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318 lines
10 KiB
Markdown
318 lines
10 KiB
Markdown
---
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id: dijkstra
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slug: /paths/bf-to-astar/dijkstra
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title: Dijkstra's algorithm
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description: |
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Moving from Bellman-Ford into the Dijsktra's algorithm.
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tags:
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- cpp
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- dynamic programming
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- greedy
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- dijkstra
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last_update:
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date: 2024-01-03
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---
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## Intro
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Let's rewind back to the small argument in the previous post about the fact that
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we can safely bound the amount of iterations with relaxations being done.
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We have said that assuming the worst-case scenario (bad order of relaxations) we
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**need** at most $\vert V \vert - 1$ iterations over all edges. We've used that
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to our advantage to _bound_ the iterations instead of the `do-while` loop that
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was a risk given the possibility of the infinite loop (when negative loops are
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present in the graph).
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:::tip
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We could've possibly used both _boolean flag_ to denote that some relaxation has
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happened and the upper bound of iterations, for some graphs that would result in
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faster termination.
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Using only the upper bound we try to relax edges even though we can't.
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:::
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Now the question arises, could we leverage this somehow in a different way? What
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if we used it to improve the algorithm instead of just bounding the iterations?
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Would that be even possible?
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**Yes, it would!** And that's when _Dijkstra's algorithm_ comes in.
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## Dijkstra's algorithm
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I'll start with a well-known meme about Dijkstra's algorithm:
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And then follow up on that with the actual backstory from Dijkstra himself:
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> What is the shortest way to travel from Rotterdam to Groningen, in general:
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> from given city to given city. It is the algorithm for the shortest path,
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> which I designed in about twenty minutes. One morning I was shopping in
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> Amsterdam with my young fiancée, and tired, we sat down on the café terrace to
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> drink a cup of coffee and I was just thinking about whether I could do this,
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> and I then designed the algorithm for the shortest path. As I said, it was
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> a twenty-minute invention. In fact, it was published in '59, three years
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> later. The publication is still readable, it is, in fact, quite nice. One of
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> the reasons that it is so nice was that I designed it without pencil and
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> paper. I learned later that one of the advantages of designing without pencil
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> and paper is that you are almost forced to avoid all avoidable complexities.
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> Eventually, that algorithm became to my great amazement, one of the
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> cornerstones of my fame.
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>
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> — Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the
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> ACM, 2001
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:::caution Precondition
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As our own naïve algorithm, Dijkstra's algorithm has a precondition that places
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a requirement of _no edges with negative weights_ in the graph. This
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precondition is required because of the nature of the algorithm that requires
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monotonically non-decreasing changes in the costs of shortest paths.
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:::
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## Short description
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Let's have a brief look at the pseudocode taken from the Wikipedia:
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```
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function Dijkstra(Graph, source):
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for each vertex v in Graph.Vertices:
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dist[v] ← INFINITY
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prev[v] ← UNDEFINED
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add v to Q
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dist[source] ← 0
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while Q is not empty:
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u ← vertex in Q with min dist[u]
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remove u from Q
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for each neighbor v of u still in Q:
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alt ← dist[u] + Graph.Edges(u, v)
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if alt < dist[v]:
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dist[v] ← alt
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prev[v] ← u
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return dist[], prev[]
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```
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Dijkstra's algorithm works in such way that it always tries to find the shortest
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paths from a vertex to which it already has a shortest path. This may result in
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finding the shortest path to another vertex, or at least some path, till further
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relaxation.
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Given that we need to **always** choose the _cheapest_ vertex, we can use a min
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heap to our advantage which can further improve the time complexity of the
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algorithm.
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## Used techniques
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This algorithm leverages the _dynamic programming_ technique that has already
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been mentioned with regards to the _Bellman-Ford_ algorithm and also _greedy_
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technique. Let's talk about them both!
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_Dynamic programming_ technique comes from the fact that we are continuously
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building on top of the shortest paths that we have found so far. We slowly build
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the shortest paths from the given source vertex to all other vertices that are
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reachable.
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_Greedy_ technique is utilized in such way that Dijkstra's algorithm always
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improves the shortest paths from the vertex that is the closest, i.e. it tries
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extending the shortest path to some vertex by appending an edge, such extended
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path may (or may not) be the shortest path to another vertex.
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:::tip Greedy algorithms
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_Greedy algorithms_ are algorithms that choose the most optimal action
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**at the moment**.
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:::
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The reason why the algorithm requires no edges with negative weights comes from
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the fact that it's greedy. By laying the requirement of non-negative weights in
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the graph we are guaranteed that at any given moment of processing outgoing
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edges from a vertex, we already have a shortest path to the given vertex. This
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means that either this is the shortest path, or there is some other vertex that
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may have a higher cost, but the outgoing edge compensates for it.
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## Implementation
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Firstly we need to have some priority queue wrappers. C++ itself offers
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functions that can be used for maintaining max heaps. They also have generalized
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version with any ordering, in our case we need reverse ordering, because we need
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the min heap.
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```cpp
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using pqueue_item_t = std::pair<int, vertex_t>;
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using pqueue_t = std::vector<pqueue_item_t>;
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auto pushq(pqueue_t& q, pqueue_item_t v) -> void {
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q.push_back(v);
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std::push_heap(q.begin(), q.end(), std::greater<>{});
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}
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auto popq(pqueue_t& q) -> std::optional<pqueue_item_t> {
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if (q.empty()) {
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return {};
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}
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std::pop_heap(q.begin(), q.end(), std::greater<>{});
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pqueue_item_t top = q.back();
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q.pop_back();
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return std::make_optional(top);
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}
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```
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And now we can finally move to the actual implementation of the Dijkstra's
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algorithm:
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```cpp
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auto dijkstra(const graph& g, const vertex_t& source)
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-> std::vector<std::vector<int>> {
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// make sure that ‹source› exists
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assert(g.has(source));
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// initialize the distances
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std::vector<std::vector<int>> distances(
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g.height(), std::vector(g.width(), graph::unreachable()));
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// initialize the visited
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std::vector<std::vector<bool>> visited(g.height(),
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std::vector(g.width(), false));
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// ‹source› destination denotes the beginning where the cost is 0
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auto [sx, sy] = source;
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distances[sy][sx] = 0;
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pqueue_t priority_queue{std::make_pair(0, source)};
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std::optional<pqueue_item_t> item{};
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while ((item = popq(priority_queue))) {
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auto [cost, u] = *item;
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auto [x, y] = u;
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// we have already found the shortest path
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if (visited[y][x]) {
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continue;
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}
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visited[y][x] = true;
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for (const auto& [dx, dy] : DIRECTIONS) {
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auto v = std::make_pair(x + dx, y + dy);
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auto cost = g.cost(u, v);
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// if we can move to the cell and it's better, relax¹ it and update queue
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if (cost != graph::unreachable() &&
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distances[y][x] + cost < distances[y + dy][x + dx]) {
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distances[y + dy][x + dx] = distances[y][x] + cost;
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pushq(priority_queue, std::make_pair(distances[y + dy][x + dx], v));
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}
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}
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}
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return distances;
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}
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```
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## Time complexity
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The time complexity of Dijkstra's algorithm differs based on the backing data
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structure.
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The original implementation doesn't leverage the heap which results in
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repetitive _look up_ of the “closest” vertex, hence we get the following
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worst-case time complexity in the _Bachmann-Landau_ notation:
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$$
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\Theta(\vert V \vert^2)
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$$
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If we turn our attention to the backing data structure, we always want the
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“cheapest” vertex, that's why we can use the min heap, given that we use
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Fibonacci heap we can achieve the following amortized time complexity:
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$$
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\mathcal{O}(\vert E \vert + \vert V \vert \cdot \log{\vert V \vert})
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$$
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:::tip Fibonacci heap
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Fibonacci heap is known as the heap that provides $\Theta(1)$ **amortized**
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insertion and $\mathcal{O}(\log{n})$ **amortized** removal of the top (either
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min or max).
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:::
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## Running the Dijkstra
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Let's run our code:
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```
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Normal cost: 1
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Vortex cost: 5
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Graph:
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#############
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#..#..*.*.**#
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##***.....**#
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#..########.#
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#...###...#.#
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#..#...##.#.#
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#..#.*.#..#.#
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#D...#....#.#
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########*.*.#
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#S..........#
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#############
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[Finite BF] Cost: 22
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[Bellman-Ford] Cost: 22
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[Dijkstra] Cost: 22
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```
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OK, so it seems to be working just fine. Now the question arises:
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> What happens when we have negative weights in our graph?
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## Busting the myth about looping Dijkstra
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One of the very common misconception about Dijkstra's algorithm is that it loops
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infinitely when you have negative weights or loops in the graph. Well, if we use
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our _propelling vortices_, not only we have the negative weights, but also the
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negative loops. Let's run our code! Our first naïve approach was actually
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looping:
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```
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Normal cost: 1
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Vortex cost: -1
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Graph:
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#############
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#..#..*.*.**#
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##***.....**#
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#..########.#
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#...###...#.#
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#..#...##.#.#
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#..#.*.#..#.#
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#D...#....#.#
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########*.*.#
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#S..........#
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#############
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[Finite BF] Cost: -240
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[Bellman-Ford] Found a negative loop
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[Bellman-Ford] Cost: -240
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[Dijkstra] Cost: 14
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```
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Well, it definitely doesn't loop. How much does `14` make sense is a different
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matter.
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:::info Variations
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There are multiple variations of the Dijkstra's algorithm. You **can** implement
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it in such way that with negative weights or loops it loops infinitely, but it
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can be countered. In our case we keep the track of the vertices that already got
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a shortest path established via the `visited`, that's how even multiple entries
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for one vertex in the heap are not an issue.
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:::
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## Summary
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Now we have an algorithm for finding the shortest path that is faster than our
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original naïve brute-force or Bellman-Ford. However we need to keep in mind its
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requirement of no negative weights for correct functioning.
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You can also see how we used our thought process of figuring out the worst-case
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time complexity for the naïve or Bellman-Ford algorithm to improve the original
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path-finding algorithms.
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