Topology Matters: Smoothed Competitiveness of Metrical Task Systems

We consider online problems that can be modeled as metrical task systems: An online algorithm resides in a graph of n nodes and may move in this graph at a cost equal to the distance. The algorithm has to service a sequence of tasks that arrive over time; each task specifies for each node a request cost that is incurred if the algorithm services the task in this particular node. The objective is to minimize the total request plus travel cost. Borodin, Linial and Saks gave a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n-1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we add some random noise to the request costs and analyze the competitive ratio of WFA on the perturbed sequence. We prove upper and matching lower bounds. Our analysis reveals that the smoothed competitive ratio of WFA is much better than its (worst case) competitive ratio and that it depends on several topological parameters of the graph underlying the metric, such as maximum degree, diameter, etc. For example, already for moderate perturbations, the smoothed competitive ratio of WFA is O(log(n)) on a clique and O(sqrt{n}) on a line. We also provide the first average case analysis of WFA. For a large class of probability distributions, we prove that WFA has O(log(D)) expected competitive ratio, where D is the maximum degree of the underlying graph

Boxicity of Line Graphs

Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). For the d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d)) \rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d) was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics, 308(23):5795-5800, 2008]. The above results are consequences of bounds that we obtain for the boxicity of fully subdivided graphs (a graph which can be obtained by subdividing every edge of a graph exactly once).Comment: 14 page

JDATATRANS for Array Obfuscation in Java Source Code to Defeat Reverse Engineering from Decompiled Codes

Software obfuscation or obscuring a software is an approach to defeat the practice of reverse engineering a software for using its functionality illegally in the development of another software. Java applications are more amenable to reverse engineering and re-engineering attacks through methods such as decompilation because Java class files store the program in a semi complied form called 'byte' codes. The existing obfuscation systems obfuscate the Java class files. Obfuscated source code produce obfuscated byte codes and hence two level obfuscation (source code and byte code level) of the program makes it more resilient to reverse engineering attacks. But source code obfuscation is much more difficult due to richer set of programming constructs and the scope of the different variables used in the program and only very little progress has been made on this front. Hence programmers resort to adhoc manual ways of obscuring their program which makes it difficult for its maintenance and usability. To address this issue partially, we developed a user friendly tool JDATATRANS to obfuscate Java source code by obscuring the array usages. Using various array restructuring techniques such as 'array splitting', 'array folding' and 'array flattening', in addition to constant hiding, our system obfuscate the input Java source code and produce an obfuscated Java source code that is functionally equivalent to the input program. We also perform a number of experiments to measure the potency, resilience and cost incurred by our tool.Comment: Manuscript submitted to ACM COMPUTE 2009 Conference,Bangalor