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Choose by edge weights and output
First decide whether a route is shortest by number of edges or by total weight. Then decide whether you need distances from one source or between all pairs of vertices. The table summarizes the standard choices. Bounds are theoretical, not results from a shared benchmark; actual runtime depends on graph structure and implementation.
| Algorithm | Task and edge-weight condition | Typical time bound | Main caveat |
|---|---|---|---|
| BFS | Single source; unweighted graph | O(V + E) | Minimizes edge count, not arbitrary weighted cost. |
| 0–1 BFS | Single source; every weight is 0 or 1 | O(E) | Requires the strict 0-or-1 weight restriction. |
| Dijkstra | Single source; all weights are nonnegative | O(V² + E) with simple selection; commonly O(E log V) with a heap on sparse graphs | Negative weights invalidate its correctness guarantee. |
| Bellman–Ford | Single source; negative edges allowed | O(VE) worst case | A source-reachable negative cycle prevents finite shortest distances for affected vertices. |
| Floyd–Warshall | All pairs; negative edges allowed if no relevant negative cycle | O(V³) time; O(V²) space | Negative cycles invalidate affected pair answers; cubic work and a full matrix can be costly. |
Here V is the number of vertices and E the number of edges. Sparse-graph heap bounds are commonly expressed as O(E log V); the precise implementation and graph representation can affect the bound.
Unweighted graphs: use BFS
In an unweighted graph, every edge has equal cost, so minimizing total cost is the same as minimizing the number of edges. Breadth-first search (BFS) explores outward from the source in layers: vertices first reached after one edge, then two, and so on. The first discovered route to a vertex therefore uses the fewest edges. BFS runs in O(V + E). See Breadth First Search.
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For an illustration, mark the source as layer 0, its neighbors as layer 1, and continue outward. Draw the predecessor edge by which each vertex was first discovered; those edges form a tree of shortest routes from the source.
Weights restricted to 0 and 1: use 0–1 BFS
When every edge weight is exactly 0 or 1, a deque adapts BFS to account for cost. On a successful relaxation, put the destination at the front of the deque if the edge costs 0, and at the back if it costs 1. This prioritizes routes that add no cost before routes that add one. The cited treatment gives O(E) time for this restricted single-source case. See 0–1 BFS.
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An illustration can label each edge 0 or 1 and show each relaxation’s deque operation: a zero-edge destination is pushed to the front, while a one-edge destination is pushed to the back. Do not use this method when other weights are possible.
Nonnegative weighted edges, one source: use Dijkstra
Dijkstra computes shortest distances from one source when every edge weight is nonnegative. Set the source distance to zero and all others to infinity. Repeatedly select the unsettled vertex with the smallest tentative distance, then relax its outgoing edges: if the route through that vertex improves a neighbor’s distance, update the distance and record the current vertex as its predecessor.
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Predecessors let you reconstruct a route after the distances are computed: start at the destination, follow predecessor links back to the source, then reverse the sequence. A simple implementation that searches linearly for the next vertex takes O(V² + E). A binary-heap implementation is commonly O(E log V) on sparse graphs. The sources explain the standard and sparse-graph variants: Dijkstra and Dijkstra on sparse graphs.
Dijkstra’s correctness depends on nonnegative weights. A negative edge can make a route to a supposedly settled vertex cheaper later, breaking the algorithm’s guarantee; use Bellman–Ford if negative weights may occur.
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Negative edges, one source: use Bellman–Ford and check cycles
Bellman–Ford allows negative edge weights. Initialize the source to zero and other distances to infinity, then scan the edges and relax reachable endpoints. With n vertices and no negative cycle reachable from the source, n − 1 full passes suffice: a shortest simple path can use at most n − 1 edges.
After those passes, scan once more. If a reachable distance can still be improved, a negative cycle is reachable from the source. Repeated travel around that cycle can keep lowering the route cost, so the cycle’s vertices—and vertices reachable from it—do not have a finite minimum distance. Bellman–Ford’s worst-case time is O(VE). See Bellman–Ford.
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SPFA is a queue-based Bellman–Ford variant that may perform better on some inputs, but its worst-case bound remains O(VE); it does not provide a guaranteed linear-time alternative.
Distances between every pair: use Floyd–Warshall
Floyd–Warshall computes all-pairs distances by maintaining a V-by-V distance matrix. For each vertex k in turn, treat k as an allowed intermediate and test whether going from i to j through k improves the current distance:
d[i][j] = min(d[i][j], d[i][k] + d[k][j])
The three nested loops take O(V³) time, and the matrix uses O(V²) space. Negative edges are allowed, but a negative cycle can make shortest-path values undefined for pairs that can reach the cycle and then leave it. When implementing the update, do not add infinity sentinels as if they represented real paths; only combine distances when both relevant subpaths exist. See Floyd–Warshall.
A practical selection checklist
- Every edge has equal cost: BFS finds routes with the fewest edges.
- Every edge costs only 0 or 1: 0–1 BFS uses a deque to prioritize zero-cost transitions.
- One source and all weights are nonnegative: Dijkstra is the standard choice; a heap is commonly suitable for sparse graphs, while simple selection can be reasonable for dense graphs.
- One source and negative edges may occur: Bellman–Ford handles them and can detect a source-reachable negative cycle.
- Distances for every pair: Floyd–Warshall is a direct option when the graph is small enough for O(V³) work and O(V²) storage.
These are asymptotic comparisons, not a universal runtime ranking: the cited references do not establish a common, reproducible benchmark across all five methods. Dijkstra dates to 1959 according to its cited reference; the Bellman–Ford reference discusses Ford’s 1956 outline and Bellman’s 1958 article, while the Floyd–Warshall reference notes 1962 publications by Floyd and Warshall and an essentially equivalent 1959 publication by Roy.
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