mirror of
https://github.com/mfocko/blog.git
synced 2024-11-24 05:51:55 +01:00
326 lines
10 KiB
Markdown
326 lines
10 KiB
Markdown
---
|
||
id: dijkstra
|
||
slug: /paths/bf-to-astar/dijkstra
|
||
title: Dijkstra's algorithm
|
||
description: |
|
||
Moving from Bellman-Ford into the Dijsktra's algorithm.
|
||
tags:
|
||
- cpp
|
||
- dynamic programming
|
||
- greedy
|
||
- dijkstra
|
||
last_update:
|
||
date: 2024-01-03
|
||
---
|
||
|
||
## Intro
|
||
|
||
Let's rewind back to the small argument in the previous post about the fact that
|
||
we can safely bound the amount of iterations with relaxations being done.
|
||
|
||
We have said that assuming the worst-case scenario (bad order of relaxations) we
|
||
**need** at most $\vert V \vert - 1$ iterations over all edges. We've used that
|
||
to our advantage to _bound_ the iterations instead of the `do-while` loop that
|
||
was a risk given the possibility of the infinite loop (when negative loops are
|
||
present in the graph).
|
||
|
||
:::tip
|
||
|
||
We could've possibly used both _boolean flag_ to denote that some relaxation has
|
||
happened and the upper bound of iterations, for some graphs that would result in
|
||
faster termination.
|
||
|
||
Using only the upper bound we try to relax edges even though we can't.
|
||
|
||
:::
|
||
|
||
Now the question arises, could we leverage this somehow in a different way? What
|
||
if we used it to improve the algorithm instead of just bounding the iterations?
|
||
Would that be even possible?
|
||
|
||
**Yes, it would!** And that's when _Dijkstra's algorithm_ comes in.
|
||
|
||
## Dijkstra's algorithm
|
||
|
||
I'll start with a well-known meme about Dijkstra's algorithm:
|
||
|
||
|
||
And then follow up on that with the actual backstory from Dijkstra himself:
|
||
|
||
> What is the shortest way to travel from Rotterdam to Groningen, in general:
|
||
> from given city to given city. It is the algorithm for the shortest path,
|
||
> which I designed in about twenty minutes. One morning I was shopping in
|
||
> Amsterdam with my young fiancée, and tired, we sat down on the café terrace to
|
||
> drink a cup of coffee and I was just thinking about whether I could do this,
|
||
> and I then designed the algorithm for the shortest path. As I said, it was
|
||
> a twenty-minute invention. In fact, it was published in '59, three years
|
||
> later. The publication is still readable, it is, in fact, quite nice. One of
|
||
> the reasons that it is so nice was that I designed it without pencil and
|
||
> paper. I learned later that one of the advantages of designing without pencil
|
||
> and paper is that you are almost forced to avoid all avoidable complexities.
|
||
> Eventually, that algorithm became to my great amazement, one of the
|
||
> cornerstones of my fame.
|
||
>
|
||
> — Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the
|
||
> ACM, 2001
|
||
|
||
:::caution Precondition
|
||
|
||
As our own naïve algorithm, Dijkstra's algorithm has a precondition that places
|
||
a requirement of _no edges with negative weights_ in the graph. This
|
||
precondition is required because of the nature of the algorithm that requires
|
||
monotonically non-decreasing changes in the costs of shortest paths.
|
||
|
||
:::
|
||
|
||
## Short description
|
||
|
||
Let's have a brief look at the pseudocode taken from the Wikipedia:
|
||
|
||
```
|
||
function Dijkstra(Graph, source):
|
||
for each vertex v in Graph.Vertices:
|
||
dist[v] ← INFINITY
|
||
prev[v] ← UNDEFINED
|
||
add v to Q
|
||
dist[source] ← 0
|
||
|
||
while Q is not empty:
|
||
u ← vertex in Q with min dist[u]
|
||
remove u from Q
|
||
|
||
for each neighbor v of u still in Q:
|
||
alt ← dist[u] + Graph.Edges(u, v)
|
||
if alt < dist[v]:
|
||
dist[v] ← alt
|
||
prev[v] ← u
|
||
|
||
return dist[], prev[]
|
||
```
|
||
|
||
Dijkstra's algorithm works in such way that it always tries to find the shortest
|
||
paths from a vertex to which it already has a shortest path. This may result in
|
||
finding the shortest path to another vertex, or at least some path, till further
|
||
relaxation.
|
||
|
||
Given that we need to **always** choose the _cheapest_ vertex, we can use a min
|
||
heap to our advantage which can further improve the time complexity of the
|
||
algorithm.
|
||
|
||
## Used techniques
|
||
|
||
This algorithm leverages the _dynamic programming_ technique that has already
|
||
been mentioned with regards to the _Bellman-Ford_ algorithm and also _greedy_
|
||
technique. Let's talk about them both!
|
||
|
||
_Dynamic programming_ technique comes from the fact that we are continuously
|
||
building on top of the shortest paths that we have found so far. We slowly build
|
||
the shortest paths from the given source vertex to all other vertices that are
|
||
reachable.
|
||
|
||
_Greedy_ technique is utilized in such way that Dijkstra's algorithm always
|
||
improves the shortest paths from the vertex that is the closest, i.e. it tries
|
||
extending the shortest path to some vertex by appending an edge, such extended
|
||
path may (or may not) be the shortest path to another vertex.
|
||
|
||
:::tip Greedy algorithms
|
||
|
||
_Greedy algorithms_ are algorithms that choose the most optimal action
|
||
**at the moment**.
|
||
|
||
:::
|
||
|
||
The reason why the algorithm requires no edges with negative weights comes from
|
||
the fact that it's greedy. By laying the requirement of non-negative weights in
|
||
the graph we are guaranteed that at any given moment of processing outgoing
|
||
edges from a vertex, we already have a shortest path to the given vertex. This
|
||
means that either this is the shortest path, or there is some other vertex that
|
||
may have a higher cost, but the outgoing edge compensates for it.
|
||
|
||
## Implementation
|
||
|
||
Firstly we need to have some priority queue wrappers. C++ itself offers
|
||
functions that can be used for maintaining max heaps. They also have generalized
|
||
version with any ordering, in our case we need reverse ordering, because we need
|
||
the min heap.
|
||
|
||
```cpp
|
||
using pqueue_item_t = std::pair<int, vertex_t>;
|
||
using pqueue_t = std::vector<pqueue_item_t>;
|
||
|
||
auto pushq(pqueue_t& q, pqueue_item_t v) -> void {
|
||
q.push_back(v);
|
||
std::push_heap(q.begin(), q.end(), std::greater<>{});
|
||
}
|
||
|
||
auto popq(pqueue_t& q) -> std::optional<pqueue_item_t> {
|
||
if (q.empty()) {
|
||
return {};
|
||
}
|
||
|
||
std::pop_heap(q.begin(), q.end(), std::greater<>{});
|
||
pqueue_item_t top = q.back();
|
||
q.pop_back();
|
||
|
||
return std::make_optional(top);
|
||
}
|
||
```
|
||
|
||
And now we can finally move to the actual implementation of the Dijkstra's
|
||
algorithm:
|
||
|
||
```cpp
|
||
auto dijkstra(const graph& g, const vertex_t& source)
|
||
-> std::vector<std::vector<int>> {
|
||
// make sure that ‹source› exists
|
||
assert(g.has(source));
|
||
|
||
// initialize the distances
|
||
std::vector<std::vector<int>> distances(
|
||
g.height(), std::vector(g.width(), graph::unreachable()));
|
||
|
||
// initialize the visited
|
||
std::vector<std::vector<bool>> visited(g.height(),
|
||
std::vector(g.width(), false));
|
||
|
||
// ‹source› destination denotes the beginning where the cost is 0
|
||
auto [sx, sy] = source;
|
||
distances[sy][sx] = 0;
|
||
|
||
pqueue_t priority_queue{std::make_pair(0, source)};
|
||
std::optional<pqueue_item_t> item{};
|
||
while ((item = popq(priority_queue))) {
|
||
auto [cost, u] = *item;
|
||
auto [x, y] = u;
|
||
|
||
// we have already found the shortest path
|
||
if (visited[y][x]) {
|
||
continue;
|
||
}
|
||
visited[y][x] = true;
|
||
|
||
for (const auto& [dx, dy] : DIRECTIONS) {
|
||
auto v = std::make_pair(x + dx, y + dy);
|
||
auto cost = g.cost(u, v);
|
||
|
||
// if we can move to the cell and it's better, relax¹ it and update queue
|
||
if (cost != graph::unreachable() &&
|
||
distances[y][x] + cost < distances[y + dy][x + dx]) {
|
||
distances[y + dy][x + dx] = distances[y][x] + cost;
|
||
pushq(priority_queue, std::make_pair(distances[y + dy][x + dx], v));
|
||
}
|
||
}
|
||
}
|
||
|
||
return distances;
|
||
}
|
||
```
|
||
|
||
## Time complexity
|
||
|
||
The time complexity of Dijkstra's algorithm differs based on the backing data
|
||
structure.
|
||
|
||
The original implementation doesn't leverage the heap which results in
|
||
repetitive _look up_ of the “closest” vertex, hence we get the following
|
||
worst-case time complexity in the _Bachmann-Landau_ notation:
|
||
|
||
$$
|
||
\Theta(\vert V \vert^2)
|
||
$$
|
||
|
||
If we turn our attention to the backing data structure, we always want the
|
||
“cheapest” vertex, that's why we can use the min heap, given that we use
|
||
Fibonacci heap we can achieve the following amortized time complexity:
|
||
|
||
$$
|
||
\mathcal{O}(\vert E \vert + \vert V \vert \cdot \log{\vert V \vert})
|
||
$$
|
||
|
||
:::tip Fibonacci heap
|
||
|
||
Fibonacci heap is known as the heap that provides $\Theta(1)$ **amortized**
|
||
insertion and $\mathcal{O}(\log{n})$ **amortized** removal of the top (either
|
||
min or max).
|
||
|
||
:::
|
||
|
||
## Running the Dijkstra
|
||
|
||
Let's run our code:
|
||
|
||
```
|
||
Normal cost: 1
|
||
Vortex cost: 5
|
||
Graph:
|
||
#############
|
||
#..#..*.*.**#
|
||
##***.....**#
|
||
#..########.#
|
||
#...###...#.#
|
||
#..#...##.#.#
|
||
#..#.*.#..#.#
|
||
#D...#....#.#
|
||
########*.*.#
|
||
#S..........#
|
||
#############
|
||
[Finite BF] Cost: 22
|
||
[Bellman-Ford] Cost: 22
|
||
[Dijkstra] Cost: 22
|
||
```
|
||
|
||
OK, so it seems to be working just fine. Now the question arises:
|
||
|
||
> What happens when we have negative weights in our graph?
|
||
|
||
## Busting the myth about looping Dijkstra
|
||
|
||
One of the very common misconception about Dijkstra's algorithm is that it loops
|
||
infinitely when you have negative weights or loops in the graph. Well, if we use
|
||
our _propelling vortices_, not only we have the negative weights, but also the
|
||
negative loops. Let's run our code! Our first naïve approach was actually
|
||
looping:
|
||
|
||
```
|
||
Normal cost: 1
|
||
Vortex cost: -1
|
||
Graph:
|
||
#############
|
||
#..#..*.*.**#
|
||
##***.....**#
|
||
#..########.#
|
||
#...###...#.#
|
||
#..#...##.#.#
|
||
#..#.*.#..#.#
|
||
#D...#....#.#
|
||
########*.*.#
|
||
#S..........#
|
||
#############
|
||
[Finite BF] Cost: -240
|
||
[Bellman-Ford] Found a negative loop
|
||
[Bellman-Ford] Cost: -240
|
||
[Dijkstra] Cost: 14
|
||
```
|
||
|
||
Well, it definitely doesn't loop. How much does `14` make sense is a different
|
||
matter.
|
||
|
||
:::info Variations
|
||
|
||
There are multiple variations of the Dijkstra's algorithm. You **can** implement
|
||
it in such way that with negative weights or loops it loops infinitely, but it
|
||
can be countered. In our case we keep the track of the vertices that already got
|
||
a shortest path established via the `visited`, that's how even multiple entries
|
||
for one vertex in the heap are not an issue.
|
||
|
||
:::
|
||
|
||
## Summary
|
||
|
||
Now we have an algorithm for finding the shortest path that is faster than our
|
||
original naïve brute-force or Bellman-Ford. However we need to keep in mind its
|
||
requirement of no negative weights for correct functioning.
|
||
|
||
You can also see how we used our thought process of figuring out the worst-case
|
||
time complexity for the naïve or Bellman-Ford algorithm to improve the original
|
||
path-finding algorithms.
|