Suppose that G = (V, E) is a directed graph with |V| = n and |E| = m unless otherwise stated. Also suppose that all edge capacities are integral and that there are no directed 2-cycles. Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation of Ford–Fulkerson. (1) Set ƒ(e) = 0 for all e E E. (2) While there exists an s-t path in Gƒ: (3) Compute the s-t path P in G that maximizes y := = {uf(e) | e € P}. (4) Augment f along P by y. Let fo, f1, … , fr be the sequence of flows that the algorithm computes (where ) = value(fo) < value(fi) < · … < value(fr) = value(f*)). Hint: The previous problem should help with (a) and (b) below. (a) Prove that value(fi) > value(f*). (b) Prove that value(f;) > value(fi-1) + ± (value(ƒ*) – value(fi-1)) for all i e [T]. (c) Prove that value(fi) > value(f*) (1– (1 – ±)') for all i € [T].
Suppose that G = (V, E) is a directed graph with |V| = n and |E| = m unless otherwise stated. Also suppose that all edge capacities are integral and that there are no directed 2-cycles. Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation of Ford–Fulkerson. (1) Set ƒ(e) = 0 for all e E E. (2) While there exists an s-t path in Gƒ: (3) Compute the s-t path P in G that maximizes y := = {uf(e) | e € P}. (4) Augment f along P by y. Let fo, f1, … , fr be the sequence of flows that the algorithm computes (where ) = value(fo) < value(fi) < · … < value(fr) = value(f*)). Hint: The previous problem should help with (a) and (b) below. (a) Prove that value(fi) > value(f*). (b) Prove that value(f;) > value(fi-1) + ± (value(ƒ*) – value(fi-1)) for all i e [T]. (c) Prove that value(fi) > value(f*) (1– (1 – ±)') for all i € [T].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please write it in detail.
![Suppose that G = (V, E) is a directed graph with |V| = n and |E| = m unless otherwise stated.
Also suppose that all edge capacities are integral and that there are no directed 2-cycles.
Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation of
Ford-Fulkerson.
(1) Set f(e) = 0 for all e e E.
(2) While there exists an s-t path in Gƒ:
(3)
Compute the s-t path P in G that maximizes y := {uf(e) | e e P}.
(4)
Augment f along P by y.
Let fo, f1, ..., fr be the sequence of flows that the algorithm computes (where
0 = value(fo) < value(fi) < · … < value(fr) = value(f*)). Hint: The previous problem
should help with (a) and (b) below.
(a) Prove that value(f1) > 1 value(f*).
(b) Prove that value(f;) > value(f;-1) + ± (value(f*) – value(fi-1)) for all i E [T].
m
(c) Prove that value(fi) > value(f*) (1– (1– 4)') for all i e [T).
m
(d) Prove that T < m ln(2 value(f*)). (In other words, this is an upper bound for the
number of iterations of this algorithm.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c12194d-d887-4cee-b26e-16e42a0f7149%2F9190f885-56cc-4a44-86be-00c1d6de9991%2Fvuw6hh_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that G = (V, E) is a directed graph with |V| = n and |E| = m unless otherwise stated.
Also suppose that all edge capacities are integral and that there are no directed 2-cycles.
Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation of
Ford-Fulkerson.
(1) Set f(e) = 0 for all e e E.
(2) While there exists an s-t path in Gƒ:
(3)
Compute the s-t path P in G that maximizes y := {uf(e) | e e P}.
(4)
Augment f along P by y.
Let fo, f1, ..., fr be the sequence of flows that the algorithm computes (where
0 = value(fo) < value(fi) < · … < value(fr) = value(f*)). Hint: The previous problem
should help with (a) and (b) below.
(a) Prove that value(f1) > 1 value(f*).
(b) Prove that value(f;) > value(f;-1) + ± (value(f*) – value(fi-1)) for all i E [T].
m
(c) Prove that value(fi) > value(f*) (1– (1– 4)') for all i e [T).
m
(d) Prove that T < m ln(2 value(f*)). (In other words, this is an upper bound for the
number of iterations of this algorithm.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
