Static and Dynamic Path Selection on Expander Graphs: A Random Walk Approach (Preliminary Version).

This paper addresses the problem of virtual circuit switching in bounded degree expander graphs. We study the static and dynamic versions of this problem. Our solutions are baaed on the rapidly mixing properties of random walks on expander graphs. In the static version of the problem an algorithm is required to route a path between each of K pairs of vertices so that no edge is used by more than g paths. A natural approach to this problem is through a multicommodity flow reduction. However, we show that the random walk approach leads to significantly stronger results than those recently obtained by Leighton and Rao [10] using the multi-commodity flow setup. In the dynamic version of the problem connection requests are continuously injected into the network, Once a connection is established it utilizes a path (a virtual circuit) for a certain time until the communication terminates and the pat h is deleted. Again each edge in the network should not be used by more than g paths at once. The dynamic version is a better model for the practical use of communication networks. Our random walk approach gives a simple and fully distributed solution for this problem. We show that if the injection to the network and the duration of connections are both controlled by Poisson processes then our algorithm achieves ●Digital Systems Research Center, 130 Lytton Ave, Palo Alto, CA 943o1 t Department of Mathematics, Carnegie-Mellon University. A portion of this work was done while the author was visiting Digital SRC. Supported in part by NSF grants CCR-9225008 and CCR9530974. t IBM Almaden Research Center, San Jose, CA 95120, and Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Pumission 10nmkc digllalflmd topics ofnll or pflll ot’thismxtcrinlfhr pemmal or clmsnmm usc is gr:mtcd ivilhoul k pro!fidcd 111:11 the copies arc not mode or distrihltcd t’orprolit or conmwrciu I adwmtagc, Ihe copyrighl notice. Ihc Iitle ol”thc puldicoliol) :In(i ils tialc appcw. and nolicc is gi\&ll that LX)pyrigh(i, b) pNllli\,iOll (>~tht :’!Vhi. ill~. “[’0LOp\ Othtr\! ist. to republish. 10 post on wrvers or 10 rcdis!r}l>tjlc IO 1ists. requires speci Iic penniwion andfor kc ,$770( ‘ 97 1:1 1’,,so. ‘1’c\m 1 ‘s:\ Copyrighl 11)97 ,-\Ckl0-XtJ7’)I-XXX-(V97,05 .,$3 5[) a steady state utilization of the network which is similar to the utilization achieved in the static case situation.