Given an arbitrary polygon P with holes, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P . This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory and present an open problem based on visibility which is a cl...

QuickSort Hoare [1962] (A) Pick a pivot element from array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the pivot itself. (C) Recursively sort the subarrays, and concatenate them. Randomized QuickSort (A) Pick a pivot element uniformly at random from the array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the...

QuickSort Hoare [1962] (A) Pick a pivot element from array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the pivot itself. (C) Recursively sort the subarrays, and concatenate them. Randomized QuickSort (A) Pick a pivot element uniformly at random from the array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the...

The behavior of the simplex algorithm is a widely studied subject. Specifically, the question of the existence of a polynomial pivot rule for the simplex algorithm is of major importance. Here, we give exponential lower bounds for three history-based pivot rules. Those rules decide their next step based on memory of the past steps. In particular, we study Zadeh’s least entered rule, Johnson’s l...

Consider the combinational complexity L(f) of Boolean functions over the basis fl={]'l]': {0, 1}2->{0, 1}}. A new method for proving linear lower bounds of size 2n is presented. we establish for a special sequence of functions [: {0, 1} "+2 g(n)+x-> {0, 1}: 2.5n <=L(f)<-6n. Also a trade-off result between circuit complexity and formula size is derived. 1. Introduction. The interest in lower bou...

It has been argued and stated that the 21st century is America's Pacific Century, the century or millennium of Asia, and century or era of China, the inevitable superpower. This paper explains the essence of the new changes when the Pivot Policy was announced by President Obama in 2011. The main question discussed is therefore the following: ‘Why (and how) did President Obama adopt (and impleme...