Sebuah graf terdiri dari simpul-simpul (vertices) dan sisi-sisi (edges) yang menghubungkan simpul-simpul tersebut.


Sebuah graf dapat tidak terarah (yang berarti tidak ada perbedaan antara kedua arah dari sisi dua arah yang menghubungkan dua buah simpul) atau sebuah graf dapat terarah (yang berarti sisi-sisi yang ada memiliki arah tertentu dari sebuah simpul ke simpul lainnya, namun belum tentu ada untuk arah sebaliknya).


Sebuah graf dapat berbobot (dengan menempatkan sebuah bobot pada tiap sisi yang berupa sebuah angka yang diasosiasikan dengan sisi tersebut) atau tidak berbobot (semua sisi memiliki bobot 1 atau semua sisi memiliki bobot konstan yang sama).


Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.
If you are an NUS student and a repeat visitor, please login.

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Kebanyakan dari masalah-masalah graf yang kita bahas di VisuAlgo berhubungan dengan graf-graf sederhana (simple graphs).


Didalam sebuah graf sederhana, tidak ada sisi yang melingkar sendiri (sisi yang menghubungkan sebuah simpul dengan simpul itu sendiri) dan tidak ada sisi-sisi ganda/paralel (sisi-sisi diantara sepasang simpul yang sama). Dengan kata lain: Cuma ada maksimum satu sisi diantara sepasang simpul yang unik.


Jumlah sisi-sisi E di dalam sebuah graf sederhana cuma bisa berkisar dari 0 sampai O(V2).


Algoritma-algoritma graf pada graf-graf sederhana lebih mudah daripada pada graf-graf tidak sederhana.


Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/[PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode.

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An undirected edge e: (u, v) is said to be incident with its two end-point vertices: u and v. Two vertices are called adjacent (or neighbor) if they are incident with a common edge. For example, edge (0, 2) is incident to vertices 0+2 and vertices 0+2 are adjacent.


Two edges are called adjacent if they are incident with a common vertex. For example, edge (0, 2) and (2, 4) are adjacent.


The degree of a vertex v in an undirected graph is the number of edges incident with vertex v. A vertex of degree 0 is called an isolated vertex. For example, vertex 0/2/6 has degree 2/3/1, respectively.


A subgraph G' of a graph G is a (smaller) graph that contains subset of vertices and edges of G. For example, a triangle {0, 1, 2} is a subgraph of the currently displayed graph.


Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). 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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Sebuah jalur (path) (dengan panjang n) dalam sebuah graf (tidak terarah) G adalah urutan simpul-simpul {v0, v1, ..., vn-1, vn} sehingga ada sisi diantara vi dan vi+1i ∈ [0..n-1] sepanjang jalur tersebut.


Jika tidak ada simpul yang diulang disepanjang jalan, kita menyebut jalur tersebut sebagai jalur sederhana (simple path).


Contohnya, {0, 1, 2, 4, 5} adalah satu jalur sederhana di graf yang sekarang ditampilkan.


Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, / to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively.

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An undirected graph G is called connected if there is a path between every pair of distinct vertices of G. For example, the currently displayed graph is not a connected graph.


An undirected graph C is called a connected component of the undirected graph G if:
1). C is a subgraph of G;
2). C is connected;
3). no connected subgraph of G has C as a subgraph and contains vertices or edges that are not in C (i.e., C is the maximal subgraph that satisfies the other two criteria).


For example, the currently displayed graph have {0, 1, 2, 3, 4} and {5, 6} as its two connected components.


A cut vertex/bridge is a vertex/edge that increases the graph's number of connected components if deleted. For example, in the currently displayed graph, there is no cut vertex, but edge (5, 6) is a bridge.

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In a directed graph, some of the terminologies mentioned earlier have small adjustments.


If we have a directed edge e: (uv), we say that v is adjacent to u but not necessarily in the other direction. For example, 1 is adjacent to 0 but 0 is not adjacent to 1 in the currently displayed directed graph.


In a directed graph, we have to further differentiate the degree of a vertex v into in-degree and out-degree. The in-degree/out-degree is the number of edges coming-into/going-out-from v, respectively. For example, vertex 1 has in-degree/out-degree of 2/1, respectively.

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In a directed graph, we extend the concept of Connected Component (CC) into Strongly Connected Component (SCC). A directed graph G is called strongly connected if there is a path in each direction between every pair of distinct vertices of G.


A directed graph SCC is called a strongly connected component of the directed graph G if:
1). SCC is a subgraph of G;
2). SCC is strongly connected;
3). no connected subgraph of G has SCC as a subgraph and contains vertices or edges that are not in SCCC (i.e., SCC is the maximal subgraph that satisfies the other two criteria).


In the currently displayed directed graph, we have {0}, {1, 2, 3}, and {4, 5, 6, 7} as its three SCCs.

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A cycle is a path that starts and ends with the same vertex.


An acyclic graph is a graph that contains no cycle.


In an undirected graph, each of its undirected edge causes a trivial cycle (of length 2) although we usually will not classify it as a cycle.


A directed graph that is also acyclic has a special name: Directed Acyclic Graph (DAG), as shown above.


There are interesting algorithms that we can perform on acyclic graphs that will be explored in this visualization page and in other graph visualization pages in VisuAlgo.

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Sebuah graf dengan properti-properti spesifik yang berhubungan dengan simpul-simpulnya dan/atau struktur sisi-sisinya bisa dipanggil dengan nama spesifiknya, seperti Pohon (seperti yang sekarang ditampilkan), Graf Komplet, Graf Bipartit, Graf Terarah Tidak-bersiklus (Directed Acyclic Graph, DAG), dan juga yang jarang digunakan: Graf Planar, Graf Garis (Line Graph), Graf Bintang (Star Graph), Graf Roda (Wheel Graph), dsb.


Dalam visualisasi ini, kita akan menyorot empat graf-graf spesial pertama nantinya.

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Graf sering sekali muncul dalam berbagai bentuk di kehidupan nyata. Bagian terpenting dari menyeesaikan masalah graf adalah bagian pemodelan graf, yaitu mereduksi masalah yang ada ke terminologi-termonologi graf: simpul-simpul, sisi-sisi, bobot-bobot, dsb.

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Jaringan Sosial (Social Network): Simpul-simpul bisa merepresentasikan orang, Sisi-sisi merepresentasikan hubungan antar orang (biasanya tidak terarah dan tidak berbobot).


Contohnya, lihat graf tidak terarah yang sedang ditampilkan. Graf ini menunjukkan 7 simpul-simpul (orang-orang) dan 8 sisi-sisi (hubungan) diantara mereka. Mungkin kita bisa bertanya pertanyaan-pertanyaan seperti berikut:

  1. Siapa saja teman(-teman) dari orang bernomor 0?
  2. Siapa yang mempunyai paling banyak teman(-teman)?
  3. Apakah ada orang yang terisolasi (orang-orang yang tidak mempunyai teman)?
  4. Apakah ada teman yang sama diantara dua orang yang tidak saling mengenal: Orang nomor 3 dan orang nomor 5?
  5. Dsb...
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Jaringan Transportasi: Simpul-simpul bisa merepresentasikan stasiun-stasiun, sisi-sisi merepresentasikan koneksi antar stasiun (biasanya berbobot).


Contohnya, lihat graf berarah berbobot yang sedang ditampilkan. Graf ini menunjukkan 5 simpul-simpul (stasiun-stasiun/tempat-tempat) dan 6 sisi-sisi (koneksi-koneksi/jalan-jalan antar stasiun-stasiun, dengan waktu tempuh berbobot positif seperti yang tertera). Misalkan kita mengemudikan sebuah mobil. Kita mungkin bertanya apa jalur yang perlu diambil untuk pergi dari stasiun 0 ke stasiun 4 supaya kita sampai di stasiun 4 dengan menggunakan waktu yang paling sedikit?


Diskusi: Bahas beberapa skenario-skenario di kehidupan nyata lainnya yang bisa dimodelkan sebagai graf.

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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Untuk beralih di antara mode-mode penggambaran graf, anda dapat memilih header yang ada. Kami memiliki:
  1. U/U = Tidak Terarah/Tidak Berbobot (default),
  2. U/W = Tidak Terarah/Berbobot,
  3. D/U = Terarah/Tidak Berbobot, dan
  4. D/W = Terarah/Berbobot.
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You can click any one of the example graphs and see its example graph drawing, which is a two-dimensional depiction of that graph. Note that the same graph can have (infinitely) many possible graph drawings.


You can further edit (add/delete/reposition the vertices or add/change the weight of/delete the edges) the currently displayed graph by clicking "Edit Graph" (read the associated Help message in that Edit Graph window).

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We limit the graphs discussed in VisuAlgo to be simple graphs. Refer to its discussion in this slide.


While now we do not really limit the number of vertices that you can draw on screen, we recommend that you draw not more than 10 vertices, ranging from vertex 0 to vertex 9 (as the Adjacency Matrix of this graph will already contain 10x10 = 100 cells). This, together with the simple graph constraint earlier, limit the number of undirected/directed edges to be 45/90, respectively.

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All example graphs can be found here. We provide seven "most relevant" example graphs per category (U/U, U/W, D/U, D/W).


Remember that after loading one of these example graphs, you can further edit the currently displayed graph to suit your needs.

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Tree, Complete, Bipartite, Directed Acyclic Graph (DAG) are properties of special graphs. As you edit the graph, these properties are checked and updated instantly.


There are other less frequently used special graphs: Planar Graph, Line Graph, Star Graph, Wheel Graph, etc, but they are not currently auto-detected in this visualization.

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Tree is a connected graph with V vertices and E = V-1 edges, acyclic, and has one unique path between any pair of vertices. Usually a Tree is defined on undirected graph.


An undirected Tree (see above) actually contains trivial cycles (caused by its bidirectional edges) but it does not contain non-trivial cycle (of length 3 or larger). A directed Tree is clearly acyclic.


As a Tree only have V-1 edges, it is usually considered a sparse graph.


We currently show our U/U: Tree example. You can go to 'Exploration Mode' and edit/draw your own trees.

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Not all Trees have the same graph drawing layout of having a special root vertex at the top and leaf vertices (vertices with degree 1) at the bottom. The (star) graph shown above is also a Tree as it satisfies the properties of a Tree.


Tree with one of its vertex designated as root vertex is called a rooted Tree.


We can always transform any Tree into a rooted Tree by designating a specific vertex (usually vertex 0) as the root, and run a DFS or BFS algorithm from the root. This process of "rooting the tree" (of a Tree that is not visually drawn as a tree yet) has a visual explanation. Imagine that each vertex is a small ball (with non-zero weight) and each edge is a rope of the same length connecting two adjacent balls. Now, if we pick the root ball/vertex and pull it up, then gravity will pull the rest of the balls downwards and that is the DFS/BFS spanning tree of the tree.

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In a rooted tree, we have the concept of hierarchies (parent, children, ancestors, descendants), subtrees, levels, and height. We will illustrate these concepts via examples as their meanings are as with real-life counterparts:

  1. The parent of 0/1/7/9/4 are none/0/1/8/3, respectively,
  2. The children of 0/1/7 are {1,8}/{2,3,6,7}/none, respectively,
  3. The ancestors of 4/6/8 are {3,1,0}/{1,0}/{0}, respectively,
  4. The lowest common ancestor between 4 and 6 is 1.
  5. The descendants of 1/8 are {2,3,4,5,6,7}/{9}, respectively,
  6. The subtree rooted at 1 includes 1, its descendants, and all associated edges,
  7. Level 0/1/2/3 members are {0}/{1,8}/{2,3,6,7,9}/{4,5}, respectively,
  8. The height of this rooted tree is its maximum level = 3.
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Untuk pohon yang berakar, kita juga dapat mendefinisikan beberapa properti-properti tambahan:


Sebuah pohon biner adalah pohon yang berakar dimana sebuah simpul memiliki paling banyak dua anak yang pas dinamakan sebagai anak kiri dan kanan. Kita sering melihat bentuk ini selama diskusi Pohon Pencarian Biner dan Tumpukan Biner.


Sebuah pohon biner penuh adalah pohon biner dimana setiap simpul bukan-daun (juga disebut sebagain simpul internal) memiliki tepat dua anak. Pohon biner yang ditunjukkan diatas memenuhi kriteria ini.


Pohon biner komplet adalah pohon biner dimana setiap tingkatan terisi penuh, kecuali mungkin tingkatan terakhir mungkin terisi sisi kirinya sebisa mungkin. Kita sering melihat bentuk ini terutama dalam pembahasan tentang Tumpukan Biner.

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Complete graph is a graph with V vertices and E = V*(V-1)/2 edges (or E = O(V2)), i.e., there is an edge between any pair of vertices. We denote a Complete graph with V vertices as KV.


Complete graph is the most dense simple graph.


We currently show our U/W: K5 (Complete) example. You can go to 'Exploration Mode' and edit/draw your own complete graphs (a bit tedious for larger V though).

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Bipartite graph is an undirected graph with V vertices that can be partitioned into two disjoint set of vertices of size m and n where V = m+n. There is no edge between members of the same set. Bipartite graph is also free from odd-length cycle.


We currently show our U/U: Bipartite example. You can go to 'Exploration Mode' and draw/edit your own bipartite graphs.


A Bipartite Graph can also be complete, i.e., all m vertices from one disjoint set are connected to all n vertices from the other disjoint set. When m = n = V/2, such Complete Bipartite Graphs also have E = O(V2).


A Tree is also a Bipartite Graph, i.e., all vertices on the even levels form one set, and all vertices on the odd levels form the other set.

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Directed Acyclic Graph (DAG) is a directed graph that has no cycle, which is very relevant for Dynamic Programming (DP) techniques.


Each DAG has at least one Topological Sort/Order which can be found with a simple tweak to DFS/BFS Graph Traversal algorithm. DAG will be revisited again in DP technique for SSSP on DAG.


We currently show our D/W: Four 0→4 Paths example. You can go to 'Exploration Mode' and draw your own DAGs.

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Ada banyak cara untuk menyimpan informasi graf kedalam sebuah struktur data graf. Dalam visualisasi ini, kami menunjukkan tiga struktur data graf: Matriks Adjacency (Adjacency Matrix), Daftar Adjacency (Adjacency List), dan Daftar Sisi (Edge List) — masing-masing dengan kekuatan-kekuatan dan kelemahan-kelemahannya tersendiri.

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Adjacency Matrix (AM) is a square matrix where the entry AM[i][j] shows the edge's weight from vertex i to vertex j. For unweighted graphs, we can set a unit weight = 1 for all edge weights.


We usually set AM[i][j] = 0 to indicate that there is no edge (i, j). However, if the graph contains 0-weighted edge, we have to use another symbol to indicate "no edge" (e.g., -1, None, null, etc).


We simply use a C++/Python/Java native 2D array/list of size VxV to implement this data structure.

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Analisa Kompleksitas Ruang: Sebuah AM sayangnya membutuhkan kompleksitas ruang yang besar yaitu O(V2), meskipun bila graf kita ternyata jarang (sparse) (yaitu tidak mempunyai banyak sisi-sisi).


Diskusi: Mengetahui besarnya kompleksitas ruang dari AM, kapankah saat yang tepat untuk menggunakannya? Atau apakah AM selalu adalah struktur data graf yang inferior dan selalu tidak boleh digunakan di setiap saat?

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Daftar Adjacency (Adjacency List, AL) adalah sebuah larik berisi V senarai (lists), satu untuk setiap simpul (biasanya dalam nomor simpul menaik) dimana untuk setiap simpul i, AL[i] menyimpan daftar dari tetangga-tetangga i. Untuk graf-graf berbobot, kita bisa menyimpan pasangan-pasangan (nomor simpul tetangga, bobot dari sisi ini).


Kami menggunakan Vector dari Vector pair (untuk graf-graf berbobot) untuk mengimplementasikan struktur data ini.
Dalam C++: vector<vector<pair<int, int>>> AL;

Dalam Python: AL = [[] for _ in range(N)]

Dalam Java: Vector<Vector<IntegerPair>> AL;

// class IntegerPair di Java seperti pair<int, int> di C++

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class IntegerPair implements Comparable<IntegerPair> {
Integer _f, _s;
public IntegerPair(Integer f, Integer s) { _f = f; _s = s; }
public int compareTo(IntegerPair o) {
if (!this.first().equals(o.first())) // this.first() != o.first()
return this.first() - o.first(); // salah karena kita mau
else // membandingkan nilai mereka,
return this.second() - o.second(); // bukan referensi mereka
}
Integer first() { return _f; }
Integer second() { return _s; }
}
// IntegerTriple mirip dengan IntegerPair, tapi mempunyai 3 parameter
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We use pairs as we need to store pairs of information for each edge: (neighbor vertex number, edge weight) where the weight field can be set to 1, 0, unused, or simply dropped for unweighted graph.


We use Vector of Pairs due to Vector's auto-resize feature. If we have k neighbors of a vertex, we just add k times to an initially empty Vector of Pairs of this vertex (this Vector can be replaced with Linked List).


We use Vector of Vectors of Pairs for Vector's indexing feature, i.e., if we want to enumerate neighbors of vertex u, we use AL[u] (C++/Python) or AL.get(u) (Java) to access the correct Vector of Pairs.

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Analisa Kompleksitas Ruang: AL mempunyai kompleksitas ruang sebesar O(V+E), yang adalah jauh lebih efisien daripada AM dan biasanya adalah struktur data graf default didalam hampir semua algoritma-algoritma graf.


Diskusi: AL adalah struktur data graf yang paling sering dipakai, tetapi diskusi beberapa skenario-skenario dimana AL sesungguhnya bukan struktur data graf yang terbaik?

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Daftar Sisi (Edge List, EL) adalah koleksi dari sisi-sisi dengan simpul-simpul yang berhubungan dan bobot-bobotnya. Biasanya, sisi-sisi ini diurutkan berdasarkan bobot menaik, contohnya seperti di bagian algoritma Kruskal untuk masalah Pohon Perentang Terkecil (Minimum Spanning Tree, MST). Tetapi dalam visualisasi ini, kita mengurutkan sisi-sisi berdasarkan nomor simpul pertama secara menaik dan jika ada yang sama, berdasarkan nomor simpul kedua secara menarik. Catat bahwa sisi-sisi dua arah dalam graf tidak-berarah/berarah didaftarkan sekali/dua-kali.


Kita menggunakan Vector dari triples untuk mengimplementasikan struktur data ini.
Dalam C++: vector<tuple<int, int, int>> EL;

Dalam Python: EL = [] 

Dalam Java: Vector<IntegerTriple> EL; 

// class IntegerTriple di Java seperti tuple<int, int, int> di C++

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Analisa Kompleksitas Ruang: EL mempunyai kompleksitas ruang sebesar O(E), yang adalah lebih efisien dari AM dan sama efisiennya dengan AL.


Diskusi: Sebutkan potensi penggunaan EL selain dalam algoritma Kruskal untuk masalah Pohon Perentang Terkecil (Minimum Spanning Tree, MST)!

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Setelah menyimpan informasi graf kita didalam sebuah struktur data graf, kita dapat menjawab beberapa pertanyaan-pertanyaan sederhana.

  1. Menghitung V,
  2. Menghitung E,
  3. Mendaftarkan tetangga-tetangga dari sebuah simpul u,
  4. Mengecek keberadaan sisi (u, v), dsb.
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Dalam AM dan AL, V hanyalah jumlah baris-baris di dalam struktur data yang bisa didapatkan dalam O(V) atau bahkan dalam O(1) — tergantung implementasi aktual.


Diskusi: Bagaimana cara menghitung V jika graf disimpan dalam sebuah EL?


Catatan: Kadang-kadang angka ini disimpan/dimutakhirkan di variable terpisah sehingga kita tidak harus menghitung ulang setiap kali — terutama jika graf kita tidak pernah/jarang berubah sejak dibuat, sehingga kita mendapatkan performa O(1), contohnya, kita dapat menyimpan data bahwa ada 7 simpul-simpul (dalam struktur data AM/AL/EL kita) untuk graf contoh diatas.

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

In an EL, E is just the number of its rows that can be counted in O(E) or even in O(1) — depending on the actual implementation. Note that depending on the need, we may store a bidirectional edge just once in the EL but on other case, we store both directed edges inside the EL.


In an AL, E can be found by summing the length of all V lists and divide the final answer by 2 (for undirected graph). This requires O(V+E) computation time as each vertex and each edge is only processed once. This can also be implemented in O(V) in some implementations.


Discussion: How to count E if the graph is stored in an AM?


PS: Sometimes this number is stored/maintained in a separate variable for efficiency, e.g., we can store that there are 8 undirected edges (in our AM/AL/EL data structure) for the example graph shown above.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Didalam sebuah AM, kita butuh untuk menelusuri seluruh kolom-kolom dari AM[u][j] ∀j ∈ [0..V-1] dan melaporkan pasangan dari (j, AM[u][j]) jika AM[u][j] tidak nol. Ini adalah O(V) — lambat.


Didalam sebuah AL, kita hanya perlu menelusuri AL[u]. Jika hanya ada k tetangga-tetangga dari simpul u, maka kita hanya perlu O(k) untuk mengenumerasi mereka — ini disebut kompleksitas waktu yang sensitif-terhadap-keluaran dan sudah merupakan yang terbaik.


Kita biasanya mendaftarkan tetangga-tetangga dalam nomor simpul menaik. Contohnya, tetangga-tetangga dari simpul 1 di graf contoh diatas adalah {0, 2, 3}, dengan urutan nomor simpul menaik.


Diskusi: Bagaimana caranya melakukan ini bila grafnya disimpan dalam sebuah EL?

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Dalam sebuah AM, kita dapat mengecek apakah AM[u][v] adalah nol. Ini adalah O(1) — yang paling cepat.


Dalam sebuah AL, kita harus mengecek apakah AL[u] berisi simpul v atau tidak. Ini adalah O(k) — lebih lambat.


Contohnya, simpul (2, 4) ada di graf contoh diatas tetapi simpul (2, 6) tidak ada.

Catat bahwa jika kita telah menemukan simpul (u, v), kita juga dapat mengakses dan/atau memutakhirkan bobotnya.


Diskusi: Bagaimana untuk melakukan ini bila grafnya disimpan dalam sebuah EL?

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Quiz: So, what is the best graph data structure?

Adjacency Matrix
Adjacency List
Edge List
It Depends


Diskusi: Kenapa?

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

You have reached the end of the basic stuffs of this relatively simple Graph Data Structures and we encourage you to explore further in the Exploration Mode by editing the currently drawn graph, by drawing your own custom graphs, or by inputting Edge List/Adjacency Matrix/Adjacency List input and ask VisuAlgo to propose a "good enough" graph drawing of that input graph.


However, we still have a few more interesting Graph Data Structures challenges for you that are outlined in this section.


Note that graph data structures are usually just the necessary but not sufficient part to solve the harder graph problems like MST, SSSP, MF, Matching, MVC, ST, or TSP.

🕑

Untuk beberapa pertanyaan-pertanyaan menarik tentang struktur data ini, silahkan latihan di modul latihan Struktur Data Graf.

🕑
Lihatlah implementasi-implementasi C++/Python/JavaOCaml berikut ini dari ketiga struktur-struktur data yang disebut dalam kuliah maya ini: Adjacency Matrix, Adjacency List, dan Edge List:  graph_ds.cpp | py | java | ml.
🕑

Cobalah selesaikan dua masalah-masalah pemrograman dasar yang membutuhkan penggunaan struktur data graf tanpa algoritma-algoritma graf apapun:

  1. UVa 10895 - Matrix Transpose dan,
  2. Kattis - flyingsafely.
🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Last but not least, there are some graphs that are so nicely structured that we do not have to actually store them in any graph data structure that we have discussed earlier.


For example, a complete unweighted graph can be simply stored with just one integer V, i.e., we just need to remember it's size and since a complete graph has an edge between any pair of vertices, we can re-construct all those V * (V-1) / 2 edges easily.


Discussion: Can you elaborate a few more implicit graphs?

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.


You have reached the last slide. Return to 'Exploration Mode' to start exploring!

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

🕑
V=0, E=0 • Pohon? No • Komplet? No • Bipartit? No • DAG? No • Tabrakan? No
Matriks Adjacency
012
0010
1101
2010
Daftar Adjacency
0: 1
1: 02
2: 1
Daftar Edge
0: 01
1: 12

Edit Graph

>
Tentang Tim Syarat Guna Kebijakan Privasi

Tentang

Initially conceived in 2011 by Associate Professor Steven Halim, VisuAlgo aimed to facilitate a deeper understanding of data structures and algorithms for his students by providing a self-paced, interactive learning platform.

Featuring numerous advanced algorithms discussed in Dr. Steven Halim's book, 'Competitive Programming' — co-authored with Dr. Felix Halim and Dr. Suhendry Effendy — VisuAlgo remains the exclusive platform for visualizing and animating several of these complex algorithms even after a decade.

While primarily designed for National University of Singapore (NUS) students enrolled in various data structure and algorithm courses (e.g., CS1010/equivalent, CS2040/equivalent (including IT5003), CS3230, CS3233, and CS4234), VisuAlgo also serves as a valuable resource for inquisitive minds worldwide, promoting online learning.

Initially, VisuAlgo was not designed for small touch screens like smartphones, as intricate algorithm visualizations required substantial pixel space and click-and-drag interactions. For an optimal user experience, a minimum screen resolution of 1366x768 is recommended. However, since April 2022, a mobile (lite) version of VisuAlgo has been made available, making it possible to use a subset of VisuAlgo features on smartphone screens.

VisuAlgo remains a work in progress, with the ongoing development of more complex visualizations. At present, the platform features 24 visualization modules.

Equipped with a built-in question generator and answer verifier, VisuAlgo's "online quiz system" enables students to test their knowledge of basic data structures and algorithms. Questions are randomly generated based on specific rules, and students' answers are automatically graded upon submission to our grading server. As more CS instructors adopt this online quiz system worldwide, it could effectively eliminate manual basic data structure and algorithm questions from standard Computer Science exams in many universities. By assigning a small (but non-zero) weight to passing the online quiz, CS instructors can significantly enhance their students' mastery of these basic concepts, as they have access to an almost unlimited number of practice questions that can be instantly verified before taking the online quiz. Each VisuAlgo visualization module now includes its own online quiz component.

VisuAlgo has been translated into three primary languages: English, Chinese, and Indonesian. Additionally, we have authored public notes about VisuAlgo in various languages, including Indonesian, Korean, Vietnamese, and Thai:

id, kr, vn, th.

Tim

Pemimpin & Penasihat Proyek (Jul 2011-sekarang)
Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

Murid-Murid S1 Peniliti 1
CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Murid-Murid Proyek Tahun Terakhir/UROP 1
Jul 2012-Dec 2013: Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy
Jun 2013-Apr 2014 Rose Marie Tan Zhao Yun, Ivan Reinaldo

Murid-Murid S1 Peniliti 2
CDTL TEG 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

Murid-Murid Proyek Tahun Terakhir/UROP 2
Jun 2014-Apr 2015: Erin Teo Yi Ling, Wang Zi
Jun 2016-Dec 2017: Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir
Aug 2021-Apr 2023: Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long, Ting Xiao, Lim Dewen Aloysius

Murid-Murid S1 Peniliti 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Murid-Murid Proyek Tahun Terakhir/UROP 3
Aug 2023-Apr 2024: Xiong Jingya, Radian Krisno, Ng Wee Han

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

Ucapan Terima Kasih
NUS CDTL gave Teaching Enhancement Grant to kickstart this project.

For Academic Year 2023/24, a generous donation from Optiver will be used to further develop VisuAlgo.

Syarat Guna

VisuAlgo is generously offered at no cost to the global Computer Science community. If you appreciate VisuAlgo, we kindly request that you spread the word about its existence to fellow Computer Science students and instructors. You can share VisuAlgo through social media platforms (e.g., Facebook, YouTube, Instagram, TikTok, Twitter, etc), course webpages, blog reviews, emails, and more.

Data Structures and Algorithms (DSA) students and instructors are welcome to use this website directly for their classes. If you capture screenshots or videos from this site, feel free to use them elsewhere, provided that you cite the URL of this website (https://visualgo.net) and/or the list of publications below as references. However, please refrain from downloading VisuAlgo's client-side files and hosting them on your website, as this constitutes plagiarism. At this time, we do not permit others to fork this project or create VisuAlgo variants. Personal use of an offline copy of the client-side VisuAlgo is acceptable.

Please note that VisuAlgo's online quiz component has a substantial server-side element, and it is not easy to save server-side scripts and databases locally. Currently, the general public can access the online quiz system only through the 'training mode.' The 'test mode' offers a more controlled environment for using randomly generated questions and automatic verification in real examinations at NUS.

List of Publications

This work has been presented at the CLI Workshop at the ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012) and this link for the short update in 2015 (to link VisuAlgo name with the previous project).

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Associate Professor Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Associate Professor Steven Halim. His contact is the concatenation of his name and add gmail dot com.

Kebijakan Privasi

Version 1.2 (Updated Fri, 18 Aug 2023).

Since Fri, 18 Aug 2023, we no longer use Google Analytics. Thus, all cookies that we use now are solely for the operations of this website. The annoying cookie-consent popup is now turned off even for first-time visitors.

Since Fri, 07 Jun 2023, thanks to a generous donation by Optiver, anyone in the world can self-create a VisuAlgo account to store a few customization settings (e.g., layout mode, default language, playback speed, etc).

Additionally, for NUS students, by using a VisuAlgo account (a tuple of NUS official email address, student name as in the class roster, and a password that is encrypted on the server side — no other personal data is stored), you are giving a consent for your course lecturer to keep track of your e-lecture slides reading and online quiz training progresses that is needed to run the course smoothly. Your VisuAlgo account will also be needed for taking NUS official VisuAlgo Online Quizzes and thus passing your account credentials to another person to do the Online Quiz on your behalf constitutes an academic offense. Your user account will be purged after the conclusion of the course unless you choose to keep your account (OPT-IN). Access to the full VisuAlgo database (with encrypted passwords) is limited to Prof Halim himself.

For other CS lecturers worldwide who have written to Steven, a VisuAlgo account (your (non-NUS) email address, you can use any display name, and encrypted password) is needed to distinguish your online credential versus the rest of the world. Your account will have CS lecturer specific features, namely the ability to see the hidden slides that contain (interesting) answers to the questions presented in the preceding slides before the hidden slides. You can also access Hard setting of the VisuAlgo Online Quizzes. You can freely use the material to enhance your data structures and algorithm classes. Note that there can be other CS lecturer specific features in the future.

For anyone with VisuAlgo account, you can remove your own account by yourself should you wish to no longer be associated with VisuAlgo tool.