms of GAP lists. `Push` and `Peek` have O(1) complexity, and `Pop` has an amortized amortised O(log n), where n is the number of items on the heap.

For a reference see [FSST86].

##### 3.4-1 PairingHeap

`‣ PairingHeap`( [isLess[, data]] ) | ( function )

Returns: A pairing heap

Constructor for pairing heaps. The optional argument isLess must be a binary function that performs comparison between two elements on the heap, and returns `true` if the first argument is less than the second, and `false` otherwise. Using the optional argument data the user can give a collection of initial values that are pushed on the stack after construction.

#### 3.5 Declarations

##### 3.5-1 IsBinaryHeapFlatRep

`‣ IsBinaryHeapFlatRep`( arg ) | ( filter )

Returns: `true` or `false`

#### 3.6 Implementation

##### 3.6-1 IsPairingHeapFlatRep

`‣ IsPairingHeapFlatRep`( arg ) | ( filter )

Returns: `true` or `false`

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