Average-Case Complexity of Shellsort

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is \Omega (pn^{1 + 1/p) for all $p \leq \log n$. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.Comment: 11 pages. Submitted to ICALP'9

Shining Light On Shadow Stacks

Control-Flow Hijacking attacks are the dominant attack vector against C/C++ programs. Control-Flow Integrity (CFI) solutions mitigate these attacks on the forward edge,i.e., indirect calls through function pointers and virtual calls. Protecting the backward edge is left to stack canaries, which are easily bypassed through information leaks. Shadow Stacks are a fully precise mechanism for protecting backwards edges, and should be deployed with CFI mitigations. We present a comprehensive analysis of all possible shadow stack mechanisms along three axes: performance, compatibility, and security. For performance comparisons we use SPEC CPU2006, while security and compatibility are qualitatively analyzed. Based on our study, we renew calls for a shadow stack design that leverages a dedicated register, resulting in low performance overhead, and minimal memory overhead, but sacrifices compatibility. We present case studies of our implementation of such a design, Shadesmar, on Phoronix and Apache to demonstrate the feasibility of dedicating a general purpose register to a security monitor on modern architectures, and the deployability of Shadesmar. Our comprehensive analysis, including detailed case studies for our novel design, allows compiler designers and practitioners to select the correct shadow stack design for different usage scenarios.Comment: To Appear in IEEE Security and Privacy 201

Fast Parallel Operations on Search Trees

Using (a,b)-trees as an example, we show how to perform a parallel split with logarithmic latency and parallel join, bulk updates, intersection, union (or merge), and (symmetric) set difference with logarithmic latency and with information theoretically optimal work. We present both asymptotically optimal solutions and simplified versions that perform well in practice - they are several times faster than previous implementations

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,...,n}.Comment: 14 page

2-stack pushall sortable permutations

In the 60's, Knuth introduced stack-sorting and serial compositions of stacks. In particular, one significant question arise out of the work of Knuth: how to decide efficiently if a given permutation is sortable with 2 stacks in series? Whether this problem is polynomial or NP-complete is still unanswered yet. In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and describe all its sortings. This result is a step to the solve the general 2-stack sorting problem in polynomial time.Comment: 41 page

A Visual Stack Based Paradigm for Visualization Environments

We present a new visual paradigm for Visualization Systems, inspired by stack-based programming. Most current implementations of Visualization systems are based on directional graphs. However directional graphs as a visual representation of execution, though initially quite intuitive, quickly grow cumbersome and difficult to follow under complex examples. Our system presents the user with a simple and compact methodology of visually stacking actions directly on top of data objects as a way of creating filter scripts. We explore and address extensions to the basic paradigm to allow for: multiple data input or data output objects to and from execution action modules, execution thread jumps and loops, encapsulation, and overall execution control. We exploit the dynamic nature of current computer graphic interfaces by utilizing features such as drag-and-drop, color emphasis and object animation to indicate action, looping, message/parameter passing; to furnish an overall better understanding of the resulting laid out execution scripts