Algorithms for near solutions to polynomial equations

Let F(x,y)$F(x,y)$ be a polynomial over a field K$K$ and m$m$ a nonnegative integer. We call a polynomial g$g$ over K$K$ an m$m$-near solution of F(x,y)$F(x,y)$ if there exists a c∈K$c\in K$ such that F(x,g)=cx^m$F(x,g)=c{x}^m$, and the number c$c$ is called an m$m$-value of F(x,y)$F(x,y)$ corresponding to g$g$. In particular, c$c$ can be 0. Hence, by viewing F(x,y)=0$F(x,y)=0$ as a polynomial equation over K[x]$K\left[x\right]$ with variable y$y$, every solution of the equation F(x,y)=0$F(x,y)=0$ in K[x]$K\left[x\right]$ is also an m$m$-near solution. We provide an algorithm that gives all m$m$-near solutions of a given polynomial F(x,y)$F(x,y)$ over K$K$, and this algorithm is polynomial time reducible to solving one variable equations over K$K$. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

Conjunctive query evaluation by search-tree revisited

The most natural and perhaps most frequently used method for testing membership of an individual tuple in a conjunctive query is based on searching trees of partial solutions, or search-trees. We investigate the question of evaluating conjunctive queries with a time-bound guarantee that is measured as a function of the size of the optimal search-tree. We provide an algorithm that, given a database D$D$, a conjunctive query Q$Q$, and a tuple a$a$, tests whether Q(a)$Q\left(a\right)$ holds in D$D$ in time bounded by a polynomial in (sn)^logk(sn)^loglogn${\left(sn\right)}^{logk}{\left(sn\right)}^{loglogn}$ and n^r$n^r$, where n$n$ is the size of the domain of the database, k$k$ is the number of bound variables of the conjunctive query, s$s$ is the size of the optimal search-tree, and r$r$ is the maximum arity of the relations. In many cases of interest, this bound is significantly smaller than the n^O(k)$n^{O\left(k\right)}$ bound provided by the naive search-tree method. Moreover, our algorithm has the advantage of guaranteeing the bound for any given conjunctive query. In particular, it guarantees the bound for queries that admit an equivalent form that is much easier to evaluate, even when finding such a form is an NP-hard task. Concrete examples include the conjunctive queries that can be non-trivially folded into a conjunctive query of bounded size or bounded treewidth. All our results translate to the context of constraint-satisfaction problems via the well-publicized correspondence between both frameworks.     

The cell probe complexity of succinct data structures

We consider time-space tradeoffs for static data structure problems in the cell probe model with word size 1 (the bit probe model). In this model, the goal is to represent n$n$-bit data with s=n+r$s=n+r$ bits such that queries (of a certain type) about the data can be answered by reading at most t$t$ bits of the representation. Ideally, we would like to keep both s$s$ and t$t$ small, but there are tradeoffs between the values of s$s$ and t$t$ that limit the possibilities of keeping both parameters small.     

Optimal number of annuli for maximizing the lifetime of sensor networks

The most effective way to maximize the lifetime of a wireless sensor network (WSN) is to allocate initial energy to sensors such that they exhaust their energy at the same time. The lifetime of a WSN as well as an optimal initial energy allocation are determined by a network design. The main contribution of the paper is to show that the lifetime of a WSN can be maximized by an optimal network design. We represent the network lifetime as a function of the number m$m$ of annuli and show that m$m$ has significant impact on network lifetime. We prove that if the energy consumed by data transmission is proportional to d^α+c$d^{\alpha }+c$, where d$d$ is the distance of data transmission and α$\alpha$ and c$c$ are some constants, then for a circular area of interest with radius R$R$, the optimal number of annuli that maximizes the network lifetime is m=R((α−1)/c)^1/α$m=R{((\alpha -1)/c)}^{1/\alpha }$ for an arbitrary sensor density function.     

Bounding the radii of balls meeting every connected component of semi-algebraic sets

We prove an explicit bound on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S⊂R^k$S\subset {\mathbb{R}}^k$ defined by a weak sign condition involving s$s$ polynomials in Z[_X1,…,_Xk]$\mathbb{Z}[X_1,\dots ,X_k]$ having degrees at most d$d$, and whose coefficients have bitsizes at most τ$\tau$. Our bound is an explicit function of s,d,k$s,d,k$ and τ$\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of S$S$ (including the unbounded components). While asymptotic bounds of the form 2^τdO(k)$2^{\tau d^{O\left(k\right)}}$ on these quantities were known before, some applications require bounds which are explicit and which hold for all values of s,d,k$s,d,k$ and τ$\tau$. The bounds proved in this paper are of this nature.     

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G=(V,E)$G=(V,E)$, with n$n$ nodes and m$m$ non-negatively real weighted edges, and a single source shortest paths tree (SPT) T$T$ of G$G$ rooted at an arbitrary node r$r$. If an edge in T$T$ is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge  , instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O(mlog²n)$O\left(m{log}^{2}n\right)$ time and O(m)$O\left(m\right)$ space algorithm to find a best swap edge for every edge of T$T$, thus improving for m=o(n²/log²n)$m=o(n^2/{log}^{2}n)$ the previously known O(n²)$O\left(n^2\right)$ time and space complexity algorithm.     

An approximation algorithm to the k-Steiner Forest problem

Given a graph G$G$, an integer k$k$, and a demand set D={(_s1,_t1),…,(_sl,_tl)}$D=\{(s_1,t_1),\dots ,(s_l,t_l)\}$, the k$k$-Steiner Forest problem finds a forest in graph G$G$ to connect at least k$k$ demands in D$D$ such that the cost of the forest is minimized. This problem was proposed by Hajiaghayi and Jain in SODA’06. Thereafter, using a Lagrangian relaxation technique, Segev et al. gave the first approximation algorithm to this problem in ESA’06, with performance ratio O(n^2/3logl)$O(n^{2/3}logl)$. We give a simpler and faster approximation algorithm to this problem with performance ratio O(n^2/3logk)$O(n^{2/3}logk)$ via greedy approach, improving the previously best known ratio in the literature.     

Adaptive timeliness of consensus in presence of crash and timing faults

The Δ$\mathrm{\Delta }$-timed uniform consensus is a stronger variant of the traditional consensus and it satisfies the following additional property: every correct process terminates its execution within a constant time Δ$\mathrm{\Delta }$ (Δ$\mathrm{\Delta }$-timeliness), and no two processes decide differently (uniformity). In this paper, we consider the Δ$\mathrm{\Delta }$-timed uniform consensus problem in presence of _fc$f_{\mathrm{c}}$ crash processes and _ft$f_{\mathrm{t}}$ timing-faulty processes, and propose a Δ$\mathrm{\Delta }$-timed uniform consensus algorithm. The proposed algorithm is adaptive in the following sense: it solves the Δ$\mathrm{\Delta }$-timed uniform consensus when at least _ft+1$f_{\mathrm{t}}+1$ correct processes exist in the system. If the system has less than _ft+1$f_{\mathrm{t}}+1$ correct processes, the algorithm cannot solve the Δ$\mathrm{\Delta }$-timed uniform consensus. However, as long as _ft+1$f_{\mathrm{t}}+1$ processes are non-crashed, the algorithm solves (non-timed) uniform consensus. We also investigate the maximum number of faulty processes that can be tolerated. We show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes requires that the system has at least _ft+1$f_{\mathrm{t}}+1$ correct processes. This impossibility result implies that the proposed algorithm attains the maximum resilience about the number of faulty processes. We also show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes cannot solve the (non-timed) uniform consensus when the system has less than _ft+1$f_{\mathrm{t}}+1$ non-crashed processes. This impossibility result implies that our algorithm attains the maximum adaptiveness.     

Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

Efficient pebbling for list traversal synopses with application to program rollback

We show how to support efficient back traversal in a unidirectional list, using small memory and with essentially no slowdown in forward steps. Using O(lgn)$O(lgn)$ memory for a list of size n$n$, the i$i$’th back-step from the farthest point reached so far takes O(lgi)$O(lgi)$ time in the worst case, while the overhead per forward step is at most ?$?$ for arbitrary small constant ?>0$?>0$. An arbitrary sequence of forward and back steps is allowed. A full trade-off between memory usage and time per back-step is presented: k$k$ vs. kn^1/k$k{n}^{1/k}$ and vice versa. Our algorithms are based on a novel pebbling technique which moves pebbles on a virtual binary, or n^1/k$n^{1/k}$-ary, tree that can only be traversed in a pre-order fashion.