1. Data Structures and Algorithms (AT70. 02) Comp. Sc. and Inf. Mgmt
Data Structures and Algorithms (AT70.02) Comp. Sc. and Inf. Mgmt. Asian Institute of Technology
Instructor: Prof. Sumanta Guha
Slide Sources: CLRS “Intro. To Algorithms” book website
(copyright McGraw Hill)
adapted and supplemented

2. CLRS “Intro. To Algorithms” Ch. 24: Single-Source Shortest Paths

3. The shortest-path weight from s to v is denoted δ(s,v)

5. Π(v) is the predecessor of v in the shortest path to v
= parent in the shortest-paths tree

6. d [v] is the shortest path estimate to v. In fact, d [v] will
always be at least as long as the actual shortest path.

8. Properties of shortest paths and relaxation
Triangle inequality: for any edge (u,v)  E, we have δ(s,v)  δ(s,u) + w(u,v).
Upper bound property: We always have d[v] ≥ δ(s,v) for all vertices v V, and once d[v] achieves the value δ(s,v), it never changes (even with further relaxations).
No-path property: If there is no path from s to v, then we always have d[v] = δ(s,v) = .
Convergence property: If s …  u  v is a shortest path in G for some u, v  V, and if d[u] = δ(s,u) at any time prior to relaxing edge (u,v), then d[v] = δ(s,v) at all times afterwards.
Path relaxation property: If p = (v0, v1, …, vk) is a shortest path from s = v0 to vk, and the edges of p are relaxed in the order (v0,v1), (v1,v2), …, (vk-1,vk), then d[vk] = δ(s,vk). This property holds regardless of any other relaxation steps that occur, even if they are intermixed with relaxations of the edges of p.
Predecessor-subgraph property: Once d[v] = δ(s,v) for all v  V, the predecessor subgraph is a shortest paths tree rooted at s. (In fact, if there is no negative-weight cycle reachable from s then the predecessor subgraph is always a tree rooted at s.)

9. If there is no negative edge
weight cycle reachable from s
then Bellman-Ford returns TRUE
and d[v] = δ(s,v) for every vertex v.
If there is a negative edge
then Bellman-Ford returns
FALSE.
Running time: (|V||E|)

10. Topologically Sorting a DAG (Directed Acyclic Graph)
A topological sort of the nodes of a DAG G is a linear ordering such that
there is no edge of G going from a node to a predecessor in the ordering.

13. // all edge weights are assumed non-negative
S = {v : d [v] = δ(s,v)}, i.e., vertices for
which the shortest path from s has been
determined.
Q is a min heap of vertices of G keyed by
their d (i.e., estimate) values.

14. Proof that Dijkstra is correct

15. x1 – x2 ≤ 0,
x1 – x5 ≤ -1,
x2 – x5 ≤ 1,
x3 – x1 ≤ 5,
x4 – x1 ≤ 4,
x4 – x3 ≤ -1,
x5 – x3 ≤ -3,
x5 – x4 ≤ -3.
Th. 24.9: Given a system Ax ≤ b of difference equations, let G = (V, E) be the
corresponding constraint graph. If G contains no negative-weight cycles, then
x = ( δ(v0, v1), δ(v0, v2), …, δ(v0, vk) )
is a feasible solution for the system. If G contains a negative-weight cycle, then
there is no feasible solution for the system.

16. Problems Ex. 24.1-1 Ex. 24.2-1 Ex. 24.3-1 Ex. 24.3-2 Ex. 24.3-4