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A Binary (Max) Heap is a complete binary tree that maintains the Max Heap property.


Binary Heap is one possible data structure to model an efficient Priority Queue (PQ) Abstract Data Type (ADT). In a PQ, each element has a "priority" and an element with higher priority is served before an element with lower priority (ties are broken with standard First-In First-Out (FIFO) rule as with normal Queue). Try clicking ExtractMax() for a sample animation on extracting the max value of random Binary Heap above.


To focus the discussion scope, we design this visualization to show a Binary Max Heap that contains distinct integers only.


Click 'Next' (on the top right)/press 'Page Down' to advance this e-Lecture slide, use the drop down list/press 'Space' to jump to a specific slide, or Click 'X' (on the bottom right)/press 'Esc' to go to Exploration mode.


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Please login if you are a repeated visitor or register for an (optional) free account first.

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Complete Binary Tree: Every level in the binary tree, except possibly the last/lowest level, is completely filled, and all vertices in the last level are as far left as possible.


Binary Max Heap property: The parent of each vertex - except the root - contains value greater than the value of that vertex. This is an easier-to-verify definition than the following alternative definition: The value of a vertex - except the leaf/leaves - must be greater than the value of its one (or two) child(ren).


Pro-tip: Since you are not logged-in, you may be a first time visitor who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown] to advance to the next slide, [PageUp] to go back to the previous slide, [Esc] to toggle between this e-Lecture mode and exploration mode.

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Priority Queue (PQ) Abstract Data Type (ADT) is similar to normal Queue ADT, but with these two major operations:

  1. Enqueue(x): Put a new element (key) x into the PQ (in some order),
  2. y = Dequeue(): Return an existing element y that has the highest priority (key) in the PQ and if ties, return the one that is inserted first, i.e. back to First-In First-Out (FIFO) behavior of a normal Queue

Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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Imagine: You are an Air Traffic Controller (ATC) working in the control tower of an airport.


You have scheduled aircraft X/Y to land in the next 3/6 minutes, respectively. Both have enough fuel for at least the next 15 minutes and both are just 2 minutes away from your airport. You observe that your airport runway is clear at the moment.


In case you do not know, aircraft can be instructed to fly in holding pattern near the airport until the designated landing time.

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You have to attend the live lecture to figure out what happens next...


There will be two options presented to you and you will have to decide:

  1. Raise AND wave your hand if you choose option A,
  2. Raise your hand but do NOT wave it if you choose option B,

If none of the two options is reasonable for you, simply do nothing.

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There are several potential usages of PQ ADT in real-life on top of what you have seen just now (only in live lecture).


Discussion: Can you mention a few other real-life situations where a PQ is needed?

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We are able to implement this PQ ADT using (circular) array or Linked List but we will have slow (i.e. in O(N)) Enqueue or Dequeue operation.



Discussion: Why?

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Now, let's view the visualisation of a (random) Binary (Max) Heap above. You should see a complete binary tree and all vertices except the root satisfy the Max Heap property (A[parent(i)] > A[i] — remember that we disallow duplicate integers here).


Quiz: Based on this Binary (Max) Heap property, where will the largest integer be located?

Can be anywhere
At the root
At one of the leaf
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Important fact to memorize at this point: If we have a Binary Heap of N elements, its height will not be taller than O(log N) since we will store it as a complete binary tree.


Simple analysis: The size N of a full (more than just a complete) binary tree of height h is always N = 2(h+1)-1, therefore h = log2(N+1)-1 ~= log2 N.


See example above with N = 7 = 2(2+1)-1 or h = log2(7+1)-1 = 2.


This fact is important in the analysis of all Binary Heap-related operations.

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A complete binary tree can be stored efficiently as a compact array A as there is no gap between vertices of a complete binary tree/elements of a compact array. To simplify the navigation operations below, we use 1-based array. VisuAlgo displays the index of each vertex as a red label below each vertex. Read those indices in sorted order from 1 to N, then you will see the vertices of the complete binary tree from top to down, left to right.


This way, we can implement basic binary tree traversal operations with simple index manipulations (with help of bit shift manipulation):

  1. parent(i) = i>>1, index i divided by 2 (integer division),
  2. left(i) = i<<1, index i multiplied by 2,
  3. right(i) = (i<<1)+1, index i multiplied by 2 and added by 1.
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In this visualization, you can perform five (5) standard Binary (Max) Heap operations:

  1. Insert(v) in O(log N)
  2. ExtractMax() in O(log N)
  3. Create(A) - O(N log N) version
  4. Create(A) - O(N) version
  5. HeapSort() - in O(N log N)

There are others possible Binary (Max) Heap operations, but currently we do not elaborate them for pedagogical reason in a certain NUS module.

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Insert(v): Insertion of a new item v into a Binary Max Heap can only be done at the last index N plus 1 to maintain the compact array = complete binary tree property. However, the Max Heap property may still be violated. This operation then fixes Max Heap property from the insertion point upwards (if necessary) and stop when there is no more Max Heap property violation. Now try clicking Insert(v) several times to insert a few random v to the currently displayed Binary (Max Heap).


The fix Max Heap property upwards operation has no standard name. We call it ShiftUp but others may call it BubbleUp or IncreaseKey operation.

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Do you understand why starting from the insertion point (index N+1) upwards (at most until the root) and swapping a vertex with its parent when there is a Max Heap property violation during insertion is always a correct strategy?

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The time complexity of this Insert(v) operation is O(log N).



Discussion: Do you understand the derivation?

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ExtractMax(): The reporting and then the deletion of the maximum element (the root) of a Binary Max Heap requires an existing element to replace the root, otherwise the Binary Max Heap (a single complete binary tree, or 林/Lín in Chinese/tree) becomes two disjoint subtrees (two copies of 木/mù in Chinese/wood). That element must be the last index N for the same reason: To maintain the compact array = complete binary tree property.


Because we promote a leaf vertex to the root vertex of a Binary Max Heap, it will very likely violates the Max Heap property. ExtractMax() operation then fixes Binary Max Heap property from the root downwards by comparing the current value with the its child/the larger of its two children (if necessary). Now try ExtractMax() on the currently displayed Binary (Max) Heap.


The fix Max Heap property downwards operation has no standard name. We call it ShiftDown but others may call it BubbleDown or Heapify operation.

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Why if a vertex has two children, we have to check (and possibly swap) that vertex with the larger of its two children during the downwards fix of Max Heap property?


Why can't we just compare with the left (or right, if exists) vertex only?

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The time complexity of this ExtractMax() operation is O(log N).



Discussion: Do you understand the derivation?

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Up to here, we have a data structure that can implement the two major operations of Priority Queue (PQ) ADT efficiently:

  1. For Enqueue(x), we can use Insert(x) in O(log N) time, and
  2. For y = Dequeue(), we can use y = ExtractMax() in O(log N) time.

However, we can do a few more operations with Binary Heap.

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Create(A): Creates a valid Binary (Max) Heap from an input array A of N integers (comma separated) into an initially empty Binary Max Heap.


There are two variants for this operations, one that is simpler but runs in O(N log N) and a more advanced technique that runs in O(N).


Pro tip: Try opening two copies of VisuAlgo on two browser windows. Execute different Create(A) versions on the worst case 'Sorted example' to see the somewhat dramatic differences of the two.

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Create(A) - O(N log N): Simply insert (that is, by calling Insert(v) operation) all N integers of the input array into an initially empty Binary Max Heap one by one.


Analysis: This operation is clearly O(N log N) as we call O(log N) Insert(v) operation N times. Let's examine the 'Sorted example' which is one of the hard case of this operation (Now try the Hard Case - O(N log N) where we show a case where A=[1,2,3,4,5,6,7] -- please be patient as this example will take some time to complete). If we insert values in increasing order into an initially empty Binary Max Heap, then every insertion triggers a path from the insertion point (a new leaf) upwards to the root.

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Create(A) - O(N): This faster version of Create(A) operation was invented by Robert W. Floyd in 1964. It takes advantage of the fact that a compact array = complete binary tree and all leaves (i.e. half of the vertices — see the next slide) are Binary Max Heap by default. This operation then fixes Binary Max Heap property (if necessary) only from the last internal vertex back to the root.


Analysis: A loose analysis gives another O(N/2 log N) = O(N log N) complexity but it is actually just O(2*N) = O(N) — details in the next few slides. Now try the Hard Case - O(N) on the same input array A=[1,2,3,4,5,6,7] and see that on the same hard case as with the previous slide (but not the one that generates maximum number of swaps), this operation is far superior than the O(N log N) version.

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Simple proof on why half of Binary (Max) Heap of N (without loss of generality, let's assume that N is even) elements are leaves are as follows:


Suppose that the last leaf is at index N, then the parent of that last leaf is at index i = N/2 (remember this slide). The left child of vertex i+1, if exists (it actually does not exist), will be 2*(i+1) = 2*(N/2+1) = N+2, which exceeds index N (the last leaf) so index i+1 must also be a leaf vertex that has no child. As Binary Heap indexing is consecutive, basically indices [i+1 = N/2+1, i+2 = N/2+2, ..., N], or half of the vertices, are leaves.

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First, we need to recall that the height of a full binary tree of size N is log2 N.


Second, we need to realise that the cost to run shiftDown(i) operation is not the gross upper bound O(log N), but O(h) where h is the height of the subtree rooted at i.


Third, there are ceil(N/2h+1) vertices at height h in a full binary tree.


On the example full binary tree above with N = 7 and h = 2, there are:
ceil(7/20+1) = 4 vertices: {44,35,26,17} at height h = 0,
ceil(7/21+1) = 2 vertices: {62,53} at height h = 1, and
ceil(7/22+1) = 1 vertex: {71} at height h = 2.

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Cost of Create(A), the O(N) version is thus:


analysis

PS: If the formula is too complicated, a modern student can also use WolframAlpha instead.

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HeapSort(): John William Joseph Williams invented HeapSort() algorithm in 1964, together with this Binary Heap data structure. HeapSort() operation (assuming the Binary Max Heap has been created in O(N)) is very easy. Simply call the O(log N) ExtractMax() operation N times. Now try HeapSort() on the currently displayed Binary (Max) Heap.


Simple Analysis: HeapSort() clearly runs in O(N log N) — an optimal comparison-based sorting algorithm.


Quiz: In worst case scenario, HeapSort() is asymptotically faster than...

Selection Sort
Insertion Sort
Bubble Sort
Merge Sort
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Although HeapSort() runs in θ(N log N) time for all (best/average/worst) cases, is it really the best comparison-based sorting algorithm?


Discussion: How about caching performance of HeapSort()?

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You have reached the end of the basic stuffs of this Binary (Max) Heap data structure and we encourage you to explore further in the Exploration Mode.


However, we still have a few more interesting Binary (Max) Heap challenges for you that are outlined in this section.


When you have cleared them all, we invite you to study more advanced algorithms that use Priority Queue as (one of) its underlying data structure, like Prim's MST algorithm, Dijkstra's SSSP algorithm, A* search algorithm (not in VisuAlgo yet), etc.

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If you are looking for an implementation of Binary (Max) Heap to actually model a Priority Queue, then there is a good news.


C++ and Java already have built-in Priority Queue implementations that very likely use this data structure. They are C++ STL priority_queue (the default is a Max Priority Queue) and Java PriorityQueue (the default is a Min Priority Queue). However, the built-in implementation may not be suitable to do some PQ extended operations efficiently (details omitted for pedagogical reason in a certain NUS module).


Python heapq exists but its performance is rather slow. OCaml doesn't have built-in Priority Queue but we can use something else that is going to be mentioned in the other modules in VisuAlgo (the reason on why the details are omitted is the same as above).


PS: Heap Sort is likely used in C++ STL algorithm partial_sort.


Nevertheless, here is our implementation of BinaryHeapDemo.cpp.

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For a few more interesting questions about this data structure, please practice on Binary Heap training module (no login is required).


However, for registered users, you should login and then go to the Main Training Page to officially clear this module and such achievement will be recorded in your user account.

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We also have a few programming problems that somewhat requires the usage of this Binary Heap data structure: UVa 01203 - Argus and Kattis - numbertree.


Try them to consolidate and improve your understanding about this data structure. You are allowed to use C++ STL priority_queue or Java PriorityQueue if that simplifies your implementation.

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Alle Schritte werden in der Status Anzeige erklärt während sie passieren
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Kontrolliere die Animation mit Hilfe deiner Tastatur! Die Tasten sind:

Leertaste: start/stop/wiederholen
Pfeiltaste rechts/links: ein Schritt vor oder zurück
-/+: senke/erhöhe die Geschwindigkeit
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Kehre zum 'Exploration Mode' zurück und beginne zu Erforschen
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Erstellen
(A) - O(N log N)

Erstellen
(A) - O(N)

Einfügen
(v)

ExtractMax()

HeapSort()

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A =

Gehen

Sorted Example

Zufällig

A =

Gehen

Sorted Example

Zufällig

v =

Gehen

Über
Mannschaft
Nutzungsbedingungen

Über

VisuAlgo wurde konzeptioniert 2011 von Dr Steven Halim als ein Tool um seinen Studenten zu helfen Datenstrukturen und Algorithmen besser zu verstehen, indem sie die Grundlagen alleine und in ihrem eigenen Tempo lernen können.
VisuAlgo enthält viele fortgeschrittene Algorithmen die auch in Dr Steven Halim's Buch ('Competitive Programming', co-author ist sein Bruder Dr Felix Halim) und mehr. Heute, können die Visualisierungen/Animationen vieler fortgeschrittener Algorithmen nur auf VisoAlgo gefunden werden.
Obwohl die Visualisierungen speziell für die verschiedenen Datenstruktur und Algorithmik Kurse der National University of Singapore (NUS) gemacht sind, freuen wir uns, als Befürworter des Online Lernens, wenn auch andere neugierige Geister unsere Visualisierungen nützlich finden.
VisuAlgo ist nicht designed um gut auf kleinen Touchscreens (z,B, Smartphones) zu funktionieren, da die Darstellung komplexer Algorithmen viele Pixel benötigt und click-and-drag Aktionen zur Interaktion. Die minimale Bildschirmauflösung für ein akzeptables Benutz Erlebnis ist 1024x768 und nur die Startseite ist einigermaßen mobilfähig.
VisuAlgo ist ein laufendes Projekt und weitere komplexe Visualisierungen werden weiterhin entwickelt.
Die aufregendste Entwicklung ist der automatisierte Fragen Generator und Überprüfer (das Online Quiz System), dass Studenten erlaubt deren Wissen über grundlegende Datenstrukturen und Algorithmen zu testen. Die Fragen werden mit der Hilfe einiger Regeln zufällig generiert und die Antworten der Studenten werden automatisch von unserem Bewertungs Server bewertet. Das Online Quiz System, wenn es von mehr Informatik Tutoren übernommen wird, sollte eigentlich grundlegende Datenstrucktur- und Algorithmikfragen in Klausuren an vielen Universitäten ersetzten. Indem man ein wenig (allerdings nicht null) Gewicht darauf legt, dass das Online Quiz bestanden wird, kann ein Informatik Tutor (stark) das Können seiner Studenten was solche grundlegenden Fragen betrifft erhöhen, da die Studenten eine nahezu unendlich Anzahl ein Trainingsfragen beantworten können bevor sie das Online Quiz machen. Der Training Modus enthält aktuell Fragen für 12 Visualisierungsmodule. Die letzten 8 werden bald folgen, sodass es für alle Visualisierungsmodule ein Online Quiz gibt.
Eine weitere aktive Abteilung ist das Internationalisierungs Sub-Projekt von VisuAlgo. Wir wollen eine Datenbank für alle Informatik Begriffe aus alle englischen Texte im VisuAlgo System anlegen. Das ist eine große Aufgabe und benötigt Crowdsourcing. Sobald das System funktionstüchtig ist, werden wir VisuAlgo Besucher dazu einladen. Besonders wenn sie keine englischen Muttersprachler sind. Aktuel, haben wir auch verschiedene Notizen in verschiedenen Sprachen über VisuAlgo:
zh, id, kr, vn, th.

Mannschaft

Projektleiter & Berater (Juli 2011 bis heute)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

Studentische Hilfskräfte 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

Abschlussprojekt/UROP Studenten 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

Abschlussprojekt/UROP Studenten 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Studentische Hilfskräfte 2 (May 2014-Jul 2014)
Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

Abschlussprojekt/UROP Studenten 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Abschlussprojekt/UROP Studenten 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

List of translators who have contributed ≥100 translations can be found at statistics page.

Danksagungen
Dieses Projekt wird durch den großzügigen Teaching Enhancement Grant des NUS Centre for Development of Teaching and Learning (CDTL) ermöglicht.

Nutzungsbedingungen

VisuAlgo ist kostenlos für die Informatik-Community dieses Planeten (natürlich auch von Leute nicht von der Erde). Wenn dir VisuAlgo gefällt, ist die einzige Bezahlung um die wir bitten, das du anderen Informatik Studenten und Tutoren von dieser Seite erzählst. =) über Facebook, Twitter, Kurs Internet Seit, Blog Eintrag, Email usw.

Bist du ein Datenstruktur oder Algorithmik Student/Tutor, darfst du diese Webseite für deine Kurse nutzen. Solltest du Screenshots (Videos) von dieser Seite machen, darfst du diese woanders verwenden, solange du die URL dieser Seite (http://visualgo.net) als Referenz angibst. Es ist allerdings NICHT erlaubt VisuAlgo (client-Side) Dateien herunter zu laden und diese auf deiner eigenen Website zu hosten, da das ein  Plagiat wäre. Es ist auch NICHT erlaubt eine Anspaltung dieser Website zu machen und Varianten von VisuAlgo zu erstellen. Eine private Nutzung einer offline Kopie (client-side) von VisuAlgo ist erlaubt.

Beachte allerdings das VisuAlgo's Online Quiz System von Natur aus eine schwere Server-seitige Komponente hat und es gibt keinen einfachen Weg die Server-seitige Scripts und Datenbanken lokal zu speichern. Aktuell kann die allgemeinen Öffentlichkeit nur den 'Trainings Modus' nutzen um an das Online Quiz System zu kommen. Der 'Test-Modus' ist eine kontrollierterte Umgebung in der zufällig generierte Fragen und automatische Überprüfung für eine echte Prüfung in NUS genutzt werden. Andere interessierte Informatik Tutoren sollten Steven kontaktieren, wenn sie auch diesen 'Test-Modus' ausprobieren wollen.

Liste der Publikationen

Diese Arbeit wurde kurz beim CLI Workshop beim ACM ICPC Weltfinale 2012 (Polen, Warschau) und bei der IOI Konferenz bei IOI 2012 (Italien, Sirmione-Montichiari). Du kannst du diesen Link klicken um unser 2012 Paper über dieses System zu lesen (Es hieß 2012 noch nicht VisuAlgo).
Diese Arbeit wurde wurde hauptsächlich von ehemaligen Studenten gemacht. Die letzten Ergebnisse sind hier: Erin, Wang Zi, Rose, Ivan.

Bug Reports oder Anfragen zu neuen Features

VisuAgo ist kein fertiges Projekt. Dr Steven Halim arbeitet aktiv daran VisuAlgo zu verbessern. Wenn du beim benutzten von VisuAlgo in einer Visualisierung/Online Quiz einen Bug findest oder ein neues Feature möchtest, kontaktiere bitte Dr Steven Halim. Sein Kontakt ist die Verkettung seines Namens und at gmail dot com.