¹/i€[k], \'1]i€[k]) functions (la)a&A ELA. Define al as the instance that we obtain from I if we scale all demands by a ≥ 1, i.e., al = (G, (la)a=A, ($i, ti)iɛ[k], (ari)iɛ[k]). Recall the definition of w(L) as defined in the Lecture Notes and assume that 0 ≤ w(L) < 1.

¹/i€[k], \'1]i€[k]) functions (la)a&A ELA. Define al as the instance that we obtain from I if we scale all demands by a ≥ 1, i.e., al = (G, (la)a=A, ($i, ti)iɛ[k], (ari)iɛ[k]). Recall the definition of w(L) as defined in the Lecture Notes and assume that 0 ≤ w(L) < 1.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Problem 1
Let L be some class of standard latency functions. Let I
(G = (V,A), (la)a=A, (Si, ti)ik], (ri)ic[k]) be a selfish routing instance with latency
functions (la)aEA ELA. Define al as the instance that we obtain from I if we
scale all demands by a ≥ 1, i.e., al = (G, (la)a=A, (Si, ti)iɛ[k], (αri)iċ[k]). Recall the
definition of w(L) as defined in the Lecture Notes and assume that 0 ≤ w(L) < 1.
Let f be a Nash flow of I and let f* be an optimal flow of al with a := (1+w(L)).
Show that C(f) <C(f*). Hint: Consider the flow x = 1f* and use the variational
inequality.

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