# The binomial-heap Reference Manual

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# The binomial-heap Reference Manual

This is the binomial-heap Reference Manual, generated automatically by Declt version 2.4 "Will Decker" on Wed Jun 20 10:49:02 2018 GMT+0.

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## 1 Introduction

``````___  __ __   ____ _    __ ___  __        _  _ ____ ___  ____
| .\ |_\| \|\|   ||\/\ |_\|  \ | |   ___ ||_|\| __\|  \ | . \
| .<_| /|  \|| . ||   \| /| . \| |__|___\| _ ||  ]_| . \| __/
|___/|/ |/\_/|___/|/v\/|/ |/\_/|___/     |/ |/|___/|/\_/|/
``````

# Abstract

Binomial-heap is a compact and succint implementation of the binomial heap data structure in Common Lisp programming language. Insertion, extremum access, extremum extraction, and union operations are performed in O(logn) time.

# Demo

``````(defvar *list* (loop repeat 20 collect (random 100)))
; => (25 50 12 53 53 55 41 71 71 41 33 8 71 57 28 4 89 96 58 25)

(defvar *heap* (make-instance 'bh:binomial-heap :test #'<))
; => #<BINOMIAL-HEAP {1002C4EC31}>

(dolist (item *list*)
(insert-key *heap* item))
; => NIL

; => -> ( 2) 25
;      -> ( 1) 89
;        -> ( 0) 96
;      -> ( 0) 58
;    -> ( 4) 4
;      -> ( 3) 12
;        -> ( 2) 41
;          -> ( 1) 53
;            -> ( 0) 55
;          -> ( 0) 71
;        -> ( 1) 25
;          -> ( 0) 50
;        -> ( 0) 53
;      -> ( 2) 8
;        -> ( 1) 41
;          -> ( 0) 71
;        -> ( 0) 33
;      -> ( 1) 57
;        -> ( 0) 71
;      -> ( 0) 28
;    NIL

(bh:get-extremum-key *heap*)
; => 4

(loop for x in (sort (copy-list *list*) (test-of *heap*))
for y = (extract-extremum-key *heap*)
unless (= x y)
collect (cons x y))
; => NIL

(let ((h1 (make-instance 'bh:binomial-heap :test #'string<))
(h2 (make-instance 'bh:binomial-heap :test #'string<))
(l1 '("foo" "bar" "baz" "mov" "mov"))
(l2 '("i" "see" "dead" "binomial" "trees")))
(dolist (l l1) (bh:insert-key h1 l))
; => -> ( 0) "mov"
;    -> ( 2) "bar"
;      -> ( 1) "baz"
;        -> ( 0) "mov"
;      -> ( 0) "foo"
; NIL
(dolist (l l2) (bh:insert-key h2 l))
; => -> ( 0) "trees"
;    -> ( 2) "binomial"
;      -> ( 1) "i"
;        -> ( 0) "see"
; NIL
(let ((h3 (bh:unite-heaps h1 h2)))
; => -> ( 1) "mov"
;      -> ( 0) "trees"
;    -> ( 3) "bar"
;      -> ( 2) "binomial"
;        -> ( 1) "i"
;          -> ( 0) "see"
;      -> ( 1) "baz"
;        -> ( 0) "mov"
;      -> ( 0) "foo"
; NIL
``````

# Caveats

Despite binomial heaps are known to perform decrease/increase key and delete operations in O(logn) time, this is practically not that easy to implement. (For the rest of this talk, I'll skip the deletion operation because of it can be achieved through setting the key field of a node to the absolute extremum -- i.e. negative infinity -- and extracting the extremum.) Consider below example.

``````--> [ Z ] -->
^
|
|
--> [ X ] -->
^^^
|||
||+----------------------------+
|+----------+        ...       |
|           |                  |
--> [ W0 ] --> [ W1 ] --> ... --> [ WN ]
``````

Suppose you decreased the key field of X and you need to bubble up X by swapping nodes in upwards direction appropriately. Because of random access is not possible in heap data structures, you need to figure out your own way of accessing to nodes -- in this example consider you have the pointers in advance to the every `BINOMIAL-TREE` in the `BINOMIAL-HEAP`. There are two ways to swap nodes:

## Swapping Key Fields

If you just swap the key fields of the nodes

``````(rotatef (key-of x) (key-of z))
``````

everything will be fine, except that the pointers to the nodes that lost their original key fields will get invalidated. Now you cannot guarantee the validity of your node pointers and hence cannot issue any more decrease key operations.

## Swapping Node Instances

If you swap the two node instances, your pointers won't get invalidated but this time you'll need to update the sibling and parent pointers as well,

``````(setf (parent-of w0) z
(parent-of w1) z
...
(parent-of wn) z)
``````

which will make your O(logn) complexity dreams fade away. (Moreover, you'll need to traverse sibling lists at levels of nodes `X` and `Y` to be able to find previous siblings to `X` and `Y` if you are not using doubly-linked-lists. But even this scheme doesn't save us from the traversal of `W1`, ..., `WN` nodes.)

## Solution

So how can we manage to perform decrease key operation in O(logn) time without invalidating any node pointers? The solution I come up with to this problem is as follows.

We can keep a separate hash table for the pointers to the nodes. When a node's key field gets modified, related hash table entry will get modified as well. And instead of returning to the user the actual `BINOMIAL-TREE` instances, we'll return to the user the key of the related hash table entry. (Consider this hash table as a mapping between the hash table keys and the pointer to the actual node instance.)

Sounds too hairy? I think so. I'd be appreciated for any sort of enlightenment of a better solution.

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## 2 Systems

The main system appears first, followed by any subsystem dependency.

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### 2.1 binomial-heap

Author

Volkan YAZICI <volkan.yazici@gmail.com>

BSD

Description

A compact binomial heap implementation.

Source

binomial-heap.asd (file)

Component

src (module)

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## 3 Modules

Modules are listed depth-first from the system components tree.

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### 3.1 binomial-heap/src

Parent

binomial-heap (system)

Location

src/

Components

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## 4 Files

Files are sorted by type and then listed depth-first from the systems components trees.

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### 4.1 Lisp

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#### 4.1.1 binomial-heap.asd

Location

binomial-heap.asd

Systems

binomial-heap (system)

Packages

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#### 4.1.2 binomial-heap/src/packages.lisp

Parent

src (module)

Location

src/packages.lisp

Packages

#### 4.1.3 binomial-heap/src/specials.lisp

Dependency

packages.lisp (file)

Parent

src (module)

Location

src/specials.lisp

Exported Definitions
Internal Definitions

#### 4.1.4 binomial-heap/src/utils.lisp

Dependency

specials.lisp (file)

Parent

src (module)

Location

src/utils.lisp

Internal Definitions

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#### 4.1.5 binomial-heap/src/operations.lisp

Dependency

utils.lisp (file)

Parent

src (module)

Location

src/operations.lisp

Exported Definitions
Internal Definitions

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## 5 Packages

Packages are listed by definition order.

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### 5.1 binomial-heap-system

Source

binomial-heap.asd

Use List
• asdf/interface
• common-lisp

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### 5.2 binomial-heap

Source

packages.lisp (file)

Nickname

bh

Use List

common-lisp

Exported Definitions
Internal Definitions

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## 6 Definitions

Definitions are sorted by export status, category, package, and then by lexicographic order.

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### 6.1 Exported definitions

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#### 6.1.1 Functions

Function: extract-extremum-key HEAP

Extracts the extremum value from the ‘BINOMIAL-HEAP’ pointed by ‘HEAP’. Function returns the ‘KEY’ field of the extracted ‘BINOMIAL-TREE’ instance.

Package
Source

operations.lisp (file)

Function: get-extremum-key HEAP

Finds the ‘BINOMIAL-TREE’ with the extremum value and its ‘KEY’ field. Function returns ‘NIL’ in case of no items found.

Package
Source

operations.lisp (file)

Function: insert-key HEAP KEY

Creates a new ‘BINOMIAL-TREE’ for ‘KEY’ and inserts this node to the ‘BINOMIAL-HEAP’ pointed by ‘HEAP’. Function returns the ‘KEY’.

Package
Source

operations.lisp (file)

Function: unite-heaps X Y

Unites given two heaps of type ‘BINOMIAL-HEAP’ into a single one. (Assuming ‘TEST’ functions of each heap are equivalent.)

Package
Source

operations.lisp (file)

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#### 6.1.2 Generic functions

Generic Function: test-of OBJECT
Generic Function: (setf test-of) NEW-VALUE OBJECT
Package
Methods
Method: test-of (BINOMIAL-HEAP binomial-heap)

Source

specials.lisp (file)

Method: (setf test-of) NEW-VALUE (BINOMIAL-HEAP binomial-heap)

automatically generated writer method

Source

specials.lisp (file)

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#### 6.1.3 Classes

Class: binomial-heap ()

Binomial heap container.

Package
Source

specials.lisp (file)

Direct superclasses

standard-object (class)

Direct methods
Direct slots
Type

list

Initargs

Writers

Slot: test
Type

function

Initargs

:test

test-of (generic function)

Writers

(setf test-of) (generic function)

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### 6.2 Internal definitions

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#### 6.2.1 Macros

Macro: prog1-let (VAR VAL) &body BODY
Package
Source

utils.lisp (file)

Macro: when-let (VAR VAL) &body BODY
Package
Source

utils.lisp (file)

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#### 6.2.2 Functions

Finds the ‘BINOMIAL-TREE’ prior to the extremum in the sibling list pointed by ‘HEAD’. Function returns ‘NIL’ in case of no extremum or extremum at the beginning.

Package
Source

operations.lisp (file)

Constructs ‘SIBLING’ slots of given list of ‘BINOMIAL-TREE’s to provide given order.

Package
Source

operations.lisp (file)

Makes ‘X’ the child of ‘Y’.

Package
Source

operations.lisp (file)

Function: merge-siblings X Y

Merges given two ‘BINOMIAL-TREE’s and their related siblings into a single ‘BINOMIAL-TREE’ sibling list.

Package
Source

operations.lisp (file)

Function: print-tree X

Utility function to print binomial tree in a human-readable(?) format.

Package
Source

operations.lisp (file)

Function: sexp->tree SEXP

Converts supplied ‘SEXP’ of ‘(KEY &KEY SIBLING CHILD)’ form into appropriate ‘BINOMIAL-TREE’ instance.

Package
Source

operations.lisp (file)

Function: sibling-list TREE

Returns reversed list of child and its consequent siblings of supplied ‘TREE’ of type ‘BINOMIAL-TREE’.

Package
Source

operations.lisp (file)

Function: tree->sexp TREE

Converts supplied ‘BINOMIAL-TREE’ into ‘(KEY &KEY SIBLING CHILD)’ compound form.

Package
Source

operations.lisp (file)

Function: unite-root-lists TEST X Y

Unites given ‘X’ and ‘Y’ ‘BINOMIAL-TREE’s and their related siblings into a single ‘BINOMIAL-TREE’.

Package
Source

operations.lisp (file)

Function: unsafe-unite-root-lists TEST X Y

Identical to ‘UNITE-ROOT-LISTS’ except that this function doesn’t handle Case 2 condition and break the loop in Case 1. (Case 1 & 2 are redundant while adding a single node to a root list.)

Package
Source

operations.lisp (file)

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#### 6.2.3 Generic functions

Generic Function: child-of OBJECT
Generic Function: (setf child-of) NEW-VALUE OBJECT
Package
Methods
Method: child-of (BINOMIAL-TREE binomial-tree)

Source

specials.lisp (file)

Method: (setf child-of) NEW-VALUE (BINOMIAL-TREE binomial-tree)

automatically generated writer method

Source

specials.lisp (file)

Generic Function: degree-of OBJECT
Generic Function: (setf degree-of) NEW-VALUE OBJECT
Package
Methods
Method: degree-of (BINOMIAL-TREE binomial-tree)

Source

specials.lisp (file)

Method: (setf degree-of) NEW-VALUE (BINOMIAL-TREE binomial-tree)

automatically generated writer method

Source

specials.lisp (file)

Generic Function: (setf head-of) NEW-VALUE OBJECT
Package
Methods

Source

specials.lisp (file)

Method: (setf head-of) NEW-VALUE (BINOMIAL-HEAP binomial-heap)

automatically generated writer method

Source

specials.lisp (file)

Generic Function: key-of OBJECT
Generic Function: (setf key-of) NEW-VALUE OBJECT
Package
Methods
Method: key-of (BINOMIAL-TREE binomial-tree)

Source

specials.lisp (file)

Method: (setf key-of) NEW-VALUE (BINOMIAL-TREE binomial-tree)

automatically generated writer method

Source

specials.lisp (file)

Generic Function: parent-of OBJECT
Generic Function: (setf parent-of) NEW-VALUE OBJECT
Package
Methods
Method: parent-of (BINOMIAL-TREE binomial-tree)

Source

specials.lisp (file)

Method: (setf parent-of) NEW-VALUE (BINOMIAL-TREE binomial-tree)

automatically generated writer method

Source

specials.lisp (file)

Generic Function: sibling-of OBJECT
Generic Function: (setf sibling-of) NEW-VALUE OBJECT
Package
Methods
Method: sibling-of (BINOMIAL-TREE binomial-tree)

Source

specials.lisp (file)

Method: (setf sibling-of) NEW-VALUE (BINOMIAL-TREE binomial-tree)

automatically generated writer method

Source

specials.lisp (file)

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#### 6.2.4 Classes

Class: binomial-tree ()

Binomial tree container.

Package
Source

specials.lisp (file)

Direct superclasses

standard-object (class)

Direct methods
Direct slots
Slot: parent
Type

binomial-heap::binomial-tree

Initargs

:parent

parent-of (generic function)

Writers

(setf parent-of) (generic function)

Slot: degree
Type

(integer 0 *)

Initargs

:degree

Initform

0

degree-of (generic function)

Writers

(setf degree-of) (generic function)

Slot: child
Type

binomial-heap::binomial-tree

Initargs

:child

child-of (generic function)

Writers

(setf child-of) (generic function)

Slot: sibling
Type

binomial-heap::binomial-tree

Initargs

:sibling

sibling-of (generic function)

Writers

(setf sibling-of) (generic function)

Slot: key
Initargs

:key

key-of (generic function)

Writers

(setf key-of) (generic function)

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## Appendix A Indexes

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### A.1 Concepts

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### A.2 Functions

Jump to: (   C   D   E   F   G   H   I   K   L   M   P   S   T   U   W
Jump to: (   C   D   E   F   G   H   I   K   L   M   P   S   T   U   W

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