Algorithms:
Dijkstra: work efficient but sequential with non-negative edge weights
Bellman-Ford: parallel algorithm with more work
Johnson: parallel algorithm that find shortest paths between pairs of vertices, no just single source.
Edge weight: a mapping between each edge to a real number. (When edge does not exist, then weight is infinity)
We allow negative edge weight. But this is non-trivial: There can be a cycle with negative total weight, leading to shortest path with weight -\infty. Even if we don't allow cycles, extension to allow negative weight is challenging.
Weight of the path: sum of weights in the path
Flavors of Shortest Path
Single-Pair Shortest Path: return a shortest path from a to b
Single-Source Shortest Path (SSSP): return a shortest path from s to every other vertex
All-Pairs Shortest Paths: find shortest paths between all pairs of vertices
SSSP+: SSSP but weights are non-negative
Sub-path property: any sub-path of a shortest path is itself a shortest path
See graph above: Suppose we want the shortest path from s to v. If an oracle tells us the shortest path to all vertices except v, then we only need O(|V|) to find the shortest path.
Notice that adding a constant to each path changes the shortest path, but multiplying does not.
Dijkstra’s Property: The overall shortest-path weight from s via a vertex in X directly to a neighbor in Y (in the frontier) is as short as any path from s to any vertex in Y.
Dijkstra's Algorithm: priority-first search:
Note that we calculate \min using priority queue. We are sure a path is shortest only when we pop out of queue. There can be multiple duplicated elements in the queue, but only the shortest will get visited.
Variants:
One variant checks whether u is already in X inside the relax
function, and if so does not inserts it into the priority queue. (This does not affect the asymptotic work bounds)
Another variant decreases the priority of the neighbors instead of adding duplicates to the priority queue. This requires a more powerful priority queue that supports a decreaseKey
function.
Note that if we use decreaseKey
, we can do priority queue operation in O(m + n \log n)
For enumerated graphs the cost of the tree tables could be improved by using adjacency sequences for the graph, and ephemeral or single-threaded sequences for the distance table, but priority queue operation still dominates the cost even when using decreaseKey
.
O(m\log n) = O(m \log m) since m \leq n^2
Heuristic must be:
consistent: h(u) \leq \delta(u, v) + h(v)
admissible: h(v) \leq \delta(v, t) (but we don't need this since we don't permit re-visit vertices, which sacrefices asymptotic bound: any consistent heuristic is also admissible.)
destination zero: h(t) = 0
Worst heuristic: h(v) = 0
Best heuristic: h(v) = \delta(v, t) (visits exactly the vertices on the shortest path)
Algorithm
// QUESTION: Can you tell which node is directly involved in creating negative cycles?
Costs with table:
finding the in-neighbors N_G^-(v): O(\log |V|)
access map D[u]
and w(u, v
): O(\log|V|)
reduce: O(|N_G(v)|) work, O(\log |N_G(v)|) span
Line 5 and Line 6 gives O((m+n)\log n) work, O(\log n) span
Line 9 tabulate
and reduce
requires O(n \log n) work, O(\log n) span
In total: O(mn\log n) work, O(n \log n) span
Costs with sequence: O(mn) work and O(n \log n) span.
Here is a good video
Graph Strategies:
adding vertices
adding edges
adding positive, negative, zero weights to edges
run algorithm twice
transpose the graph
Here is a good video on metric space.
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