Polylogarithmic Algorithm Research Articles

The frequent items problem, under polynomial decay, in the streaming model

We consider the problem of estimating the frequency count of data stream elements under polynomial decay functions. In these settings every element in the stream is assigned with a time-decreasing weight, using a non-increasing polynomial function. Decay functions are used in applications where older data is less significant, less interesting or even less reliable than recent data. Consider a data stream of N elements drawn from a universe U . We propose three poly-logarithmic algorithms for the problem. The first one, deterministic, uses O ( 1 ϵ 2 log N ( log log N + log U ) ) bits, where ϵ ∈ ( 0 , 1 ) is the approximation parameter. The second one, probabilistic, uses O ( 1 ϵ 2 log N δ log 1 ϵ ) bits or O ( 1 ϵ 2 log N δ log N ) bits, depending on the decay function parameter, where δ ∈ ( 0 , 1 ) is the probability of failure. The third one, deterministic in the stochastic model, uses O ( 1 ϵ log U ) bits or O ( 1 ϵ 2 log N ) bits, also depending on the decay parameter as will be described in this paper. This variant of the problem is important and has many applications. To our knowledge, it has never been studied before.

Maintaining dynamic minimum spanning trees: An experimental study

We report our findings on an extensive empirical study on the performance of several algorithms for maintaining minimum spanning trees in dynamic graphs. In particular, we have implemented and tested several variants of the polylogarithmic algorithm by Holm et al., sparsification on top of Frederickson’s algorithm, and other (less sophisticated) dynamic algorithms. In our experiments, we considered as test sets several random, semi-random and worst-case inputs previously considered in the literature together with inputs arising from real-world applications (e.g., a graph of the Internet Autonomous Systems).

On synthesizing cube and tree for parallel processing

In this paper we propose a VLSI implementable architecture called Cube Connected Tree having advantageous properties of both tree and hypercube. This structure has a fixed low degree of nodes for any size of the network unlike the hypercube where the node degree is dependent on the size of the hypercube. The degree-diameter product metric [26]of CCT is low compared to that of a hypercube of comparable size. It overcomes the data congestion problem near the root of the binary tree by having multiple roots in the structure, thereby enhancing the I/O bandwidth of the system. The complexity of the VLSI layout of this structure has been addressed within the grid model of Thompson [12]. By using spare links and PEs, fault tolerance capabilities of the system have been enhanced. Easy programmability of this structure has been demonstrated by designing polylogarithmic algorithms for sorting and discrete Fourier transform.