Partitioning random graphs into large cycles

AbstractLet r ⩾ 1 be a fixed positive integer. We give the limiting distribution for the probability that the vertices of a random graph can be partitioned equitably into r cycles

On the value of a random minimum spanning tree problem

AbstractSuppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let Ln be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: limn → ∞E(Ln) = ζ(3)/D where ζ(3) = Σk = 1∞ 1/k3 = 1.202… and for any ε > 0 limn → ∞ Pr(|Ln− ζ(3)/D|) > ε) = 0

On the value of a random minimum spanning tree problem

AbstractSuppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let Ln be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: limn → ∞E(Ln) = ζ(3)/D where ζ(3) = Σk = 1∞ 1/k3 = 1.202… and for any ε > 0 limn → ∞ Pr(|Ln− ζ(3)/D|) > ε) = 0

Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

On an optimization problem with nested constraints

AbstractWe describe algorithms for solving the integer programming problem maximise ∑j=1n⨍j(xj),subject to ∑jϵSixj⩽bi, i=1,…,m,xj⩾0, j=1,…,n, where the ⨍i are concave nondecreasing and the Si form a nested collection of sets. For the general problem, we present an algorithm of time-complexity O(n log2 n log b), where b is less than the largest of the bi. We also examine the case in which all ⨍i are identical and give an algorithm requiring O(n + m log m) time. Both algorithms use only O(n) space

On the independence and chromatic numbers of random regular graphs

AbstractLet Gr denote a random r-regular graph with vertex set {1, 2, …, n} and α(Gr) and χ(Gr) denote respectively its independence and chromatic numbers. We show that with probability going to 1 as n → ∞ respectively |δ(Gr) − 2nr(logr − log logr + 1 − log 2)|⩽γnr and |χ(Gr) − r2 log r − 8r log logr(log)2| ⩽ 8r log log r(log r)2 provided r = o(nθ), θ < 13, 0 < ε < 1, are constants, and r ≥ rε, where rε depends on ε only

Wear Minimization for Cuckoo Hashing: How Not to Throw a Lot of Eggs into One Basket

We study wear-leveling techniques for cuckoo hashing, showing that it is possible to achieve a memory wear bound of $\log\log n+O(1)$ after the insertion of $n$ items into a table of size $Cn$ for a suitable constant $C$ using cuckoo hashing. Moreover, we study our cuckoo hashing method empirically, showing that it significantly improves on the memory wear performance for classic cuckoo hashing and linear probing in practice.Comment: 13 pages, 1 table, 7 figures; to appear at the 13th Symposium on Experimental Algorithms (SEA 2014

A probabilistic analysis of the next fit decreasing bin packing heuristic

A probabilistic analysis is presented of the Next Fit Decreasing bin packing heuristic, in which bins are opened to accomodate the items in order of decreasing size

Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within $A$. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall expected time and space requirement in the average case are proven to be $O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.Comment: 11 pages, 1 figur