algorithms(paths): ‹BF to A*› (#13)

algorithms/11-paths/2024-01-01-bf-to-astar/01-bf.md

@@ -0,0 +1,573 @@
+---
+id: bf
+slug: /paths/bf-to-astar/bf
+title: BF
+description: |
+  Solving the shortest path problem with a naïve approach that turns into
+  something.
+tags:
+- cpp
+- brute force
+- bellman ford
+- dynamic programming
+last_update:
+  date: 2024-01-01
+---
+
+## Basic idea
+
+We will _ease in_ with our own algorithm to find the shortest path. We will
+start by thinking about the ways we can achieve that. If we didn't have the `*`
+cells, we could've easily run a BFS[^1] and be done with it. Maybe it is a good
+place to start, or isn't, there is only one way to find out though.
+
+_How does the BFS work?_ We know the vertex where we start and we know the
+vertex we want to find the shortest path to. Given this knowledge we
+incrementally visit all of our neighbours and we do that over and over until the
+destination is found[^2]. Could we leverage this somehow?
+
+## Naïve approach
+
+Well, we could probably start with all vertices being _unreachable_ (having the
+highest possible price) and try to improve what we've gotten so far until there
+are no improvements. That sounds fine, we shall implement this. Since we are
+going on repeat, we will name this function `bf()` as in _brute-force_, cause it
+is trying to find it the hard way:
+```cpp
+const static std::vector<vertex_t> DIRECTIONS =
+    std::vector{std::make_pair(0, 1), std::make_pair(0, -1),
+                std::make_pair(1, 0), std::make_pair(-1, 0)};
+
+auto bf(const graph& g, const vertex_t& source, const vertex_t& destination)
+    -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we need to improve the paths as long as possible
+  bool improvement_found;
+  do {
+    // reset the flag at the beginning
+    improvement_found = false;
+
+    // go through all of the vertices
+    for (int y = 0; y < g.height(); ++y) {
+      for (int x = 0; x < g.width(); ++x) {
+        // skip the cells we cannot reach
+        if (distances[y][x] == graph::unreachable()) {
+          continue;
+        }
+
+        // go through the neighbours
+        auto u = std::make_pair(x, y);
+        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
+          if (cost != graph::unreachable() &&
+              distances[y][x] + cost < distances[y + dy][x + dx]) {
+            distances[y + dy][x + dx] = distances[y][x] + cost;
+            improvement_found = true;
+          }
+        }
+      }
+    }
+  } while (improvement_found);
+
+  return distances[destination.second][destination.first];
+}
+```
+
+:::info Relaxation
+
+I have made a brief mention of the relaxation in the comment in the code. You've
+been probably thought that **relaxation of an edge** means that you found
+a better solution to the problem.
+
+In general it is an approximation technique that _reduces_ the problem of
+finding the path `u → x1 → … → xn → v` to subproblems
+`u → x1, x1 → x2, …, xn → v` such that the sum of the costs of each step is
+**minimal**.
+
+:::
+
+### Correctness
+
+_Is our solution correct?_ It appears to be correct… We have rather complicated
+map and our algorithm has finished in an instant with the following output:
+```
+Normal cost: 1
+Vortex cost: 5
+Graph:
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#D...#....#.#
+########*.*.#
+#S..........#
+#############
+Cost: 22
+```
+
+If you have a better look at the map, you will realize that the cost `22` is the
+one path skipping the `*` cells, since they cost more than going around.
+
+We can play around a bit with it. The `*` cells can be even vortices that pull
+you in with a negative price and let you _propel_ yourself out :wink: Let's
+change their cost to `-1` then. Let's check what's the fastest path to the cell.
+```
+Normal cost: 1
+Vortex cost: -1
+Graph:
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#D...#....#.#
+########*.*.#
+#S..........#
+#############
+```
+
+And we're somehow stuck… The issue comes from the fact that _spinning around_ in
+the vortices allows us to lower the cost infinitely. That's why after each
+iteration there is still a possibility to lower the cost, hence the algorithm
+doesn't finish. _What can we do about this?_
+
+:::tip
+
+This algorithm is correct as long as there are no negative loops, i.e. ways how
+to lower the cost infinitely. Therefore we can also just lay a precondition that
+requires no negative loops to be present.
+
+:::
+
+### Fixing the infinite loop
+
+Our issue lies in the fact that we can endlessly lower the cost. Such thing must
+surely happen in some kind of a loop. We could probably track the relaxations
+and once we spot repeating patterns, we know we can safely terminate with _some_
+results at least.
+
+This approach will not even work on our 2D map, let alone any graph. Problem is
+that the _negative loops_ lower the cost in **each** iteration and that results
+in lowering of the costs to the cells that are reachable from the said loops.
+That's why this problem is relatively hard to tackle, it's not that easy to spot
+the repeating patterns algorithmically.
+
+On the other hand, we can approach this from the different perspective. Let's
+assume the worst-case scenario (generalized for any graph):
+> Let $K_n$ be complete graph. Let $P$ be the shortest path from $v_1$ to $v_n$
+> such that $P$ has $n - 1$ edges, i.e. the shortest path between the two chosen
+> vertices visits all vertices (not necessarily in order) and has the lowest
+> cost.
+>
+> In such scenario assume the worst-case ordering of the relaxations (only one
+> _helpful_ relaxation per iteration). In this case, in each iteration we find
+> the next edge on our path $P$ as the last. This means that we need
+> $\vert V \vert - 1$ iterations to find the shortest path $P$.
+>
+> Because we have laid $P$ as the shortest path from $v_1$ to $v_n$ and it
+> visits all vertices, its prefixes are the shortest paths from $v_1$ to any
+> other vertex in our graph.
+>
+> Therefore, we can safely assume that any relaxation after $\vert V \vert - 1$
+> iterations, is the effect of a negative loop in the graph.
+
+_How can we leverage this?_ We will go through the edges only as many times as
+cells we have. Let's adjust the code to fix the looping:
+```cpp
+auto bf_finite(const graph& g, const vertex_t& source,
+               const vertex_t& destination) -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int y = 0; y < g.height(); ++y) {
+      for (int x = 0; x < g.width(); ++x) {
+        // skip the cells we cannot reach
+        if (distances[y][x] == graph::unreachable()) {
+          continue;
+        }
+
+        // go through the neighbours
+        auto u = std::make_pair(x, y);
+        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
+          if (cost != graph::unreachable() &&
+              distances[y][x] + cost < distances[y + dy][x + dx]) {
+            distances[y + dy][x + dx] = distances[y][x] + cost;
+          }
+        }
+      }
+    }
+  }
+
+  return distances[destination.second][destination.first];
+}
+```
+
+And we get the following result:
+```
+Normal cost: 1
+Vortex cost: -1
+Graph:
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#D...#....#.#
+########*.*.#
+#S..........#
+#############
+Cost: -236
+```
+
+The negative cost means that there is a way to _propel_ ourselves via some
+vortices. Let's adjust the cost of _vortices_ back to the original `5` and check
+whether our modified algorithm works as it did before. And it surely does yield
+the `22` as before.
+
+:::tip Refactoring
+
+You can definitely notice some _deep nesting_ in our code, to counter this
+phenomenon I will convert the looping over `x` and `y` to one variable that can
+be decomposed to `x` and `y`. It is a very common practice when working with 2D
+arrays/lists to represent them as 1D. In our case:
+
+```
+i : 0 → width * height - 1
+x = i % width
+y = i / width
+```
+
+:::
+
+## Bellman-Ford
+
+If you have ever attended any Algorithms course that had path-finding in its
+syllabus, you probably feel like you've seen the algorithm above before[^3]… And
+yes, the first algorithm I have proposed is a very dumb version of the
+_Bellman-Ford_ algorithm, it's dumb, because it loops :wink: After our “looping”
+prevention we got to the point that is almost the _Bellman-Ford_ with the one
+exception that it doesn't report whether there are any negative cycles, it just
+ends.
+
+Let's have a look at a proper implementation of the Bellman-Ford algorithm:
+```cpp
+auto bellman_ford(const graph& g, const vertex_t& source)
+    -> std::vector<std::vector<int>> {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      int y = v / g.width();
+      int x = v % g.width();
+
+      // skip the cells we cannot reach
+      if (distances[y][x] == graph::unreachable()) {
+        continue;
+      }
+
+      // go through the neighbours
+      auto u = std::make_pair(x, y);
+      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
+        if (cost != graph::unreachable() &&
+            distances[y][x] + cost < distances[y + dy][x + dx]) {
+          distances[y + dy][x + dx] = distances[y][x] + cost;
+        }
+      }
+    }
+  }
+
+  // now we check for the negative loops
+  bool relaxed = false;
+  for (int v = g.height() * g.width() - 1; !relaxed && v >= 0; --v) {
+    int y = v / g.width();
+    int x = v % g.width();
+
+    // skip the cells we cannot reach
+    if (distances[y][x] == graph::unreachable()) {
+      continue;
+    }
+
+    // go through the neighbours
+    auto u = std::make_pair(x, y);
+    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
+      if (cost != graph::unreachable() &&
+          distances[y][x] + cost < distances[y + dy][x + dx]) {
+        relaxed = true;
+        std::cerr << "Found a negative loop\n";
+        break;
+      }
+    }
+  }
+
+  return distances;
+}
+```
+
+And if we run it with our negative cost of entering vortices:
+```
+[Bellman-Ford] Found a negative loop
+[Bellman-Ford] Cost: -240
+```
+
+### On the Bellman-Ford
+
+You might be surprised that we have managed to iterate from a brute-force method
+that mindlessly tries to find a better path until there are no better paths left
+all the way to the Bellman-Ford algorithm.
+
+I always say that Bellman-Ford is a _smart_ brute-force. BF is also an algorithm
+that leverages _dynamic programming_. You might wonder how can it utilize DP if
+it is “technically” a brute-force technique. Table with the shortest distances
+is the thing that makes it DP.
+
+> I might not know the shortest path yet, but I do remember all of other paths,
+> and I can improve them, if possible.
+
+That's where the beauty of both _dynamic programming_ and _relaxing_ gets merged
+together and does its magic.
+
+Proof of the correctness of the BF is done via induction to the number of
+iterations. I would suggest to try to prove the correctness yourself and
+possibly look it up, if necessary.
+
+Also the correctness of the BF relies on the conclusion we've made when fixing
+the infinite-loop on our naïve BF solution.
+
+## Time complexity
+
+Let's have a short look at the time complexities of the presented algorithms:
+
+1. naïve approach: given that there are no negative loops, we are bound by the
+   worst-case ordering of the relaxations which results in
+   $$
+   \mathcal{O}(\vert V \vert \cdot \vert E \vert)
+   $$
+
+2. our naïve approach with the fixed count of iterations instead of the
+   `do-while` loop results in the same worst-case time complexity:
+   $$
+   \Theta(\vert V \vert \cdot \vert E \vert)
+   $$
+
+3. and finally the well-known Bellman-Ford's algorithm time complexity:
+   $$
+   \Theta(\vert V \vert \cdot \vert E \vert)
+   $$
+
+## Small refactor
+
+Since we are literally copy-pasting the body of the loops just for the sake of
+relaxing, we can factor that part out into a separate function:
+```cpp
+static auto _check_vertex(const graph& g,
+                          std::vector<std::vector<int>>& distances, int v,
+                          bool check_only = false) -> bool {
+  bool improvement_found = false;
+
+  // unpack the vertex coordinates
+  int y = v / g.width();
+  int x = v % g.width();
+
+  // skip the cells we cannot reach
+  if (distances[y][x] == graph::unreachable()) {
+    return false;
+  }
+
+  // go through the neighbours
+  auto u = std::make_pair(x, y);
+  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
+    if (cost != graph::unreachable() &&
+        distances[y][x] + cost < distances[y + dy][x + dx]) {
+      if (check_only) {
+        return true;
+      }
+
+      distances[y + dy][x + dx] = distances[y][x] + cost;
+      improvement_found = true;
+    }
+  }
+
+  return improvement_found;
+}
+```
+
+This function can be also used for checking the negative loops at the end of the
+BF by using the `check_only` parameter to signal that we just want to know if
+there would be any edge relaxed instead of performing the relaxation itself.
+
+Then we can also see the differences between the specific versions of our
+path-finding algorithms in a clear way:
+```cpp
+auto bf(const graph& g, const vertex_t& source, const vertex_t& destination)
+    -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we need to improve the paths as long as possible
+  bool improvement_found;
+  do {
+    // reset the flag at the beginning
+    improvement_found = false;
+
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      improvement_found = _check_vertex(g, distances, v) || improvement_found;
+    }
+  } while (improvement_found);
+
+  return distances[destination.second][destination.first];
+}
+
+auto bf_finite(const graph& g, const vertex_t& source,
+               const vertex_t& destination) -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      _check_vertex(g, distances, v);
+    }
+  }
+
+  return distances[destination.second][destination.first];
+}
+
+auto bellman_ford(const graph& g, const vertex_t& source)
+    -> std::vector<std::vector<int>> {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      _check_vertex(g, distances, v);
+    }
+  }
+
+  // now we check for the negative loops
+  for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+    if (_check_vertex(g, distances, v, true)) {
+      std::cerr << "[Bellman-Ford] Found a negative loop\n";
+      break;
+    }
+  }
+
+  return distances;
+}
+
+```
+
+---
+
+:::tip
+
+You might've noticed that I've been using abbreviation _BF_ interchangeably for
+both _Bellman-Ford_ and _brute-force_. If you think about the way Bellman-Ford
+algorithm works, you should realize that in the worst case it's updating the
+shortest path till there no shorter path exists, so in a sense, you could really
+consider it a brute-force algorithm.
+
+:::
+
+
+[^1]: [Breadth-first search](https://en.wikipedia.org/wiki/Breadth-first_search)
+[^2]: Of course, there are some technicalities like keeping track of the visited
+      vertices to not taint the shortest path by already visited vertices.
+[^3]: or at least you should, LOL

algorithms/11-paths/2024-01-01-bf-to-astar/02-dijkstra.md

@@ -0,0 +1,318 @@
+---
+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:
+![Dijkstra's algorithm meme](/img/algorithms/paths/bf-to-astar/dijkstra-meme.jpg)
+
+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.

algorithms/11-paths/2024-01-01-bf-to-astar/03-astar.md

@@ -0,0 +1,224 @@
+---
+id: astar
+slug: /paths/bf-to-astar/astar
+title: A* algorithm
+description: |
+  Moving from Dijkstra's algorithm into the A* algorithm.
+tags:
+- cpp
+- dynamic programming
+- astar
+last_update:
+  date: 2024-01-03
+---
+
+## Intro
+
+Let's start by the recap of what we've achieved so far:
+1. We have implemented a naïve brute-force algorithm that tries to relax paths
+   as long as there are any paths to be relaxed.
+2. Then we have fixed an issue caused by negative loops that can result in
+   a non-terminating run of our brute-force method. At this moment we have made
+   some small arguments why are bounding is enough and doesn't prevent any
+   shortest path to _not be_ discovered.
+3. Finally we have converted our bounded brute-force algorithm into the
+   Bellman-Ford algorithm.
+4. We have mentioned the worst-case time complexity of our bounded naïve
+   approach and also the Bellman-Ford algorithm. Our worst-case depended on the
+   fact that we assumed the worst possible ordering of the relaxations. However
+   we could also try to relax in the most ideal ordering which could result in a
+   faster algorithm and that's how we got to the Dijkstra's algorithm.
+
+Now the question is, could we improve the Dijkstra's algorithm to get even
+better results? And the answer is _maybe_!
+
+Dijkstra's algorithm chooses the next cheapest vertex for relaxing. This is good
+as long as there is no additional information. However, imagine a roadmap of
+some country. If you're in the middle of the map and you want to go south, it
+doesn't make much sense for you to go to the north (in the opposite direction),
+but a little bit might make sense, so that you can switch to highway and go much
+faster.
+
+The important question here is how to _influence_ the algorithm, so that it does
+choose the path that _makes more sense_ rather than the one that costs the
+least.
+
+## A* description
+
+The _A* algorithm_ can be considered a modification of Dijkstra's algorithm. The
+cost is still the same, we cannot change it, right? However when we pick the
+vertices from the heap, we can influence the order by some _heuristic_. In this
+case, we introduce a function that can suggest how feasible the vertex is.
+
+## Roadmap heuristic
+
+Let's have a look at the heuristic we could use for the roadmap example. There
+are roads (the edges) and towns (the vertices). Cost could be an average time to
+travel the road. What heuristic could we use to influence our algorithm to
+choose a better ordering of the vertices when relaxing?
+
+In the former example we've said that it doesn't make much sense to go in the
+opposite direction than our goal is… We could choose the distance from our goal
+as the heuristic, e.g. right now we're 100 km away from our goal, using this
+road makes us 50 km away and using the other road we will be 200 km away.
+
+## Heuristic for our map
+
+Our map is a bit simpler, but we can use a very similar principle. We will use
+the _Manhattan distance_, which is defined in a following way:
+$$
+\vert x_a - x_b \vert + \vert y_a - y_b \vert
+$$
+
+Since we cannot move in diagonals, it makes sense to maintain the distance in
+the actual steps from the goal.
+
+## Passing the heuristic
+
+In our case, when we're using C++, we can just template the function that will
+calculate the shortest path and pass the heuristic as a parameter.
+
+## Implementation
+
+Actual implementation is very easy once we have the Dijkstra's algorithm:
+```cpp
+auto astar(const graph& g, const vertex_t& source, const auto& h)
+    -> 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 + h(source), 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] + h(v), v));
+      }
+    }
+  }
+
+  return distances;
+}
+```
+
+## Running on our map
+
+For this algorithm I will also show the example of a call:
+```cpp
+distances = astar(g, std::make_pair(1, 9), [](const auto& u) {
+  auto [x, y] = u;
+  return std::abs(1 - x) + std::abs(7 - y);
+});
+std::cout << "[A*] Cost: " << distances[7][1] << "\n";
+```
+
+First argument to the function is the graph itself. Second argument is the
+source vertex where we start. And finally the lambda returns
+_Manhattan distance_ to the goal.
+
+And we get the following result:
+```
+Normal cost: 1
+Vortex cost: 5
+Graph:
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#D...#....#.#
+########*.*.#
+#S..........#
+#############
+[Finite BF] Cost: 22
+[Bellman-Ford] Cost: 22
+[Dijkstra] Cost: 22
+[A*] Cost: 22
+```
+
+## Comparison
+
+Now you may wonder how does it compare to the previous algorithms. Supposedly it
+should be faster. Let's add counters and debugging output when we update
+distance to our goal. And now if we run our code, we get the following output:
+```
+Normal cost: 1
+Vortex cost: 5
+Graph:
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#D...#....#.#
+########*.*.#
+#S..........#
+#############
+Relaxing path to goal in 40. relaxation
+Relaxing path to goal in 68. relaxation
+Relaxing path to goal in 89. relaxation
+[Finite BF] Cost: 22
+Relaxing path to goal in 40. relaxation
+Relaxing path to goal in 68. relaxation
+Relaxing path to goal in 89. relaxation
+[Bellman-Ford] Cost: 22
+Relaxing path to goal in 41. iteration
+[Dijkstra] Cost: 22
+Relaxing path to goal in 31. iteration
+[A*] Cost: 22
+```
+
+From the output we can easily deduce that for both brute-force and Bellman-Ford,
+which are in our case identical, we actually relax three times and for the last
+time in the 89th iteration.
+
+Dijkstra's algorithm manages to find the shortest path to our goal already in
+the 41st iteration.
+
+And finally after introducing some heuristic, we could find the shortest path
+in the 31st iteration.
+
+:::danger
+
+Please keep in mind that choosing bad heuristic can actually lead to worse
+results than using no heuristic at all.
+
+:::
+
+## Summary
+
+And there we have it. We have made our way from the brute-force algorithm all
+the way to more optimal ones. Hopefully we could notice how the small
+improvements of the already existing algorithms made them much better.

algorithms/11-paths/2024-01-01-bf-to-astar/index.md

@@ -0,0 +1,171 @@
+---
+id: index
+slug: /paths/bf-to-astar
+title: From BF to A*
+description: |
+  Figuring out shortest-path problem from the BF to the A* algorithm.
+tags:
+- cpp
+- brute force
+- bellman ford
+- dynamic programming
+- dijkstra
+- greedy
+- a star
+last_update:
+  date: 2024-01-01
+---
+
+## Intro
+
+We will delve into the details and ideas of the most common path-finding
+algorithms. For the purpose of demonstrating some “features” of the improved
+algorithms, we will use a 2D map with some rules that will allow us to show cons
+and pros of the shown algorithms.
+
+Let's have a look at the example map:
+```
+#############
+#..#..*.*.**#
+##***.....**#
+#..########.#
+#...###...#.#
+#..#...##.#.#
+#..#.*.#..#.#
+#....#....#.#
+########*.*.#
+#...........#
+#############
+```
+
+We can see three different kinds of cells:
+1. `#` which represent walls, that cannot be entered at all
+2. `*` which represent vortices that can be entered at the cost of 5 coins
+3. `.` which represent normal cells that can be entered for 1 coin (which is the
+   base price of moving around the map)
+
+Let's dissect a specific position on the map to get a better grasp of the rules:
+```
+ .
+#S*
+ .
+```
+We are standing in the cell marked with `S` and we have the following options
+* move to the north (`.`) with the cost of 1 coin,
+* move to the west (`#`) **is not** allowed because of the wall,
+* move to the east (`*`) is allowed with the cost of 5 coins, and finally
+* move to the south (`.`) with the cost of 1 coin.
+
+:::info
+
+Further on I will follow the same scheme for marking cells with an addition of
+`D` to denote the _destination_ to which we will be finding the shortest path.
+
+:::
+
+## Boilerplate
+
+For working with this map I have prepared a basic structure for the graph in C++
+that will abstract some of the internal workings of our map, namely:
+* remembers the costs of moving around
+* provides a simple function that returns price for moving **directly** between
+  two positions on the map
+* allows us to print the map out, just in case we'd need some adjustments to be
+  made
+
+We can see the `graph` header here:
+```cpp
+#ifndef _GRAPH_HPP
+#define _GRAPH_HPP
+
+#include <cmath>
+#include <limits>
+#include <ostream>
+#include <utility>
+#include <vector>
+
+using vertex_t = std::pair<int, int>;
+
+struct graph {
+  graph(const std::vector<std::vector<char>>& map)
+      : map(map),
+        _height(static_cast<int>(map.size())),
+        _width(map.empty() ? 0 : static_cast<int>(map[0].size())) {}
+
+  static auto unreachable() -> int { return UNREACHABLE; }
+  static auto normal_cost() -> int { return NORMAL_COST; }
+  static auto vortex_cost() -> int { return VORTEX_COST; }
+
+  auto cost(const vertex_t& u, const vertex_t& v) const -> int {
+    auto [ux, uy] = u;
+    auto [vx, vy] = v;
+
+    auto hd = std::abs(ux - vx) + std::abs(uy - vy);
+    switch (hd) {
+      // ‹u = v›; staying on the same cell
+      case 0:
+        return 0;
+      // ‹u› and ‹v› are neighbours
+      case 1:
+        break;
+      // ‹u› and ‹v› are not neighbouring cells
+      default:
+        return UNREACHABLE;
+    }
+
+    // boundary check
+    if (vy < 0 || vy >= _height || vx < 0 || vx >= _width) {
+      return UNREACHABLE;
+    }
+
+    switch (map[vy][vx]) {
+      case '#':
+        return UNREACHABLE;
+      case '*':
+        return VORTEX_COST;
+      default:
+        return NORMAL_COST;
+    }
+  }
+
+  auto width() const -> int { return _width; }
+  auto height() const -> int { return _height; }
+  auto has(const vertex_t& v) const -> bool {
+    auto [x, y] = v;
+    return (0 <= y && y < _height) && (0 <= x && x < _width);
+  }
+
+  friend std::ostream& operator<<(std::ostream& os, const graph& g);
+
+ private:
+  std::vector<std::vector<char>> map;
+  int _height, _width;
+
+  const static int UNREACHABLE = std::numeric_limits<int>::max();
+  // XXX: modify here to change the price of entering the vortex
+  const static int VORTEX_COST = 5;
+  const static int NORMAL_COST = 1;
+};
+
+std::ostream& operator<<(std::ostream& os, const graph& g) {
+  for (const auto& row : g.map) {
+    for (const char cell : row) {
+      os << cell;
+    }
+    os << "\n";
+  }
+
+  return os;
+}
+
+#endif /* _GRAPH_HPP */
+```
+
+:::info Source code
+
+You can find all the source code referenced in this series
+[here](pathname:///files/algorithms/paths/bf-to-astar.tar.gz).
+
+:::
+
+Let's finally start with some algorithms!

static/files/algorithms/paths/bf-to-astar/astar.hpp

@@ -0,0 +1,59 @@
+#ifndef _ASTAR_HPP
+#define _ASTAR_HPP
+
+#include <algorithm>
+#include <cassert>
+#include <functional>
+#include <optional>
+#include <utility>
+#include <vector>
+
+#include "graph.hpp"
+
+auto astar(const graph& g, const vertex_t& source, const auto& h)
+    -> 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 + h(source), 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] + h(v), v));
+      }
+    }
+  }
+
+  return distances;
+}
+
+#endif /* _ASTAR_HPP */

static/files/algorithms/paths/bf-to-astar/bf.hpp

@@ -0,0 +1,136 @@
+#ifndef _BF_HPP
+#define _BF_HPP
+
+#include <cassert>
+#include <iostream>
+#include <utility>
+#include <vector>
+
+#include "graph.hpp"
+
+static auto _check_vertex(const graph& g,
+                          std::vector<std::vector<int>>& distances, int v,
+                          bool check_only = false) -> bool {
+  bool improvement_found = false;
+
+  // unpack the vertex coordinates
+  int y = v / g.width();
+  int x = v % g.width();
+
+  // skip the cells we cannot reach
+  if (distances[y][x] == graph::unreachable()) {
+    return false;
+  }
+
+  // go through the neighbours
+  auto u = std::make_pair(x, y);
+  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
+    if (cost != graph::unreachable() &&
+        distances[y][x] + cost < distances[y + dy][x + dx]) {
+      if (check_only) {
+        return true;
+      }
+
+      distances[y + dy][x + dx] = distances[y][x] + cost;
+      improvement_found = true;
+    }
+  }
+
+  return improvement_found;
+}
+
+auto bf(const graph& g, const vertex_t& source, const vertex_t& destination)
+    -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we need to improve the paths as long as possible
+  bool improvement_found;
+  do {
+    // reset the flag at the beginning
+    improvement_found = false;
+
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      improvement_found = _check_vertex(g, distances, v) || improvement_found;
+    }
+  } while (improvement_found);
+
+  return distances[destination.second][destination.first];
+}
+
+auto bf_finite(const graph& g, const vertex_t& source,
+               const vertex_t& destination) -> int {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // ‹destination› must be within the bounds
+  assert(g.has(destination));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      _check_vertex(g, distances, v);
+    }
+  }
+
+  return distances[destination.second][destination.first];
+}
+
+auto bellman_ford(const graph& g, const vertex_t& source)
+    -> std::vector<std::vector<int>> {
+  // ‹source› must be within the bounds
+  assert(g.has(source));
+
+  // we need to initialize the distances
+  std::vector<std::vector<int>> distances(
+      g.height(), std::vector(g.width(), graph::unreachable()));
+
+  // ‹source› destination denotes the beginning where the cost is 0
+  auto [sx, sy] = source;
+  distances[sy][sx] = 0;
+
+  // now we only iterate as many times as cells that we have
+  for (int i = g.height() * g.width(); i > 0; --i) {
+    // go through all of the vertices
+    for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+      _check_vertex(g, distances, v);
+    }
+  }
+
+  // now we check for the negative loops
+  for (int v = g.height() * g.width() - 1; v >= 0; --v) {
+    if (_check_vertex(g, distances, v, true)) {
+      std::cerr << "[Bellman-Ford] Found a negative loop\n";
+      break;
+    }
+  }
+
+  return distances;
+}
+
+#endif /* _BF_HPP */

static/files/algorithms/paths/bf-to-astar/dijkstra.hpp

@@ -0,0 +1,58 @@
+#ifndef _DIJKSTRA_HPP
+#define _DIJKSTRA_HPP
+
+#include <algorithm>
+#include <cassert>
+#include <functional>
+#include <optional>
+#include <utility>
+#include <vector>
+
+#include "graph.hpp"
+
+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;
+}
+
+#endif /* _DIJKSTRA_HPP */

static/files/algorithms/paths/bf-to-astar/graph.hpp

@@ -0,0 +1,108 @@
+#ifndef _GRAPH_HPP
+#define _GRAPH_HPP
+
+#include <cmath>
+#include <limits>
+#include <ostream>
+#include <utility>
+#include <vector>
+
+using vertex_t = std::pair<int, int>;
+
+struct graph {
+  graph(const std::vector<std::vector<char>>& map)
+      : map(map),
+        _height(static_cast<int>(map.size())),
+        _width(map.empty() ? 0 : static_cast<int>(map[0].size())) {}
+
+  static auto unreachable() -> int { return UNREACHABLE; }
+  static auto normal_cost() -> int { return NORMAL_COST; }
+  static auto vortex_cost() -> int { return VORTEX_COST; }
+
+  auto cost(const vertex_t& u, const vertex_t& v) const -> int {
+    auto [ux, uy] = u;
+    auto [vx, vy] = v;
+
+    auto md = std::abs(ux - vx) + std::abs(uy - vy);
+    switch (md) {
+      // ‹u = v›; staying on the same cell
+      case 0:
+        return 0;
+      // ‹u› and ‹v› are neighbours
+      case 1:
+        break;
+      // ‹u› and ‹v› are not neighbouring cells
+      default:
+        return UNREACHABLE;
+    }
+
+    // boundary check
+    if (vy < 0 || vy >= _height || vx < 0 || vx >= _width) {
+      return UNREACHABLE;
+    }
+
+    switch (map[vy][vx]) {
+      case '#':
+        return UNREACHABLE;
+      case '*':
+        return VORTEX_COST;
+      default:
+        return NORMAL_COST;
+    }
+  }
+
+  auto width() const -> int { return _width; }
+  auto height() const -> int { return _height; }
+  auto has(const vertex_t& v) const -> bool {
+    auto [x, y] = v;
+    return (0 <= y && y < _height) && (0 <= x && x < _width);
+  }
+
+  friend std::ostream& operator<<(std::ostream& os, const graph& g);
+
+ private:
+  std::vector<std::vector<char>> map;
+  int _height, _width;
+
+  const static int UNREACHABLE = std::numeric_limits<int>::max();
+  // XXX: modify here to change the price of entering the vortex
+  const static int VORTEX_COST = 5;
+  const static int NORMAL_COST = 1;
+};
+
+std::ostream& operator<<(std::ostream& os, const graph& g) {
+  for (const auto& row : g.map) {
+    for (const char cell : row) {
+      os << cell;
+    }
+    os << "\n";
+  }
+
+  return os;
+}
+
+const static std::vector<vertex_t> DIRECTIONS =
+    std::vector{std::make_pair(0, 1), std::make_pair(0, -1),
+                std::make_pair(1, 0), std::make_pair(-1, 0)};
+
+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);
+}
+
+#endif /* _GRAPH_HPP */

static/files/algorithms/paths/bf-to-astar/main.cpp

@@ -0,0 +1,50 @@
+#include <iostream>
+#include <string>
+#include <utility>
+#include <vector>
+
+#include "astar.hpp"
+#include "bf.hpp"
+#include "dijkstra.hpp"
+#include "graph.hpp"
+
+auto line_to_vector(const std::string& l) -> std::vector<char> {
+  return std::vector(l.begin(), l.end());
+}
+
+auto main() -> int {
+  graph g{std::vector{
+      line_to_vector(std::string("#############")),
+      line_to_vector(std::string("#..#..*.*.**#")),
+      line_to_vector(std::string("##***.....**#")),
+      line_to_vector(std::string("#..########.#")),
+      line_to_vector(std::string("#...###...#.#")),
+      line_to_vector(std::string("#..#...##.#.#")),
+      line_to_vector(std::string("#..#.*.#..#.#")),
+      line_to_vector(std::string("#....#....#.#")),
+      line_to_vector(std::string("########*.*.#")),
+      line_to_vector(std::string("#...........#")),
+      line_to_vector(std::string("#############")),
+  }};
+  std::cout << "Normal cost: " << g.normal_cost() << "\n";
+  std::cout << "Vortex cost: " << g.vortex_cost() << "\n";
+  std::cout << "Graph:\n" << g;
+
+  // finding the distances from the bottom left corner to the 2 rows above
+  auto cost = bf_finite(g, std::make_pair(1, 9), std::make_pair(1, 7));
+  std::cout << "[Finite BF] Cost: " << cost << "\n";
+
+  auto distances = bellman_ford(g, std::make_pair(1, 9));
+  std::cout << "[Bellman-Ford] Cost: " << distances[7][1] << "\n";
+
+  distances = dijkstra(g, std::make_pair(1, 9));
+  std::cout << "[Dijkstra] Cost: " << distances[7][1] << "\n";
+
+  distances = astar(g, std::make_pair(1, 9), [](const auto& u) {
+    auto [x, y] = u;
+    return std::abs(1 - x) + std::abs(7 - y);
+  });
+  std::cout << "[A*] Cost: " << distances[7][1] << "\n";
+
+  return 0;
+}

static/files/algorithms/paths/bf-to-astar/makefile

@@ -0,0 +1,8 @@
+CXX=c++
+CXXFLAGS=-std=c++20 -Wall -Wextra -g
+
+all: format
+	$(CXX) $(CXXFLAGS) main.cpp -o main
+
+format:
+	clang-format -i -style=google *.hpp *.cpp