The document describes the single-source shortest paths algorithm for directed acyclic graphs (DAGs). It involves topologically sorting the vertices of the DAG, initializing distances from the source vertex s, and then relaxing edges in topologically sorted order. This guarantees that when a vertex u is relaxed, the shortest path distances from s to its neighbors will be accurate. The algorithm runs in O(V+E) time. It is used to find critical paths in PERT charts by finding the longest path after negating or reversing edge weights.

1. Single-Source Shortest Paths in a
Directed Acyclic Graph
Dr. Kiran K
Assistant Professor
Department of CSE
UVCE
Bengaluru, India.

2. Algorithm
Given a Weighted, Directed Acyclic Graph (DAG) G = (V, E), the Shortest Paths
from a single source is computed by Relaxing the edges according to a Topological
Sort of its vertices.
TOPOLOGICAL-SORT (G)
Call DFS (G) to compute finishing times
v.f for each vertex v.
As each vertex is finished, insert it onto
the front of a linked list.
Return the linked list of vertices.
DAG-SHORTEST-PATHS (G, w, s)
Topologically Sort the vertices of G
INITIALIZE-SINGLE-SOURCE (G, s)
For each vertex u, taken in
Topologically Sorted order
For each vertex v є G.Adj [u]
RELAX (u, v, w)

3. Algorithm…
Running Time:
Topological Sort - Ө (V + E)
INITIALIZE-SINGLE-SOURCE - Ө (V)
Relax - Ө (1)
DAG-SHORTEST-PATHS - Ө (V + E)
(The algorithm relaxes all the edges
exactly once in the for loop for all the
vertices in the graph.)
INITIALIZE-SINGLE-SOURCE (G, s)
For (Each Vertex v є G.V)
v.d = ꝏ
v.π = NIL
s.d = 0
RELAX (u, v, w)
If (v.d > (u.d + w (u, v))
v.d = u.d + w (u, v)
v.π = u

4. • v.d : Shortest Path Estimate; Estimate of the shortest path from the source s to a
vertex v.
• v.π : Predecessor of v in the path from s to v.
• v.f : Timestamp recording when the search finishes examining v’s adjacency list
• δ (s, v) : Shortest path from the source s to vertex v.
• Relax (u, v) : Process of testing whether the shortest path to v (from s) found so far
can be improved by going through u and if so, updating v.d and v.π accordingly.
Note:
• Relax (u, v) is equivalent to finding whether the shortest path to v found so far is
minimum or the path through u is minimum. i.e,
v.d = min (v.d, u.d + w (u, v))
Algorithm…

5. Example
Vertex: r
Relax (r, s): s.d = min (s.d, r.d + w (r, s)) = min (0, ꝏ + 5) = 0
Relax (r, t): t.d = min (t.d, r.d + w (r, t)) = min (ꝏ, ꝏ + 3) = ꝏ

6. Example…
Vertex: s
Relax (s, t): t.d = min (t.d, s.d + w (s, t)) = min (ꝏ, 0 + 2) = 2; t.π = s
Relax (s, x): x.d = min (x.d, s.d + w (s, x)) = min (ꝏ, 0 + 6) = 6; x.π = s

7. Example…
Vertex: t
Relax (t, x): x.d = min (x.d, t.d + w (t, x)) = min (6, 2 + 7) = 6
Relax (t, y): y.d = min (y.d, t.d + w (t, y)) = min (ꝏ, 2 + 4) = 6; y.π = t
Relax (t, z): z.d = min (z.d, t.d + w (t, z)) = min (ꝏ, 2 + 2) = 4; z.π = t

8. Example…
Vertex: x
Relax (x, y): y.d = min (y.d, x.d + w (x, y)) = min (6, 6 + (-1)) = 5; y.π = x
Relax (x, z): z.d = min (z.d, x.d + w (x, z)) = min (4, 6 + 1) = 4

9. Example…
Vertex: y
Relax (y, z): v.z = min (v.z, v.y + w (y, z)) = min (4, 5 + (-2)) = 3; z.π = y

10. Example…

11. Example…
The Shortest Paths from source s to all the vertices is:
Destination Path Cost
r No Path
r.π = NIL
ꝏ
s - 0
t s → t
t.π = s
2
x s → x
x.π = s
6
y s → x → y
x.π = s, y.π = x
5
(6 + (-1))
z s → x → y → z
x.π = s, y.π = x, z.π = y
3
(6 + (-1) + (-2))

12. If a weighted, directed graph G = (V, E) has source vertex s and no cycles, then at the
termination of the DAG-SHORTEST-PATHS procedure, v.d = δ (s, v) for all vertices v є V,
and the predecessor subgraph Gπ is a Shortest-Paths Tree.
Proof:
(1) v.d = δ (s, v):
(i) If v is Not Reachable from s
v.d = δ (s, v) = ꝏ (No-Path Property) (T1)
(ii) If v Reachable from s
Let p = <v0, v1, . . . , vk>, v0 = s and vk = v be the Shortest Path from s to v.
The algorithm processes the vertices in Topologically Sorted Order (T2)
(T2) → The edges are relaxed in the order (v0, v1), (v1, v2), . . . , (vk - 1, vk) (T3)
(T3) → vi.d = δ (s, vi) for i = 1, 2, . . . , k (Path-Relaxation Property) (T4)
(T1) & (T4) → v.d = δ (s, v) (T5)
(2) Gπ is a Shortest-Paths Tree:
(T5) → Gπ is a shortest-paths tree rooted at s (Predecessor-Subgraph Property)
Theorem

13. Determining Critical Paths in PERT (Program Evaluation and Review Technique) chart
analysis.
• Edges represent Jobs to be performed.
• Edge Weights represent the Times required to perform particular jobs.
• If edge (u, v) enters vertex v and edge (v, x) leaves v, then job (u, v) must be performed
before job (v, x).
• A Path through this dag represents a Sequence of Jobs that must be performed in a particular
order.
• Critical path
 Longest path through the dag, corresponding to the longest time to perform any sequence of jobs.
 Thus, the weight of a critical path provides a lower bound on the total time to perform all the
jobs.
 Finding a critical path:
(1) Negating the edge weights and running DAG-SHORTEST-PATHS, or
(2) Replacing ꝏ by - ꝏ in INITIALIZE-SINGLE-SOURCE, > by < in RELAX and running
DAG-SHORTEST-PATHS
Application
u v x

14. Appendix
No-path property
If there is no path from s to v, then v.d = δ (s, v) = ꝏ always.
Path-relaxation property
If p = <v0, v1, . . . , vk> is a shortest path from s = v0 to vk , then vk.d = δ (s, vk) if the
edges of p are Relaxed in the order (v0, v1), (v1, v2), . . . , (vk - 1, vk).
Predecessor-subgraph property
Once v.d = δ (s, v) for all v є V , the predecessor subgraph is a shortest-paths tree
rooted at s.

15. References:
• Thomas H Cormen. Charles E Leiserson, Ronald L Rivest, Clifford Stein,
Introduction to Algorithms, Third Edition, The MIT Press Cambridge,
Massachusetts London, England.