Showing papers by "Andrei Z. Broder published in 1994"

Balanced allocations (extended abstract)

Abstract: Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability lnn/lnlnn(1 + o(1)) balls in it. Suppose instead, that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only lnlnn/ln2 + O(1) balls - exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and on-line load balancing.

Existence and Construction of Edge-Disjoint Pathson Expander Graphs

Abstract: Given an expander graph $G=(V,E)$ and a set of $q$ disjoint pairs of vertices in $V$, the authors are interested in finding for each pair $(a_i, b_i)$ a path connecting $a_i$ to $b_i$ such that the set of $q$ paths so found is edge disjoint. (For general graphs the related decision problem is NP complete.) The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\kappa$ depends only on the expansion properties of the input graph, and not on $n$. Furthermore, a randomized $o(n^3)$ time algorithm, and a random $\cal NC$ algorithm for constructing these paths is presented. (Previous existence proofs and construction algorithms allowed only up to $n^\epsilon$ pairs, for some $\epsilon\ll \frac{1}{3}$, and strong expanders [D. Peleg and E. Upfal, Combinatorica, 9 (1989), pp.~289--313.].) In passing, an algorithm is developed for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.

Trading Space for Time in Undirected $s-t$ Connectivity

Abstract: Aleliunas et al. [20th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1979, pp. 218--223] posed the following question: "The reachability problem for undirected graphs can be solved in log space and $O(mn)$ time [$m$ is the number of edges and $n$ is the number of vertices] by a probabilistic algorithm that simulates a random walk, or in linear time and space by a conventional deterministic graph traversal algorithm. Is there a spectrum of time-space trade-offs between these extremes?" This question is answered in the affirmative for sparse graphs by presentation of an algorithm that is faster than the random walk by a factor essentially proportional to the size of its workspace. For denser graphs, this algorithm is faster than the random walk but the speed-up factor is smaller.

Finding hidden hamiltonian cycles

Abstract: Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “high” the cycle. Is it possible to unravel the structures, that is, to efficiently find a Himiltonian cycle in G? We describe an O(n3 log n)-step algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots” of G by a collection of disjoint paths. (This is the hard part to analyze). Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems. © 1994 John Wiley & Sons, Inc.

Optimal construction of edge-disjoint paths in random graphs

Abstract: Given a graph G =( V;E) with n vertices, m edges, and a family of pairs of vertices in V , we are interested in nding for each pair (ai;bi) a path connecting ai to bi such that the set of paths so found is edge disjoint. (For arbitrary graphs the problem isNP-complete, although it is inP if is xed.) We present a polynomial time randomized algorithm for nding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn;m for all edge densities above the connectivity threshold. (The graph is chosen rst; then an adversary chooses the pairs of endpoints.) Our results give the rst tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.

Near‐perfect token distribution

Abstract: Suppose that n tokens are arbitrarily placed on the n nodes of a graph. At each parallel step one token may be moved from each node to an adjacent node. An algorithm for the near‐perfect token distribution problem redistributes the tokens in a finite number of steps, so that, at the end, no more than O(1) tokens reside at each node. (In perfect distribution, at the end, exactly one token resides at each node.) In this paper we present a simple algorithm that works for all extrovert graphs, a new property which we define and study. In terms of connectivity requirements, extrovert graphs are roughly in‐between expanders and compressors. Our results lead to an optimal solution for the near‐perfect token distribution problem on almost all cubic graphs. The new solution is conceptually simpler than previous algorithms, and applies to graphs of minimum possible degree. © 1994 John Wiley & Sons, Inc.