Lecture 23 - Review

Question 1

What’s the running time of Fast exponentiation? How is this obtained?

Answer 1

Q(logn)

Observation: x^2k = (x^2)^k and x^(2k+1) = x * (x^2)^k

Question 2

What are the three proof techniques we learned?

Answer 2

1. Loop invariants.
2. Contradiction
3. Cut and Paste

Question 3

Prove that any subpath of a shortest path is a shortest path.

Answer 3

Suppose this path p is a shortest path from u to v.
Then δ(u,v) = w(p) = w(pux) + w(pxy) + w(pyv).
Now suppose there exists a shorter path p'xy.
Then w(p’xy)

Question 4

Prove the following Theorem 1: Let (S, V-S) be any cut that respects A, and let (u, v) be
a light edge crossing (S, V-S). Then, (u, v) is safe for A.

Answer 4

Lec 23 - Slide 29

Question 5

Prove the correctness of Djikstra’s.

DIJKSTRA(V, E,w,s):
INIT-SINGLE-SOURCE(V,s)
S ← ∅
Q ← V
while Q ≠ ∅ do
........u ← EXTRACT-MIN(Q)
........S ← S ∪ {u}
........for each vertex v∈Adj[u] do
................RELAX(u,v,w)

Answer 5

Initialization
Initially, S = ∅ ⟹ True

Loop Invariant:
At the start of each iteration:
d[v] = δ(s,v) ∀v ∈ S.

Termination:
Stops when Q=∅ ⇒
d[v] = δ(s,v) ∀ v ∈ V
(by Loop Invariant Property)

Maintenance:
Show that d[u] = δ(s,u) when
u is added to S in each
Iteration.

(check original lecture slide for proof)

Question 6

Give the steps to calculate a min cut.

Answer 6

To find a min cut compute a max flow.
To find the cut run BFS (or DFS) from s on the residual graph.
The reachable vertices define the (min) cut.