ece250: add algs

docs/2a/ece250.md

@@ -162,3 +162,119 @@ fn sort(A: Vec) {
 ### Directed acyclic graphs
 
 a DAG is acyclic if and only if there are no **back edges** — edges from a child to an ancestor.
+
+### Bellman-Ford
+
+The Bellman-Ford algorithm allows for negative edges and detects negative cycles.
+
+```rust
+fn bf(G: Graph, s: Node) {
+	let mut distance = Vec::new(INFINITY);
+	let mut adj_list = Vec::from(G);
+	
+	distance[s] = 0;
+	
+	for i in 1..G.vertices.len()-1 {
+		for (u,v) in G.edges {
+			if distance[v] > distance[u] + adj_list[u][v] {
+				distance[v] = distance[u] + adj_list[u][v];
+			}
+		}
+	}
+	for (u, v) in G.edges {
+		if distance[v] > distance[u] + adj_list[u][v] {
+			return false;
+		}
+	}
+	return true;
+}
+```
+
+### Topological sort
+
+This is used to find the shortest path in a DAG simply by DFS.
+
+```rust
+fn shortest_path(G: Graph, s: Node) {
+	let nodes: Vec<Node> = top_sort(G);
+	let mut adj_list = G.to_adj_list();
+	let mut distance = Vec::new(INFINITY);
+	
+	for v in nodes {
+		for adjacent in adj_list[v] {
+			if distance[adjacent] > distance[v] + adjacent[v] {
+				distance[v] = distance[adjacent] + adjacent[v];
+			}
+		}
+	}
+}
+
+### Minimum spanning tree
+
+!!! definition
+    - A **cut** $(S, V-S)$ is a partition of vertices into disjoint sets $S$ and $V-S$.
+    - An edge $u,v\in E$ **crosses the cut** $(S,V-S)$ if t`he endpoints are on different sides of the cut.
+    - A cut **respects** a set of edges $A$ if and only if no edge in $A$ crosses the cut.
+    - A **light edge** is the minimum of all edges that could cross the cut. There can be more than one light edge per cut.
+
+A **spanning tree** of $G$ is a subgraph that contains all of its vertices. An MST minimises the sum of all edges in the spanning tree.
+
+To create an MST:
+
+1. Add edges from the spanning tree to an empty set, maintaining that the set is always a subset of an MST (only "safe edges" are added)
+
+The **Prim-Jarnik algorithm** grows a tree one vertex at a time. $A$ is a subset of the already computed portion of $T$, and all vertices outside $A$ have a weight of infinity if there is no edge.
+
+```rust
+// r is the start vertex
+fn create_mst_prim(G: Graph, r: Vertex) {
+	// clean all vertices
+	for vertex in G.vertices.iter_mut() {
+		vertex.min_weight = INFINITY;
+		vertex.parent = None;
+	}
+	
+	let Q = BinaryHeap::from(G.vertices); // priority queue
+	
+	while let Some(u) = Q.pop() {
+		for v in u.adjacent_vertices.iter_mut() {
+			if Q.contains(v) && v.edge_to(u).weight < v.min_weight {
+				v.min_weight = v.edge_to(u).weight;
+				Q."modify_key"(v);
+				v.parent = u;
+			}
+		}
+	}
+	
+}
+```
+
+**Kruskal's algorithm** is objectively better by relying on edges instead.
+
+```rust
+fn create_mst_kruskal(G: Graph) -> HashSet<Edge> {
+	let mut A = HashSet::new();
+	let mut S = DisjointSet::new();  // vertices set
+	
+	for v in G.vertices.iter() {
+		S.add_as_new_set(v);
+	}
+	G.edges.sort(|edge| edge.weight);
+	
+	for (from, dest) in G.edges {
+		if S.find_set_that_contains(from) != S.find_set_that_contains(dest) {
+			A.insert((from, to));
+			let X = S.pop(from);
+			let Y = S.pop(to);
+			S.insert({X.union(Y)});
+		}
+	}
+	return A;
+}
+```
+
+The time complexity is $O(E\log V)$.
+
+### All pairs shortest path
+
+Also known as an adjacency matrix extended such that each point represents the minimum distance from one edge to that other edge.