Dynamic tree shortcut with constant degree

LNCS v.9188 entitled: Computing and Combinatorics: 21st International Conference, COCOON 2015, Beijing, China, August 4-6, 2015, ProceedingsGiven a rooted tree with n nodes, the tree shortcut problem is to add a set of shortcut edges to the tree such that the shortest path from each node to any of its ancestors is of length O(log n) and the degree increment of each node is constant. We consider in this paper the dynamic version of the problem, which supports node insertion and deletion. For insertion, a node can be inserted as a leaf node or an internal node by sub-dividing an existing edge. For deletion, a leaf node can be deleted, or an internal node can be merged with its single child. We propose an algorithm that maintains a set of shortcut edges in O(log n) time for an insertion or deletion.postprin

Faster Deterministic Fully-Dynamic Graph Connectivity

We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in $O(\log^2n/\log\log n)$ amortized time and connectivity queries in $O(\log n/\log\log n)$ worst-case time, where $n$ is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have $O(\log^2n)$ amortized update time and $O(\log n/\log\log n)$ worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer machine with standard $AC^0$ instructions.Comment: To appear at SODA 2013. 19 pages, 1 figur

The Max-Distance Network Creation Game on General Host Graphs

In this paper we study a generalization of the classic \emph{network creation game} in the scenario in which the $n$ players sit on a given arbitrary \emph{host graph}, which constrains the set of edges a player can activate at a cost of $\alpha \geq 0$ each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its \emph{maximum} distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of $\Omega (\sqrt{ n / (1+\alpha)})$ for any $\alpha = o(n)$. Notice that this implies a counter-intuitive lower bound of $\Omega(\sqrt{n})$ for very small values of $\alpha$ (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either $k$-regular (for any constant $k \geq 3$), or a 2-dimensional grid, the PoA is still $\Omega(1+\min\{\alpha, \frac{n}{\alpha}\})$, which is proven to be tight for $\alpha=\Omega(\sqrt{n})$. On the positive side, if $\alpha \geq n$, we show the PoA is $O(1)$. Finally, in the case in which the host graph is very sparse (i.e., $|E(H)|=n-1+k$, with $k=O(1)$), we prove that the PoA is $O(1)$, for any $\alpha$.Comment: 17 pages, 4 figure

Fully Dynamic Connectivity in O(log⁡n(log⁡log⁡n)2)O(\log n(\log\log n)^2) Amortized Expected Time

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with $O(\log n(\log\log n)^2)$ amortized expected update time and $O(\log n/\log\log\log n)$ worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011)

Incremental 22-Edge-Connectivity in Directed Graphs

In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ and $w$ are not $2$-edge-connected, we can produce in constant time a "witness" of this property, by exhibiting an edge that is contained in all paths from $v$ to $w$ or in all paths from $w$ to $v$. (ii) Report in $O(n)$ time all the $2$-edge-connected blocks of $G$. To the best of our knowledge, this is the first dynamic algorithm for $2$-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.Comment: Full version of paper presented at ICALP 201

Fast and simple connectivity in graph timelines

In this paper we study the problem of answering connectivity queries about a \emph{graph timeline}. A graph timeline is a sequence of undirected graphs $G_1,\ldots,G_t$ on a common set of vertices of size $n$ such that each graph is obtained from the previous one by an addition or a deletion of a single edge. We present data structures, which preprocess the timeline and can answer the following queries: - forall$(u,v,a,b)$ -- does the path $u\to v$ exist in each of $G_a,\ldots,G_b$? - exists$(u,v,a,b)$ -- does the path $u\to v$ exist in any of $G_a,\ldots,G_b$? - forall2$(u,v,a,b)$ -- do there exist two edge-disjoint paths connecting $u$ and $v$ in each of $G_a,\ldots,G_b$ We show data structures that can answer forall and forall2 queries in $O(\log n)$ time after preprocessing in $O(m+t\log n)$ time. Here by $m$ we denote the number of edges that remain unchanged in each graph of the timeline. For the case of exists queries, we show how to extend an existing data structure to obtain a preprocessing/query trade-off of $\langle O(m+\min(nt, t^{2-\alpha})), O(t^\alpha)\rangle$ and show a matching conditional lower bound.Comment: 21 pages, extended abstract to appear in WADS'1

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

A Random Structure for Optimum Cache Size Distributed hash table (DHT) Peer-to-Peer design

We propose a new and easily-realizable distributed hash table (DHT) peer-to-peer structure, incorporating a random caching strategy that allows for {\em polylogarithmic search time} while having only a {\em constant cache} size. We also show that a very large class of deterministic caching strategies, which covers almost all previously proposed DHT systems, can not achieve polylog search time with constant cache size. In general, the new scheme is the first known DHT structure with the following highly-desired properties: (a) Random caching strategy with constant cache size; (b) Average search time of $O(log^{2}(N))$; (c) Guaranteed search time of $O(log^{3}(N))$; (d) Truly local cache dynamics with constant overhead for node deletions and additions; (e) Self-organization from any initial network state towards the desired structure; and (f) Allows a seamless means for various trade-offs, e.g., search speed or anonymity at the expense of larger cache size.Comment: 13 pages, 2 figures, preprint versio

Succinct Representations of Permutations and Functions

We investigate the problem of succinctly representing an arbitrary permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for any i and any (positive or negative) integer power k. A representation taking (1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant \epsilon <= 1. A representation taking the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time. We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed quickly for any i and any integer power k. We give a representation that takes (1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and computes arbitrary positive powers in constant time. It can also be used to compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time. We place emphasis on the redundancy, or the space beyond the information-theoretic lower bound that the data structure uses in order to support operations efficiently. A number of lower bounds have recently been shown on the redundancy of data structures. These lower bounds confirm the space-time optimality of some of our solutions. Furthermore, the redundancy of one of our structures "surpasses" a recent lower bound by Golynski [Golynski, SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the Proceedings of ICALP 2003 and 2004. However, all results in this version are improved over the earlier conference versio