1.0 - Negative-Weight Graphs

1.1 - Graphs without Negative Weight Edges

Let

$$p_n=v_1\rightarrow v_2\rightarrow v_3\rightarrow \cdots\rightarrow v_n$$

be any shortest path from $v_1$ to $v_n$ in a weighted graph with no negative weight edges

For all $v_i\in v_1, v_2, \cdots, v_{n-1}$, we must have that the path prefix

$$p_i=v_1\rightarrow v_2\rightarrow v_3\rightarrow\cdots\rightarrow v_i$$

is a shortest path from $v_1$ to $v_i$ and $\text{weight}(p_i)\le\text{weight}(p_n)$

$\text{distance}(v_1, v_i)\le \text{distance}(v_1, v_n)$

1.2 - Graphs with Negative-Weight Edges

• What if our graph has negative weights?

• Even if the graph has no negative weight cycles, Dijkstra’s algorithm might not relax edges in the right order.

• For example $(u,v)$ might be relaxed before the shortest path to $u$ has been found.

• Graphs with negative weights might have negative weight cycles

• Even if a graph with negative weights has no negative-weight cycles, we need to work out how to relax edges so that we are guaranteed to find shortest paths.

1.3 - Bellman-Ford Solution

🌱 Shortest paths (of length of at least $V-1$) with negative edge weights of length $n$ can be found after relaxing all vertices $n-1$ times (regardless of order of edge relaxations)

• After initialisation, we know that we have found all shortest paths that contain $\le 0$ edges

• After relaxing each edge in the graph 1 time, we have found all shortest paths that contain $\le 1$ edges

• After relaxing each edge in the graph 2 times, we have found all shortest paths that contain $\le 2$ edges.

• After relaxing each edge in the graph $V-1$ times, we have found all shortest paths that contain $\le V-1$ edges.

1.4 - Bellman-Ford Algorithm

• The Bellman-Ford algorithm uses this idea to find:
  • Single-source shortest paths on directed graphs
  • That may have negative edges
• Moreover, it:
  • Detects negative-weight cycles (reachable from the source vertex), returning false if one is found (as shortest paths are not defined valid in negative-weight cycles)

bellman_ford(G, w, s)
	// G is the graph, w is the weight function, s is the source vertex
	init_single_source(G, s)
  // Relax each edge V - 1 times to find shortest paths
	for i = 1 to |G.V| - 1
			for each edge (u, v) ∈ G.E
					Relax(u, v, w)
  // Check for negative-weighte cycles reachable from s
	for each edge (u, v) ∈ G.E
			// If the relaxation condition is met, a negative-weight cycle
			//     is detected
			if v.d > u.d + w(u, v)
					// Negative-weight cycle detected
					return false
	return true

1.4.1 - Bellman Ford Algorithm Analysis

• $\Theta(V)$ time for initialisation
• $\Theta(VE)=\Theta(V^3)$ for the main relaxation loop
• $\Theta(E)=\Theta(V^2)$ for the negative-weight cycle checking algorithm

1.5 - Priority-First Search

• Dijkstra’s algorithm for finding single-source shortest paths and
• Prim’s algorithm for finding minimum spanning trees

are specialisations of a priority-first search

1.5.1 - What is a Priority-First Search

• In a priority-first search, vertices are visited in order of their priority.
• For each vertex $v\in G.V$, we calculate:
  • $v.\text{key}$ A vertex’s priority, relative to source vertex $s$
  • $v.\pi$ The predecessor to v, in a priority-first search tree
• The algorithm uses $Q$, a priority queue of unvisited vertices with key $v.key$

PriorityFirstSearch(G, s, ...)
    init_single_source(G, s)
    Q = G.V
    while Q != ∅
			u = extract_min(Q)
      for each vertex v in G.adj[u]
          relax(u, v, ...)
init_single_source(G, s)
    for each vertex v in G.V
        v.key = infinity
        v.pi = nil
    s.key = 0

relax(u, v, ...)
    if v.key > PRIORITY
        v.key = PRIORITY
        v.pi = u

1.5.2 - Prim’s Algorithm as a Priority-First Search

• Instantiate PRIORITY = weight(u, v)
• $G.V-Q$ is the subset of vertices in the MST under construction $(T)$
• $v.\text{key}$ is the least weight edge connecting v to T
• $v.\pi$ is the least vertex adjacent to v on that least-weight edge

At the end, the priority-first search tree is a minimum spanning tree.

1.5.3 - Dijkstra’s Algorithm

• Instantiate PRIORITY = u.key + w(u, v)
• G.V - Q is the set of vertices already visited in order of their distance from s.
• v.key is the length of the shortest path from s to v that only uses edges adjacent to visited vertices
• v.pi is the predecessor of v on the shortest path from s to v that only used edges adjacent to visited vertices

At the end, the priority-first search tree is a shortest-path tree