arXiv subjects

Benjamin Doerr

Publications and source records attributed to Benjamin Doerr.

143 records · Page 8Linked to original sources

arXiv2011

Black-Box Complexities of Combinatorial Problems

Black-box complexity is a complexity theoretic measure for how difficult a problem is to be optimized by a general purpose optimization algorithm. It is thus one of the few means trying to understand which problems are tractable for genetic algorithms and other randomized search heuristics. Most previous work on black-box complexity is on artificial test functions. In this paper, we move a step forward and give a detailed analysis for the two combinatorial problems minimum spanning tree and single-source shortest paths. Besides giving interesting bounds for their black-box complexities, our work reveals that the choice of how to model the optimization problem is non-trivial here. This in particular comes true where the search space does not consist of bit strings and where a reasonable definition of unbiasedness has to be agreed on.

cs.NE↗

arXiv2011

Multiplicative Drift Analysis

In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time $O(n \log n)$, where $n$ is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most ${(1+o(1)) 1.39 \euler n\ln (n)}$, again using multiplicative drift analysis. We also prove a corresponding lower bound of ${(1-o(1))e n\ln(n)}$ which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours.

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arXiv2010

Quasirandom Rumor Spreading: An Experimental Analysis

We empirically analyze two versions of the well-known "randomized rumor spreading" protocol to disseminate a piece of information in networks. In the classical model, in each round each informed node informs a random neighbor. In the recently proposed quasirandom variant, each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list. While for sparse random graphs a better performance of the quasirandom model could be proven, all other results show that, independent of the structure of the lists, the same asymptotic performance guarantees hold as for the classical model. In this work, we compare the two models experimentally. This not only shows that the quasirandom model generally is faster, but also that the runtime is more concentrated around the mean. This is surprising given that much fewer random bits are used in the quasirandom process. These advantages are also observed in a lossy communication model, where each transmission does not reach its target with a certain probability, and in an asynchronous model, where nodes send at random times drawn from an exponential distribution. We also show that typically the particular structure of the lists has little influence on the efficiency.

cs.DS↗

arXiv2010

Faster Black-Box Algorithms Through Higher Arity Operators

We extend the work of Lehre and Witt (GECCO 2010) on the unbiased black-box model by considering higher arity variation operators. In particular, we show that already for binary operators the black-box complexity of \leadingones drops from $Θ(n^2)$ for unary operators to $O(n \log n)$. For \onemax, the $Ω(n \log n)$ unary black-box complexity drops to O(n) in the binary case. For $k$-ary operators, $k \leq n$, the \onemax-complexity further decreases to $O(n/\log k)$.

cs.NE↗

arXiv2010

Asymptotically Optimal Randomized Rumor Spreading

We propose a new protocol solving the fundamental problem of disseminating a piece of information to all members of a group of n players. It builds upon the classical randomized rumor spreading protocol and several extensions. The main achievements are the following: Our protocol spreads the rumor to all other nodes in the asymptotically optimal time of (1 + o(1)) \log_2 n. The whole process can be implemented in a way such that only O(n f(n)) calls are made, where f(n)= ω(1) can be arbitrary. In contrast to other protocols suggested in the literature, our algorithm only uses push operations, i.e., only informed nodes take active actions in the network. To the best of our knowledge, this is the first randomized push algorithm that achieves an asymptotically optimal running time.

cs.DS↗

arXiv2010

Non-Existence of Linear Universal Drift Functions

Drift analysis has become a powerful tool to prove bounds on the runtime of randomized search heuristics. It allows, for example, fairly simple proofs for the classical problem how the (1+1) Evolutionary Algorithm (EA) optimizes an arbitrary pseudo-Boolean linear function. The key idea of drift analysis is to measure the progress via another pseudo-Boolean function (called drift function) and use deeper results from probability theory to derive from this a good bound for the runtime of the EA. Surprisingly, all these results manage to use the same drift function for all linear objective functions. In this work, we show that such universal drift functions only exist if the mutation probability is close to the standard value of $1/n$.

cs.NE↗

arXiv2010

Optimizing Monotone Functions Can Be Difficult

Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotone. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant $c$ in the mutation probability $p(n) = c/n$ can make a decisive difference. We show that if $c < 1$, then the (1+1) evolutionary algorithm finds the optimum of every such function in $Θ(n \log n)$ iterations. For $c=1$, we can still prove an upper bound of $O(n^{3/2})$. However, for $c > 33$, we present a strictly monotone function such that the (1+1) evolutionary algorithm with overwhelming probability does not find the optimum within $2^{Ω(n)}$ iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors.

cs.NE↗

arXiv2010

Quasi-Random Rumor Spreading: Reducing Randomness Can Be Costly

We give a time-randomness tradeoff for the quasi-random rumor spreading protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete graphs. In this protocol, the goal is to spread a piece of information originating from one vertex throughout the network. Each vertex is assumed to have a (cyclic) list of its neighbors. Once a vertex is informed by one of its neighbors, it chooses a position in its list uniformly at random and then informs its neighbors starting from that position and proceeding in order of the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009] showed that after $(1+o(1))(\log_2 n + \ln n)$ rounds, the rumor will have been broadcasted to all nodes with probability $1 - o(1)$. We study the broadcast time when the amount of randomness available at each node is reduced in natural way. In particular, we prove that if each node can only make its initial random selection from every $\ell$-th node on its list, then there exists lists such that $(1-\varepsilon) (\log_2 n + \ln n - \log_2 \ell - \ln \ell)+\ell-1$ steps are needed to inform every vertex with probability at least $1-O\bigl(\exp\bigl(-\frac{n^\varepsilon}{2\ln n}\bigr)\bigr)$. This shows that a further reduction of the amount of randomness used in a simple quasi-random protocol comes at a loss of efficiency.

cs.DS↗

arXiv2010

Randomized Rounding for Routing and Covering Problems: Experiments and Improvements

Following previous theoretical work by Srinivasan (FOCS 2001) and the first author (STACS 2006) and a first experimental evaluation on random instances (ALENEX 2009), we investigate how the recently developed different approaches to generate randomized roundings satisfying disjoint cardinality constraints behave when used in two classical algorithmic problems, namely low-congestion routing in networks and max-coverage problems in hypergraphs. We generally find that all randomized rounding algorithms work well, much better than what is guaranteed by existing theoretical work. The derandomized versions produce again significantly better rounding errors, with running times still negligible compared to the one for solving the corresponding LP. It thus seems worth preferring them over the randomized variants. The data created in these experiments lets us propose and investigate the following new ideas. For the low-congestion routing problems, we suggest to solve a second LP, which yields the same congestion, but aims at producing a solution that is easier to round. Experiments show that this reduces the rounding errors considerably, both in combination with randomized and derandomized rounding. For the max-coverage instances, we generally observe that the greedy heuristics also performs very good. We develop a strengthened method of derandomized rounding, and a simple greedy/rounding hybrid approach using greedy and LP-based rounding elements, and observe that both these improvements yield again better solutions than both earlier approaches on their own. For unit disk max-domination, we also develop a PTAS. Contrary to all other algorithms investigated, it performs not much better in experiments than in theory; thus, unless extremely good solutions are to be obtained with huge computational resources, greedy, LP-based rounding or hybrid approaches are preferable.

cs.DS↗

arXiv2010

Deterministic Random Walks on Regular Trees

Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\Z^d$ and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite $k$-ary tree ($k \ge 3$), we show that for any deviation $D$ there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least $D$ more chips than expected in the random walk model. However, to achieve a deviation of $D$ it is necessary that at least $\exp(Ω(D^2))$ vertices contribute by being occupied by a number of chips not divisible by $k$ at a certain time.

math.CO↗

arXiv2007

Deterministic Random Walks on the Two-Dimensional Grid

Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. We analyze the difference between Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time, the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8, if all vertices serve their neighbors in clockwise or counterclockwise order and 7.3 otherwise. This result in particular shows that the order in which the neighbors are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

math.CO↗

arXiv2006

Hereditary Discrepancies in Different Numbers of Colors II

We bound the hereditary discrepancy of a hypergraph $\HH$ in two colors in terms of its hereditary discrepancy in $c$ colors. We show that $\herdisc(\HH,2) \le K c \herdisc(\HH,c)$, where $K$ is some absolute constant. This bound is sharp.

cs.DM↗

arXiv2006

Unbiased Matrix Rounding

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics. We improve earlier solutions of different authors in two ways. For rounding matrices of size $m \times n$, we reduce the runtime from $O((m n)^2 Second, our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. This is particularly useful in the statistics application of controlled rounding. The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased, that is, for all entries $y_{ij}$ of our rounding, we have $E(y_{ij}) = x_{ij}$, where $x_{ij}$ is the corresponding entry of the input matrix.

cs.DS↗

arXiv2006

Discrepancy of Symmetric Products of Hypergraphs

For a hypergraph ${\mathcal H} = (V,{\mathcal E})$, its $d$--fold symmetric product is $Δ^d {\mathcal H} = (V^d,\{E^d |E \in {\mathcal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${disc}(Δ^d {\mathcal H},2) \le {disc}({\mathcal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${disc}(Δ^d {\mathcal H},c) = Ω_d({disc}({\mathcal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\mathcal H}^d$, which satisfies ${disc}({\mathcal H}^d,c) = O_{c,d}({disc}({\mathcal H},c)^d)$.

math.CO↗

arXiv2006

Improved Bounds and Schemes for the Declustering Problem

The declustering problem is to allocate given data on parallel working storage devices in such a manner that typical requests find their data evenly distributed on the devices. Using deep results from discrepancy theory, we improve previous work of several authors concerning range queries to higher-dimensional data. We give a declustering scheme with an additive error of $O_d(\log^{d-1} M)$ independent of the data size, where $d$ is the dimension, $M$ the number of storage devices and $d-1$ does not exceed the smallest prime power in the canonical decomposition of $M$ into prime powers. In particular, our schemes work for arbitrary $M$ in dimensions two and three. For general $d$, they work for all $M\geq d-1$ that are powers of two. Concerning lower bounds, we show that a recent proof of a $Ω_d(\log^{\frac{d-1}{2}} M)$ bound contains an error. We close the gap in the proof and thus establish the bound.

cs.DM↗

arXiv2006

Deterministic Random Walks on the Integers

Jim Propp's P-machine, also known as the "rotor router model" is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c_1, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L), for the L_2 average of a contiguous set of intervals even to O(sqrt{log L}). All these bounds are tight.

math.CO↗

arXiv2004

Balanced Partitions of Vector Sequences

Let $d, r \in \N$, $\|\cdot\|$ any norm on $\R^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,...$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1, ..., V_r$ in a balanced manner with respect to the partial sums: For all $n \in \N$, $\ell \le r$, we have $\|\sum_{i \le k, v_i \in V_\ell} v_i - \tfrac 1r \sum_{i \le k} v_i\| \le 2.0005 d$. A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of Bárány and Grinberg (1981) to partitions in arbitrarily many classes.

math.CO↗