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In the Single-Source Shortest Paths (SSSP) problem, we aim to find the shortest paths weights (and the actual paths) from a particular single-source vertex to all other vertices in a directed weighted graph (if such paths exist).


The SSSP problem is a(nother) very well-known Computer Science (CS) problem that every CS students worldwide need to be aware of and hopefully master.


The SSSP problem has several different efficient (polynomial) algorithms (e.g., Bellman-Ford, BFS, DFS, Dijkstra — 2 versions, and/or Dynamic Programming) that can be used depending on the nature of the input directed weighted graph, i.e. weighted/unweighted, with/without (negative weight) cycle, or structurally special (a tree/a DAG).


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If you are an NUS student and a repeat visitor, please login.

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SSSP is one of the most frequent graph problem encountered in real-life. Every time we want to move from one place (usually our current location) to another (our destination), we will try to pick a short — if not the shortest — path.


SSSP algorithm(s) is embedded inside various map software like Google Maps and in various Global Positioning System (GPS) tool.


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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Input 1: A directed weighted graph G(V, E), not necessarily connected, where V/vertices can be used to describe intersections, junctions, houses, landmarks, etc and E/edges can be used to describe streets, roads, avenues with proper direction and weight/cost.


Input 2: As the name implies, the SSSP problem has another input: A source vertex sV.


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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The objective of the SSSP problem is to find the shortest path weight from s to each vertex uV, denoted as δ(s, u) (δ is pronounced as 'delta') and also the actual shortest path from s to u.


The path weight of a path p is simply the summation of edge weights along that path.


The weight of the shortest path from s to s is trivial: 0.
The weight of the shortest path from s to any unreachable vertex is also trivial: +∞.


PS: The weight of the shortest path from s to v where (s, v) ∈ E does not necessarily the weight of w(s, v). See the next few slides to realise this.


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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The outputs of all six (6) SSSP algorithms for the SSSP problem discussed in this visualization are these two arrays/Vectors:

  1. An array/Vector D of size V (D stands for 'distance')
    Initially, D[u] = 0 if u = s; otherwise D[u] = +∞ (a large number, e.g. 109)
    D[u] decreases as we find better (shorter) paths
    D[u]δ(s, u) throughout the execution of SSSP algorithm
    D[u] = δ(s, u) at the end of SSSP algorithm
  2. An array/Vector p of size V (p stands for 'parent'/'predecessor'/'previous')
    p[u] = the predecessor on best path from source s to u
    p[u] = NULL (not defined, we can use a value like -1 for this)
    This array/Vector p describes the resulting SSSP spanning tree
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Initially, D[u] = +∞ (practically, a large value like 109) ∀uV\{s}, but D[s] = D[0] = 0.
Initially, p[u] = -1 (to say 'no predecessor') ∀uV.


Now click Dijkstra(0) — don't worry about the details as they will be explained later — and wait until it is over (approximately 10s on this small graph).


At the end of that SSSP algorithm, D[s] = D[0] = 0 (unchanged) and D[u] = δ(s, u)uV
e.g. D[2] = 6, D[4] = 7 (these values are stored as red text under each vertex).
At the end of that SSSP algorithm, p[s] = p[0] = -1 (the source has no predecessor), but p[v] = the origin of the red edges for the rest, e.g. p[2] = 0, p[4] = 2.


Thus, if we are at s = 0 and want to go to vertex 4, we will use shortest path 0 → 2 → 4 with path weight 7.

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Some graphs contain negative weight edge(s) (not necessarily cyclic) and/or negative weight cycle(s). For example (fictional): Suppose you can travel forward in time (normal, edges with positive weight) or back in time by passing through time tunnel (special wormhole edges with negative weight), as the example shown above.


On that graph, the shortest paths from the source vertex s = 0 to vertices {1, 2, 3} are all ill-defined. For example 1 → 2 → 1 is a negative weight cycle as it has negative total path (cycle) weight of 15-42 = -27. Thus we can cycle around that negative weight cycle 0 → 1 → 2 → 1 → 2 → ... forever to get overall ill-defined shortest path weight of -∞.


However, notice that the shortest path from the source vertex s = 0 to vertex 4 is ok with δ(0, 4) = -99. So the presence of negative weight edge(s) is not the main issue. The main issue is the presence of negative weight cycle(s) reachable from source vertex s.

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The main operation for all SSSP algorithms discussed in this visualization is the relax(u, v, w(u, v)) operation with the following pseudo-code:

relax(u, v, w_u_v)
if D[v] > D[u]+w_u_v // if the path can be shortened
D[v] = D[u]+w_u_v // we 'relax' this edge
p[v] = u // remember/update the predecessor
// update some other data structure(s) as necessary

For example, see relax(1,2,4) operation on the figure below: relax operation example

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There are two different sources for specifying an input graph:

  1. Edit Graph: You can draw, edit, or import any directed weighted graph as the input graph.
  2. Example Graphs: You can select from the list of our selected example graphs to get you started. These example graphs have different characteristics.
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In this visualization, we will discuss 6 (SIX) SSSP algorithms.


We will start with the O(V×E) Bellman-Ford algorithm first as it is the most versatile (but also the slowest) SSSP algorithm. We will then discuss 5 (FIVE) other algorithms (including two variants of Dijkstra's algorithm) that solve special-cases of SSSP problem in a much faster manner.

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The general purpose Bellman-Ford algorithm can solve all kinds of valid SSSP problem variants (expect one — the one that is ill-defined anyway, to be discussed soon), albeit with a rather slow O(V×E) running time. It also has an extremely simple pseudo-code:

for i = 1 to |V|-1 // O(V) here, so O(V×E×1) = O(V×E)
for each edge(u, v) ∈ E // O(E) here, e.g. by using an Edge List
relax(u, v, w(u, v)) // O(1) here

Without further ado, let's see a preview of how it works on the example graph above by clicking BellmanFord(0) (≈30s, and for now, please ignore the additional loop at the bottom of the pseudo-code).

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Bellman-Ford algorithm can be made to run slightly faster on normal input graph, from the worst case of O(V×E) to just O(k×E) where k is the number of iterations of the outer loop of Bellman-Ford.


Discussion: How to do this? Is the speed-up significant?

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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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To convince the worldwide audience that Bellman-Ford algorithm works, let's temporarily move from visualization mode to proof mode for a few slides.


Theorem 1: If G = (V, E) contains no negative weight cycle, then the shortest path p from source vertex s to a vertex v must be a simple path.


Recall: A simple path is a path p = {v0, v1, v2, ..., vk}, (vi, vi+1) ∈ E, ∀ 0 ≤ i ≤ (k-1) and there is no repeated vertex along this path.

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  1. Suppose the shortest path p is not a simple path
  2. Then p must contains one (or more) cycle(s) (by definition of non-simple path)
  3. Suppose there is a cycle c in p with positive weight (e.g., greenbluegreen on the left image) cycle
  4. If we remove c from p, then we will have a shorter 'shortest path' than our shortest path p
  5. A glaring contradiction, so p must be a simple path
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  1. Even if c is actually a cycle with zero (0) total weight — it is possible according to our Theorem 1 assumption: no negative weight cycle (see the same greenbluegreen but on the right image), we can still remove c from p without increasing the shortest path weight of p cycle
  2. In conclusion, p is a simple path (from point 5) or can always be made into a simple path (from point 6)

In another word, shortest path p has at most |V|-1 edges from the source vertex s to the 'furthest possible' vertex v in G (in terms of number of edges in the shortest path — see the Bellman-Ford Killer example above).

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Theorem 2: If G = (V, E) contains no negative weight cycle, then after Bellman-Ford algorithm terminates, we will have D[u] = δ(s, u), ∀ uV.


For this, we will use Proof by Induction and here are the starting points:


Consider the shortest path p from source vertex s to vertex vi where vi is defined as a vertex which the actual shortest path to reach it requires i hops (edges) from source vertex s. Recall from Theorem 1 that p will be simple path as we have the same assumption of no negative weight cycle.

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  1. Initially, D[v0] = δ(s, v0) = 0, as v0 is just the source vertex s
  2. After 1 pass through E, we have D[v1] = δ(s, v1)
  3. After 2 pass through E, we have D[v2] = δ(s, v2)
  4. ...
  5. After k pass through E, we have D[vk] = δ(s, vk)
  6. When there is no negative weight cycle, the shortest path p is a simple path (see Theorem 1), thus the last iteration should be iteration |V|-1
  7. After |V|-1 pass through E, we have D[v|V|-1] = δ(s, v|V|-1), regardless the ordering of edges in E — see the Bellman-Ford Killer example above
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Try running BellmanFord(0) on the 'Bellman-Ford Killer' example above. There are V = 7 vertices and E = 6 edges but the edge list E is configured to be at its worst possible order. Notice that after (V-1)×E = (7-1)*6 = 36 operations (~40s, be patient), Bellman-Ford will terminate with the correct answer and there is no way we can terminate Bellman-Ford algorithm earlier.

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The only input graph that Bellman-Ford algorithm has issue is the input graph with negative weight cycle reachable from the source vertex s.


However, Bellman-Ford can be used to detect if the input graph contains at least one negative weight cycle reachable from the source vertex s by using the corollary of Theorem 2: If at least one value D[u] fails to converge after |V|-1 passes, then there exists a negative-weight cycle reachable from the source vertex s.


Now run BellmanFord(0) on the example graph that contains negative edges and a negative weight cycle. Please concentrate on the loop at the bottom of the pseudo-code.

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Sometimes, the actual problem that we face is not the general form of the original problem. Therefore in this e-Lecture, we want to highlight five (5) special cases involving the SSSP problem. When we encounter any one of them, we can solve it with different and (much) faster algorithm than the generic O(V×E) Bellman-Ford algorithm. They are:

  1. On Unweighted Graphs: O(V+E) BFS,
  2. On Graphs without negative weight: O((V+E) log V) Dijkstra's algorithm,
  3. On Graphs without negative weight cycle: O((V+E) log V) Modified Dijkstra's,
  4. On Tree: O(V+E) DFS/BFS,
  5. On Directed Acyclic Graphs (DAG): O(V+E) Dynamic Programming (DP)
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The O(V+E) Breadth-First Search (BFS) algorithm can solve special case of SSSP problem when the input graph is unweighted (all edges have unit weight 1, try BFS(5) on example: 'CP3 4.3' above) or positive constant weighted (all edges have the same constant weight, e.g. you can change all edge weights of the example graph above with any positive constant weight of your choice).

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When the graph is unweighted — this appears quite frequently in real life — the SSSP problem can be viewed as a problem of finding the least number of edges traversed from the source vertex s to other vertices.


The BFS spanning tree from source vertex s produced by the fast O(V+E) BFS algorithm — notice the + sign — precisely fits the requirement.


Compared with the O(V×E) of Bellman-Ford — notice the × sign — it is a no-brainer to use BFS for this special case of SSSP problem.

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Compared to the standard BFS in Graph Traversal module, we need to perform simple modifications to make BFS able to solve the unweighted version of the SSSP problem:

  1. First, we change the Boolean array visited into an Integer array D.
  2. At the start of BFS, instead of setting visited[u] = false, we set D[u] = 1e9 (a large number to symbolise +∞ or even -1 to symbolise 'unvisited' state, but we cannot use 0 as D[0] = 0) ∀uV\{s}; Then we set D[s] = 0
  3. We change the BFS main loop from
    if (visited[v] = 0) { visited[v] = 1 ... } // v is unvisited
    to
    if (D[v] = 1e9) { D[v] = D[u]+1 ... } // v is 1 step away from u
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However, BFS will very likely produce wrong answer when run on weighted graphs as BFS is not actually designed for to solve the weighted version of SSSP problem. There may be a case that taking a path with more number of edges used produces lower total overall path weight than taking a path with minimum number of edges used — which is the output of BFS algorithm.


In this visualization, we will allow you to run BFS even on 'wrong' input graph for pedagogical purpose, but we will display a warning message at the end of the algorithm. For example, try BFS(0) on the general graph above and you will see that vertices {3,4} will have wrong D[3] and D[4] values (and also p[3] and p[4] values).


We will soon see Dijkstra's algorithm (2 implementation variants) for solving certain weighted SSSP problems in a faster way than the general Bellman-Ford algorithm.

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The O((V+E) log V) Dijkstra's algorithm is the most frequently used SSSP algorithm for typical input: Directed weighted graph that has no negative weight edge, formally: ∀edge(u, v) ∈ E, w(u, v) ≥ 0. Such weighted graph (especially the positive weighted ones) is very common in real life as travelling from one place to another always use positive time unit(s). Try Dijkstra(0) on one of the Example Graphs: CP4 4.16 shown above.

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Dijkstra's algorithm maintains a set R(esolved) — other literature use set S(olved) but set S and source vertex s are too close when pronounced — of vertices whose final shortest path weights have been determined. Initially R = {}, empty.


Then, it repeatedly selects vertex u in {V\R} (can also be written as {V-R}) with the minimum shortest path estimate (the first vertex selected is u = s as initially only d[s] = 0 and the other vertices have d[u] = ∞), adds u to R, and relaxes all outgoing edges of u. Detailed proof of correctness of this Dijkstra's algorithm is usually written in typical Computer Science algorithm textbooks and we replicate it in the next few slides. For a simpler intuitive visual explanation on why this greedy strategy works, see this.


This entails the use of a Priority Queue as the shortest path estimates keep changing as more edges are processed. The choice of relaxing edges emanating from vertex with the minimum shortest path estimate first is greedy, i.e., use the "best so far", but we will see later that it can be proven (with loop invariants) that it will eventually ends up with an optimal result — if the graph has no negative weight edge.

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In Dijkstra's algorithm, each vertex will only be extracted from the Priority Queue (PQ) once. As there are V vertices, we will do this maximum O(V) times.


ExtractMin() operation runs in O(log V) whether the PQ is implemented using a Binary Min Heap or using a balanced BST like AVL Tree.


Therefore this part is O(V log V).

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Every time a vertex is processed, we relax its neighbors. In total, E edges are processed.


If by relaxing edge(u, v), we have to decrease D[v], we call the O(log V) DecreaseKey() operation in Binary Min Heap (harder to implement as C++ STL priority_queue/Python heapq/Java PriorityQueue does not support this operation efficiently yet) or simply delete the old entry and then re-insert a new entry in balanced BST like AVL Tree (which also runs in O(log V), but this is much easier to implement, just use C++ STL set/Java TreeSet — unfortunately not natively supported in Python).


Therefore, this part is O(E log V).


Thus in overall, Dijkstra's algorithm runs in O(V log V + E log V) = O((V+E) log V) time, which is much faster than the O(V×E) Bellman-Ford algorithm.

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To show the correctness of Dijkstra's algorithm on non-negative weighted graph, we need to use loop invariant: a condition which is True at the start of every iteration of the loop.


We want to show:
- Initialization: The loop invariant is true before the first iteration.
- Maintenance: If the loop invariant is true for iteration x, it remains true for iteration x+1.
- Termination: When the algorithm ends, the loop invariant helps the proof of correctness.

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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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When the input graph contains at least one negative weight edge — not necessarily negative weight cycle — Dijkstra's algorithm can produce wrong answer.


Try Dijkstra(0) on one of the Example Graphs: CP3 4.18.


At the end of the execution of Dijkstra's algorithm, vertex 4 has wrong D[4] value as the algorithm started 'wrongly' thinking that subpath 0 → 1 → 3 is the better subpath of weight 1+2 = 3, thus making D[4] = 6 after calling relax(3,4,3). However, the presence of negative weight -10 at edge 2 → 3 makes the other subpath 0 → 2 → 3 eventually the better subpath of weight 10-10 = 0 although it started worse with path weight 10 after the first edge 0 → 2. This better D[3] = 0 is never propagated further due to the greedy nature of Dijkstra's algorithm, hence D[4] is wrong.

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Dijkstra's algorithm can also be implemented differently. The O((V+E) log V) Modified Dijkstra's algorithm can be used for directed weighted graphs that may have negative weight edges but no negative weight cycle.


Such input graph appears in some practical cases, e.g., travelling using an electric car that has battery and our objective is to find a path from source vertex s to another vertex that minimizes overall battery usage. As usual, during acceleration (or driving on flat/uphill road), the electric car uses (positive) energy from the battery. However, during braking (or driving on downhill road), the electric car recharges (or use negative) energy to the battery. There is no negative weight cycle due to kinetic energy loss.


For example, try ModifiedDijkstra(0) on one of the Example Graphs: CP3 4.18 that has troubled the original version of Dijkstra's algorithm (see previous slide).

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The key idea is the 'usage modification' done to C++ STL priority_queue/Python heapq/Java PriorityQueue to allow it to perform the required 'DecreaseKey' operation efficiently, i.e., in O(log V) time.


The technique is called 'Lazy Update' where we leave the 'outdated/weaker/bigger-valued information' in the Min Priority Queue instead of deleting it straight-away. As the items are ordered from smaller values to bigger values in a Min PQ, we are guaranteeing ourself that we will encounter the smallest/most-up-to-date item first before encountering the weaker/outdated item(s) later - which by then can be easily ignored.

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On non-negative weighted graphs, the behavior of Modified Dijkstra's implementation is exactly the same as the Original Dijkstra's so we can use the same time complexity analysis of O((V+E) log V).


PS: We note that when we use the Modified Dijkstra's algorithm, there can be more items (up to E) in the Priority Queue than if we use the Original Dijkstra's algorithm (up to V). However, since O(log E) = O(log V^2) = O(2 log V) = O(log V), we still treat the Priority Queue operations as O(log V).


However, if the graph has at least one negative weight edge, the analysis is harder.

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When the input graph contains at least one negative weight edge but no negative weight cycle — the modified Dijkstra's algorithm produces correct answer.


Try ModifiedDijkstra(0) on one of the Example Graphs: CP3 4.18 that causes problem for Dijkstra(0).


At the end of the execution of ModifiedDijkstra's algorithm, vertex 4 has correct D[4] value as although the modified Dijkstra's algorithm also started 'wrongly' thinking that subpath 0 → 1 → 3 is the better subpath of weight 1+2 = 3, thus making D[4] = 6 after calling relax(3,4,3). Here, the modified Dijkstra's algorithm continues propagating D[3] = 0 after it founds out that the other subpath 0 → 2 → 3 is eventually the better subpath of weight 10-10 = 0. Hence D[4] is eventually correct again. However, this is at the expense of potentially running (much more) operations than O((V+E) log V).

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Unfortunately, running ModifiedDijkstra(0) on the graph with negative weight cycle as shown on one of the Example Graphs: CP3 4.17 above will cause an endless loop (the animation is very long but we limit the number of loop to be 100 edges processed so your web browser will not hang).

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Try ModifiedDijkstra(0) on the extreme corner case above that is very hard to derive without proper understanding of this algorithm and was part of Asia Pacific Informatics Olympiad (APIO) 2013 task set by Steven.


The Modified Dijkstra's algorithm will terminate with correct answer, but only after running exponential number of operations (each carefully constructed triangle raises the number of required operations by another power of two). Thus we cannot prematurely terminate Modified Dijkstra's in this worst case input situation.


However, such extreme corner case is rare and thus in practice, Modified Dijkstra's algorithm can be used on directed graphs that have some negative weighted edges as long as the graph has no negative weight cycle reachable from the source vertex s.

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The O(V) Depth-First Search (DFS) algorithm can solve special case of SSSP problem, i.e. when the input graph is a (weighted) Tree.


In a Tree, there is only one unique and acylic path that connects two distinct vertices. Thus the unique path that connects the source vertex s to any another vertex uV is actually also the shortest path. For example, try DFS(0) on the Tree above.


Notice that for a (weighted) Tree, we can also use BFS. For example, try BFS(0) on the same Tree above.


Discussion: Why DFS (and also BFS) runs in O(V) instead of O(V+E) if the input is a (weighted) Tree?

🕑

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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DFS will very likely produce wrong answer when run on any other graph that is not a Tree. We will display a warning message for such cases although we do not prevent you from trying this feature for pedagogical purpose.


For example, try DFS(0) on the general graph above and you will see that vertex {4} will have wrong D[4] value (and also wrong p[4] value) as DFS(0) goes deep 0 → 1 → 3 → 4 first, backtrack all the way to vertex 0 and eventually visit 0 → 2 but edge 2 → 4 cannot be processed as vertex 4 has been visited by DFS earlier.

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The O(V+E) Dynamic Programming algorithm can solve special case of SSSP problem, i.e. when the input graph is a Directed Acyclic Graph (DAG) thus we can find at least one topological order of the DAG and process the edge relaxation according to this topological order.


For example, try DP(0) on the example DAG above. First, it computes one (there are other) possible topological order using either the O(V+E) DFS or the BFS/Kahn's algorithm outlined in Graph Traversal module. For example, assume one topological order is {0,2,1,3,4,5}. Then, it relaxes the outgoing edges of vertices listed in that topological order. After just one O(V+E) pass, we will have correct D[u] values ∀uV.

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On the Modified Dijkstra's killer example shown above, DP(0) works fast as the graph is actually a DAG, albeit having negative weight edge. As the graph is a DAG, there will not be any negative weight cycle to worry about.


However, DP will not work for any non DAG as non DAG contains at least one cycle and thus no topological order can be found within that cycle.

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DP algorithm for solving SSSP on DAG is also called one-pass Bellman-Ford algorithm as it replaces the outermost V-1 loop (we do not know the correct order so we just repeat until the maximum possible) with just one topological order pass (we know that this is (one of) the correct order(s) of this DAG).


Compare DP(0) (relax E edges just once — according to topological order of its vertices) versus BellmanFord(0) (relax E edges in random order, V-1 times) on the same example DAG above.

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We have lots of other stuffs on top of this basic explanation of SSSP algorithms for SSSP problems.


Meanwhile, you are allowed to use/modify our implementation code for Bellman-Ford/Bellman-Ford-Moore/Dijkstra's Algorithms:
bellman_ford.cpp/bellman_ford_moore.cpp/dijkstra.cpp
bellman_ford.java/bellman_ford_moore.java/dijkstra.java
bellman_ford.py/bellman_ford_moore.py/dijkstra.py
bellman_ford.ml/bellman_ford_moore.ml/dijkstra.ml

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For a few more interesting questions about this SSSP problem and its various algorithms, please practice on SSSP 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 (after clearing the other pre-requisites modules) 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 the correct SSSP algorithm: Kattis - hidingplaces and Kattis - shortestpath1.


Try to solve them and then try the many more interesting twists/variants of this interesting SSSP problem.


Advertisement: Buy Competitive Programming textbook to read more on this interesting problem.


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.

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About

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.

Team

Project Leader & Advisor (Jul 2011-present)
Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

Undergraduate Student Researchers 1
CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Final Year Project/UROP students 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

Undergraduate Student Researchers 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

Final Year Project/UROP students 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

Undergraduate Student Researchers 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Final Year Project/UROP students 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.

Acknowledgements
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.

Terms of use

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.

Privacy Policy

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.