L11: Algebraic Path Problems with applications to Internet Routing Lecture 15. Path Weight with functions on arcs?

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1 L11: Algebraic Path Problems with applications to Internet Routing Lecture 15 Timothy G. Griffin Computer Laboratory University of Cambridge, UK Michaelmas Term, 2016 tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 151 / 21 Path Weight with functions on arcs? For graph G pv, Eq, and arc path p pu 0, u 1 qpu 1, u 2 q pu k 1, u k q. Functions on arcs: two natural ways to do this... Weight function w : E Ñ ps Ñ Sq. Let f j wpu j 1, u j q. wa L ppq f 1 pf 2 p f k paq qq pf 1 f 2 f k qpaq wa R ppq f k pf k 1 p f 1 paq qq pf k f k 1 f 1 qpaq How can we make this work for path problems? tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 152 / 21

2 Algebra of Monoid Endomorphisms (AME) (See Gondran and Minoux 2008) Let ps,, 0q be a commutative monoid. ps,, F Ď S Ñ S, 0q is an algebra of monoid endomorphisms (AME) P F, f p0q P c P S, f pb cq f pbq f pcq I will declare these as g P F, f g P F (closed) Di P P S, ipsq s Dω P P N, ωpnq 0 Note: as with semirings, we may have to drop some of these axioms in order to model Internet routing... tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 153 / 21 So why do we want AMEs? Each (closed with ω and i) AME can be viewed as a semiring of functions. Suppose ps,, F, 0q is an algebra of monoid endomorphisms. We can turn it into a semiring F pf, ˆ,, ω, iq where pf ˆ gqpaq f paq gpaq and pf gqpaq f pgpaqq. But functions are hard to work with... All algorithms need to check equality over elements of a semiring f g P S, f paq gpaq S can be very large, or infinite... tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 154 / 21

3 How do we represent a set of functions F Ď S Ñ S? Assume we a set L and a function P L Ñ ps Ñ Sq. We normally write l s rather than plqpsq. We think of l P L as the index for a function f l psq l s. In this way pl, q can be used to represent the set of functions F tf l λs.pl sq l P Lu. tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 155 / 21 Indexed Algebra of Monoid Endomorphisms (IAME) Let ps,, 0q be a commutative and idempotent monoid. A (left) IAME ps, L,,, 0q P L Ñ ps Ñ P L, l 0 0 Dl P P S, l s s Dl P P S, l s P m P S, l pn mq pl nq pl mq When we need closure? Not very often! If needed, it would 1, l 2 P L, Dl 3 P P S, l 3 s l 1 pl 2 sq tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 156 / 21

4 IAME of Matrices Given a left IAME ps, L,,, 0q define the left IAME of matrices pm n psq, M n plq,,, Jq. For all i, j we have Jpi, jq 0. For A P M n plq and B, C P M n psq define pb Cqpi, jq Bpi, jq Cpi, jq pa Bqpi, jq à 1ďqďn Api, qq Bpq, jq tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 157 / 21 Solving (some) equations. Left version here... We will be interested in solving for L equations of the form L pa Lq B Let A 0 B B A k`1 B A pa k Bq and A pkq B A 0 B A 1 B A 2 B A k B A B A 0 B A 1 B A 2 B A k B tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 158 / 21

5 Key result (again) q stability If there exists a q such that for all B, A pqq B A pq`1q B, then A is q-stable. Therefore, A B A pqq B. Theorm If A is q-stable, then L A pbq solves the equation L pa Lq B. tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routingc 2015 Lecture 159 / 21 Something familiar : Lexicographic product ps, L S, S, S q ˆpT, L T, T, T q psˆt, L SˆL T, S ˆ T, Sˆ T q Theorem Where DppS, L S, S, S q ˆ pt, L T, T, T qq ðñ DpS, L S, S, S q ^DpT, L T, T, T q ^ pcps, L S, S q _KpT, L T, T qq DpS, L,, b P S, l P L, l pa bq pl aq pl bq CpS, L, b P S, l P L, l a l b ùñ a b KpS, L, b P S, l P L, l a l b tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

6 Something new: Functional Union ps, L 1,, 1 q `m ps, L 2,, 2 q ps, L 1 Z L 2,, 1 Z 2 q Where Fact inlplq p 1 Z 2 q s l 1 s inrplq p 1 Z 2 q s l 2 s DppS, L 1,, 1 q `m ps, L 2,, 2 qq ðñ DpS, L 1,, 1 q ^DpS, L 2,, 2 q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21 Left and Right rightps, q ps, tru,, rightq leftps, q ps, S,, leftq The following are always hold. R right s s s 1 left s 2 s 1 DprightpS, qq IPpS, q ñ DpleftpS, qq CprightpS, qq KpleftpS, qq tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

7 Scoped Product (Think ibgp/ebgp) ps, L S, S, S q Θ pt, L T, T, T q pps, L S, S, S q ˆ leftpt, T qq `m prightps, S q ˆ pt, L T, T, T qq Between regions (λ P L S, s P S, t 1, t 2 P T) inlpλ, t 2 q ps, t 1 q pλ S s, t 2 q Within regions (λ P L T, s P S, t P T) inrpr, λq ps, tq ps, λ T tq tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21 Theorem. If IPpT, T q, then pdpps, L S, S, S q Θ pt, L T, T, T qq ðñ DpS, L S, S, S q ^DpT, L T, T, T qq DpppS, L S, S, S q ˆ leftpt, T qq `m prightps, S q ˆ pt, L T, T, T qqq ðñ DppS, L S, S, S q ˆ leftpt, T qq ^ DpprightpS, S qq ˆ pt, L T, T, T qq ðñ DpS, L S, S, S q ^DpleftpT, T qq ^ pcps, L S, S q _KpleftpT, T qqq ^ DprightpS, S qq ^DpT, L T, T, T q ^ pcprightps, S qq _KpT, L T, T qq ðñ DpS, L S, S, S q ^DpT, L T, T, T q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

8 Homework 3 : Shortest Paths with Gains and Losses spgl pr ˆR,, q Where ps 1, t 1 q ps 2, t 2 q $ & % ps 1, t 1 min t 2 q (if s 1 s 1 ) ps 1, t 1 q (if s 1 s 1 min s 2 s 2 ) ps 2, t 2 q (if s 1 s 1 min s 2 s 2 ) pc 1, n 1 q pc 2, n 2 q pc 1 ` n 1 c 2, n 1 n 2 q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture 15 / 21 Homework 3 : Calculating path weights in spgl ppc 1, n 1 q pc 2, n 2 qq pc 3, n 3 q pc 1 ` n 1 c 2, n 1 n 2 q pc 3, n 3 q pc 1 ` n 1 c 2 ` n 1 n 2 c 3, n 1 n 2 n 3 q pc 1, n 1 q pc 2, n 2 q pc 3, n 3 q pc k, n k q pc 1 ` n 1 c 2 ` n 1 n 2 c 3 ` ` n 1 n 2 n 3 n k 1 c k, n 1 n 2 n 3 n k q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

9 Homework 3 : Shortest Paths with Discounting Assume 0 ă δ ă 1. spd pr ˆN,, q Where ps 1, t 1 q ps 2, t 2 q $ & % ps 1, t 1 min t 2 q (if s 1 s 1 ) ps 1, t 1 q (if s 1 s 1 min s 2 s 2 ) ps 2, t 2 q (if s 1 s 1 min s 2 s 2 ) pc 1, n 1 q pc 2, n 2 q pc 1 ` c 2 δ n 1, n 1 ` n 2 q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21 Homework 3 : Calculating path weights in spd ppc 1, n 1 q pc 2, n 2 qq pc 3, n 3 q pc 1 ` c 2 δ n 1, n 1 ` n 2 q pc 3, n 3 q pc 1 ` c 2 δ n 1 ` c 3 δ pn 1`n 2 q, n 1 ` n 2 ` n 3 q pc 1 ` c 2 δ n 1 ` c 3 δ n 1δ n 2, n 1 ` n 2 ` n 3 q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

10 Homework 3 : Can we generalise this kind of construction? Given semigroups ps, b S q and pt, b T q, and a function Write t s instead of ptqpsq. P T Ñ ps Ñ Sq Define the semi-direct product on S ˆ T as ps 1, t 1 q ps 2, t 2 q ps 1 b S pt 1 s 2 q, t 1 b T t 2 q Homework 3 : A new construction Assume is commutative and selective. ps, S, b S q pt, T, b T q ps ˆ T,, q where T ˆ T (lexicographic product). tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21 Homework 3 : Does this work? spgl Let then n c n ˆ c spgl pr, min, `q pr, min, ˆq spd Assume 0 ă δ ă 1 and n c c ˆδ n, then spd pr, min, `q pn, min, `q tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21

11 Homework 3 : Problems Question 1 (30 marks) When is ps ˆ T, q a semigroup? Question 2 (35 marks) When is ps, S, b S q pt, T, b T q left distributive? Question 3 (35 marks) When is ps, S, b S q pt, T, b T q right distributive? tgg22 (cl.cam.ac.uk) L11: Algebraic Path Problems with applications to Internet T.G.Griffin Routing c 2015 Lecture / 21