On the insertion time of random walk cuckoo hashing

Cuckoo Hashing is a hashing scheme invented by Pagh and Rodler. It uses $d\geq 2$ distinct hash functions to insert items into the hash table. It has been an open question for some time as to the expected time for Random Walk Insertion to add items. We show that if the number of hash functions $d=O(1)$ is sufficiently large, then the expected insertion time is $O(1)$ per item.Comment: 9 page

On edge disjoint spanning trees in a randomly weighted complete graph

Assume that the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ edges weights. We consider the expected minimum total weight $\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show that $\mu_k\approx k^2$. Most of the paper is concerned with the case $k=2$. We show that \m_2 tends to an explicitly defined constant and that $\mu_2\approx 4.1704288\ldots$.Comment: Fixed minor issue

Minimum-cost matching in a random graph with random costs

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$, suppose that each edge $e\in E$ is given an independent uniform exponential rate one cost. Let $C(G)$ denote the random variable equal to the length of the minimum cost perfect matching, assuming that $G$ contains at least one. We show that w.h.p. if $d=np\gg(\log n)^2$ then w.h.p. {\bf E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}. This generalises the well-known result for the case $G=K_{n,n}$. We also show that w.h.p. {\bf E}[C(G_{n,p})] =(1+o(1))\frac{\p^2}{12p} along with concentration results for both types of random graph.Comment: Replaces an earlier paper where $G$ was an arbitrary regular bipartite grap

On random k-out sub-graphs of large graphs

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $\delta(G)\geq (\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $\phi(n)$ (or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$

The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

We consider a random walk process, introduced by Orenshtein and Shinkar [Tal Orenshtein and Igor Shinkar, 2014], which prefers to visit previously unvisited edges, on the random r-regular graph G_r for any odd r >= 3. We show that this random walk process has asymptotic vertex and edge cover times 1/(r-2)n log n and r/(2(r-2))n log n, respectively, generalizing the result from [Cooper et al., to appear] from r = 3 to any larger odd r. This completes the study of the vertex cover time for fixed r >= 3, with [Petra Berenbrink et al., 2015] having previously shown that G_r has vertex cover time asymptotic to rn/2 when r >= 4 is even

Asymmetric trends and European monetary policy in t he Post-Bretton Woods Era

In public debate the crisis of the eurozone has been laid on the footstep of imprudent government finance. This paper argues that the depth and longevity of the crisis instead is due to ‘asymmetric trends’ that are inherent in the eurozone. We show this, first, by a test of the ‘one-size-fits-all’ ECB monetary policy. The results provide an estimate of how ECB at the same time fuelled some ‘bubble economies’ and put on a deflationary pressure in other economies. Second, we measure how the higher inflation rate in the periphery eroded its international competitiveness under the restriction of the ‘irrevocably fixed exchange rates’. This is compared with the development during the preceding half century and periods with more flexible exchange rates. Before the EMS crisis of the early 1990s, the EMS had its ‘soft’ and its ‘hard’ phase. During the ‘soft’ phase, so called realignments of exchange rates adjusted for diverging trends in inflation and competitiveness. The ‘hard’ phase ended with the relaxation of the narrow band for the exchange rates. Again, a ‘soft’ phase followed until the ‘irrevocably’ fixing of the exchange rates that launched the euro. The catch-up and convergence of incomes within Western Europe have been largely enhanced by exchange rate adjustments. A pendulum between ‘soft’ and ‘hard’ phases have characterized European monetary policy and significantly conditioned European economic growth. A bottom line is that, ironically, the Maastricht aim of further integration actually is counteracted by the economic mechanisms of the monetary unification