This paper considers an algorithm suggested by A. Butrimenko for a network communications problem. It is shown here that the algorithm converges, and to the optimal solution, when started from a particular initial solution

Bell, D.E.

International Institute for Applied Systems Analysis (IIASA)

On a Network Algorithm of ButrimenkoBell, D.E.IIASA Working PaperWP-74-0081974 Bell, D.E. (1974) On a Network Algorithm of Butrimenko. IIASA Working Paper. WP-74-008 Copyright © 1974 by the author(s). http://pure.iiasa.ac.at/163/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at ON A NETWORK ALGORITHM OF BUTRIMENKOD. E. BellFebruary 1974 WP-74-8Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor-mation purposes. The views expressedare those of the author, and do notnecessarily reflect those of IIASA.AbstractThis paper considers an algorithm suggested byA. Butrimenko for a network communications problem. It isshown here that the algorithm converges, and to the optimalsolution, when started from a particular initial sOlution.Contents1. Description of the Problem2. An Algorithm3. Solutions to the Recursion4. The Optimal Policy5. Proof of Convergence to the Optimal Solution6. Conjecture7. Finite Convergence for Acyclic NetworksOn a Network Algorithm of Butrimenko1. Description of the ProblemGiven a directed network with n nodes, a distinguished nodecalled the destination, a set of probabilities {Pij} where arc(i,j) is "open" with probability p .. and "closed" otherwise, thelJobject is to find the probability at any given node of reachingthe destination. Travel in the network is sequential, aftertransferring from one node to another the arc probabilities areindependent. When at a node, it is known which arcs directedout of the node are open and which are closed.For example, consider the following network:node 2node 1destinationnode 3Starting at node 1, suppose first that the policy is tomove to node 2 if possible, node 3 otherwise. Then the probabilityof reaching the destination is2"53 3'Ii" + "5 1• "3 • 4 = 23"5 502.The reverse policy (node 3 first) would be slightly better13"4 2'5 + '3 2"5 3:: 711 15The questions are, what is the best policy and how maythe associated probabilities be calculated.2. An AlgorithmDr. Butrimenko suggests the following iterative algorithm.000Start with an initial guess P :: (Pl, ... ,Pn) of the node proba-bilities, with ｐ ｾ ･ ｳ ｴ ｩ ｮ ｡ ｴ ｩ ｯ ｮ :: 1. Suppose the nodes are indexedso thatThenwhere the term in pX- l is omitted. ｐ ｾ ･ ｳ ｴ ｩ ｮ ｡ ｴ ｩ ｯ ｮ Ｚ Ｚ 1 for allvalues of t. Butrimenko asks whether this iterative scheme willconverge, and if it does, whether the solution is meaningful. Itwill be shown here that for the starting solutions(i)(ii)PC:> :: 11PC:> :: 01for all nodes ifor all nodes except the destinationthe algorithm converges and in case (i) to the optimal solutionfor the given problem.3. SOlutions to the RecursionIt will be useful to have a compact notation for therecursion of section 2. Letwhere p .. = 0 and11where ｰ ｾ is the estimate at iteration t of the probability of1reaching the destination from node i. Define a function f bythe relationwhere 0 is a permutation chosen so that3 ....Note that we can always take 0(1) = destination.4.This notation reduces the recursion of section 2 tot t-lPi = f(Pi' P )which, if it converges, leads to the relation( * )It will be shown later that this has at least one solution,but at present it is not clear that it need have exactly onesOlution. If it has more than one solution, then even if thealgorithm converges, it may not be known whether it has convergedto the optimal solution.4. The Optimal PolicySuppose that, situated at node i, it is known for eachj ｾ i the maximum probability P. of reaching the destination fromJj. Since, at node i, we know which arcs are open and which not,how should we proceed? It is clear that we should choose theopen arc which leads to the node having maximum probability. Ifthat is not clear, the following proof should help.Lemma. The optimal policy at a node is to move to the node which,amongst those available, affords the best chance of reaching thedestination.5 .Proof. Consider the case where only two arcs lead out of anode. The first, open with probability pPQleads to a node from which the probability of reaching thedestination is P and the second with probabilities q, Qrespectively. Suppose P > Q. If the policy of the lemma isadopted, there is a probability ofpP + (1 - p) qQ (1)of gaining the destination. Otherwise the probability isqQ + (1 - q) pP (2)But (1) is larger than (2) if and only if P > Q. Now considera situation with three arcs with parameters P, Q, R and assumeP > Q > R.The ordering (P,Q,R) gives a probabilitypP + (1 - p)[qQ + (1 - q) rRJ6.which is greater thanpP + (l - p) erR + (1 - r) qQJsince Q > R.Hence (P,Q,R) > (P,R,Q) and by repetition of such pairwiseinterchanges(P,Q,R) > (P,R,Q) > (R,P,Q) > (R,Q,P)and(P,Q,R) > (Q,P,R) > (Q,R,P) > (R,Q,P)showing that (P,Q,R) is the best policy. This process may berepeated for any number of arcs. IIThe lemma was somewhat labored but it is important toestablish the optimality of that policy. Note then that thispolicy gives the maximum probability of reaching the destinationfrom i, P., as1which is the limiting situation for the recurrence relation givenin section 2, that is, equation (*).Since this relation holds for all nodes i, excepting onlythe destination node, it is demonstrated that the optimalsolution to the given problem satisfies recursion (3).5. Proof of Convergence to the Optimal SolutionThe approach here will be to show that a particularsequence {pt } has the properties(i) each pt is an upper bound for all solutions to (*),(ii) it converges to a solution of (*).Thus the algorithm will have converged to the optimal solutionof the problem.7.Theorem 1. If P satisfiesP. = f(p., P)1 1for all iand P < pt then,P. < ｰｾＫｬ = f(P1., pt)1 1Proof. Iff(Pi' P) = r a .. P .j lJ Jand f (p. , pt ) r t= b .. P.1 j lJ Jthen r a .. P.1J Jt< r a .. P.1J J by assumption8.and tr a .. P. <1J Jtr b .. P.1J J by the lemmaso that P. < ｰｾＫｬ1 1as required. IIHence if an initial value p? = 1 for all i is chosen, this is1evidently an upper bound for the optimal solution; hence, thealgorithm produces a sequence of upper bounds.Theorem 2.Proof. ｰ ｾ Ｋ ｬ = r a .. ｐ ｾ say,1 1J Jand tr a .. P.1J Jt-l< r a .. P.1J J by assumption,but t-l t-lr a .. P. < f(p., P )1J J 1 by the lemmaand so ｰ ｾ Ｋ ｬ < ｐｾ1 1 IIMain Theorem. From the starting solution Pi = 1 for all i,the algorithm converges and does so to the optimal solution.Proof. By Theorem 2, the sequence {pt} is monotonicallyco. •decreasing and thus converges to some solution P Slnce 1t isbounded below by zero. co •By Theorem 1, P 1S an upper bound forall solutions to the recursion (*). By the lemma, the optimalsolution satisfies the recursion.solution. IIcoHence, P is the optimal6. Conjecture,Starting from an initial solution ｰ ｾ = 0 for all i, it_may be shown that the resulting sequence {pt} is a sequenceof lower bounds for all solutions to (*) and converges to onesuch solution. Hence, amongst the set of such solutions oneis an upper bound and one is a lower bound. This state ofaffairs suggests that the two bounding solutions may indeedbe the same solution and that there is exactly one solution to(*). This would show that if the algorithm converged, it woulddo so to the optimal solution. It remains to show whether thealgorithm converges for any starting solution.7. Finite Convergence for Acyclic NetworksFor a certain class of networks, the conjecture is trueand, in addition, the algorithm converges finitely.9.Theorem 3. For an acyclic network, the algorithm converges inat most n-l steps and to the optimal solution, from any initialovalues of P ..JProof. The idea is that after the kth iteration, ｰ ｾ =J00P. forJat least k+l values of j.Let So = {destination node}Sl = {set of nodes whose only outgoing arcsgo to the destination node}1and in general,Sk = {set of nodes whose outgoing arcs arek-lonly incident with nodes in US,}.i=O l,. k 00, •The clalm lS that P. = P. for all JESk . The case k = 0 lSJ Jautomatic since ｐ ｾ ･ ｳ ｴ ｩ ｮ ｡ ｴ ｩ ｯ ｮ is constantly 1.Assume, using induction, that it is true for k = 0,1, ... ,toBut for iESt +l , the recurrence relationis only in terms of those Pj withthe sets Sk; hence, pl+l = pl+2 =iESt +l ·fixed values by definition ofpool" and thus ｰ ｾ Ｋ ｬ = ｰｾ for alll lIt now only remains to show thatfor allSuppose that for some k, Sk+l = Sk and ISkl < n. Let Skbe those nodes not in Sk' Since ｓ ｫ ｾ Sk+l is empty, each nodeof Sk has an outgoing arc to another node in Sk' Starting atany node iESk construct a sequence (i,i l ,i2 , ... ) wherep.. > 0 and iSESk for all j. Since Sk is finite, somelsls+li = i for s ｾ t and Sk contains a cycle, which is contradictor.s tHence the algorithm converges finitely to a solution whichis independent of the initial values. I INote that the algorithm for the acyclic case is a one00 • •stage process since each P. may be obtalned sequentlallyJimmediately.11.

On a Network Algorithm of Butrimenko

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