Sorting and selection on dynamic data

Abstract

We formulate and study a new computational model for dynamic data. In this model, the data changes gradually and the goal of an algorithm is to compute the solution to some problem on the data at each time step, under the constraint that it only has limited access to the data each time. As the data is constantly changing and the algorithm might be unaware of these changes, it cannot be expected to always output the exact right solution; we are interested in algorithms that guarantee to output an approximate solution. In particular, we focus on the fundamental problems of sorting and selection, where the true ordering of the elements changes slowly. We provide algorithms with performance close to the optimal in expectation and with high probability. (C) 2010 Elsevier B.V. All rights reserved

Algorithms on evolving graphs

“…At each time step t, the weights of the edges induce an ordering π t on the edge set E. The ordering π t+1 is obtained from π t by swapping a random consecutive pair. As argued in [2], this is perhaps the most natural model for gradual change with ordinal weights, and as we will see, an ordinal weighted model is rich enough to capture the evolving MST problem.…”

“…We follow the general framework defined in [2] for algorithms on dynamic data. In the case of evolving graphs, this model can be described as follows.…”

“…For weighted graphs, we assume an ordinal model of change similar to the one in [2]: the edge set is fixed (i.e., Et = E for all t), and only the weights change. At each time step t, the weights of the edges induce an ordering π t on the edge set E. The ordering π t+1 is obtained from π t by swapping a random consecutive pair.…”

“…Our first algorithm is as follows: we use the simple sorting algorithm of [2] to keep an approximationπ t of the real ordering π t of all the O(n 2 ) edges. At each time step, our first algorithm returns the tree built using Kruskal's greedy algorithm onπ t (i.e., the MST for orderingπ t ).…”

“…To address some of the challenges of interest that are not captured by the aforementioned models, we propose a new computational framework on graphs where the underlying graph changes over time and an algorithm can only observe these changes by probing the graph in a limited way. This evolving graph framework is inspired by the work of Anagnostopoulos et al [2], who studied sorting and selection problems in an evolving (scalar) data setting. In our framework, the graph changes over time without notifying the algorithm and therefore the algorithm needs to periodically probe the graph to learn about new changes and update its solution accordingly.…”