Salvage-Embeddings of Complete Trees

A salvage-embedding (S-embedding) maps a complete binary tree G into a larger complete binary tree H some fraction of whose leaves have been labeled good. The S-embedding maps leaves of G one-to-one to good leaves of H; it may be many-to-one on internal nodes. We measure the quality of an S-embedding by its harvest, the fraction H of good leaves of H that are images of leaves of G, and its congestion, the largest number of edges of G that get “routed” across the same edge of H. We consider three scenarios for an N -leaf tree H. In the worst-case scenario: if the fraction G of H’s leaves are labeled good, then one can S-embed an HGN -leaf G in H with congestion log logN+ a constant depending only on G and H, no matter how the good leaves are distributed; this congestion cannot be lowered by more than a small constant factor. In the expected-case scenario, where leaves of H are labeled good or not, independently, with fixed probability: with probability exceeding 1−N−Ω(1), for any 0 < H ≤ 1/8, one can S-embed an HN -leaf G in H with congestion O(log log logN); we do not know if this congestion can be lowered. In the salvaging scenario: we present an algorithm that, in time O(N(logN)3C+2), S-embeds in a given a leaf-labeled H the largest possible G, subject to the prespecified bound C on congestion. This work is motivated by the problem of salvaging a fault-free subnetwork of a leaf-tree machine — a tree architecture whose leaves hold “full-power” PEs and whose nonleaf nodes hold “rudimentary” PEs that route messages and perform simple combining tasks. ————————————⋆ A preliminary version of this paper was presented at CONPAR/92, Lyon, France, September 1-4, 1992. Authors’ Mailing Addresses: S.N. Bhatt: Dept. of Computer Science, Yale University, New Haven, CT 06520 F.R.K. Chung: Bell Communications Research, 435 South St., Morristown, NJ 07960 F.T. Leighton: Dept. of Mathematics and Lab. for Computer Science, MIT, Cambridge, MA 02139 A.L. Rosenberg: Dept. of Computer Science, University of Massachusetts, Amherst, MA 01003