Barbara Geissmann

Collaborative Delivery with Energy-Constrained Mobile Robots

We consider the problem of collectively delivering some message from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent has a limited energy which constrains the distance it can move. Hence multiple agents need to collaborate to move the message, each agent handing over the message to the next agent to carry it forward. Given the positions of the agents in the graph and their respective budgets, the problem of finding a feasible movement schedule for the agents can be challenging. We consider two variants of the problem: in non-returning delivery, the agents can stop anywhere; whereas in returning delivery, each agent needs to return to its starting location, a variant which has not been studied before. We first provide a polynomial-time algorithm for returning delivery on trees, which is in contrast to the known (weak) \(\mathrm {NP}\)-hardness of the non-returning version. In addition, we give resource-augmented algorithms for returning delivery in general graphs. Finally, we give tight lower bounds on the required resource augmentation for both variants of the problem. In this sense, our results close the gap left by previous research.

Recurring Comparison Faults: Sorting and Finding the Minimum

In a faulty environment, comparisons between two elements with respect to an underlying linear order can come out right or go wrong. A wrong comparison is a recurring comparison fault if comparing the same two elements yields the very same result each time we compare the elements. We examine the impact of such faults on the elementary problems of sorting a set of distinct elements and finding a minimum element in such a set. The more faults occur, the worse the approaches to solve these problems can become and we parametrize our analysis by an upper bound \(k\) on the number of faults.

Cache Oblivious Minimum Cut

We show how to compute the minimum cut of a graph cache-efficiently. Let B be the width of a cache line and M be the size of the cache. On a graph with V vertices and E edges, we give a cache oblivious algorithm that incurs \(O(\lceil \frac{E}{B} (\log ^4 E) \log _{M/B} E\rceil )\) cache misses and a simpler one that incurs \(O(\lceil \frac{V^2}{B} \log ^3 V\rceil )\) cache misses.

Sorting by swaps with noisy comparisons

We study sorting of permutations by random swaps if the comparison operator is noisy. The noise is not associated with the underlying fitness but is inherent to the comparison operator. This type of fitness-independent noise has not been studied before in the community but is prototypical for comparison-based evolutionary algorithms, which often do not need to compute or approximate explicit fitness values. As quality measure, we compute the average fitness of the stationary distribution. To measure runtime, we compute the minimal number of steps after which the expected fitness approximates the average fitness of the stationary distribution. As mutations, we allow swaps of any two elements which have distance at most r. We give theoretical results for the extreme cases r = 1 and r = n, and experimental results for intermediate cases. We find a trade-off between faster convergence (for large r) and better average quality of the solution after convergence (for small r).

Finding Robust Minimum Cuts

We study the minimum cut problem in the presence of uncertainty and show how to apply a novel robust optimization approach, which aims to exploit the similarity in subsequent graph measurements or similar graph instances, without posing any assumptions on the way they have been obtained. With experiments we show that the approach works well when compared to other approaches that are also oblivious towards the relationship between the input datasets.

Optimal Dislocation with Persistent Errors in Subquadratic Time.

We study the problem of sorting N elements in presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability p, but repeating the same comparison gives always the same result. The best known algorithms for this problem have running time O(N^2) and achieve an optimal maximum dislocation of O(log N) for constant error probability. Note that no algorithm can achieve dislocation o(log N), regardless of its running time. In this work we present the first subquadratic time algorithm with optimal maximum dislocation: Our algorithm runs in tilde{O}(N^{3/2}) time and guarantees O(log N) maximum dislocation with high probability. Though the first version of our algorithm is randomized, it can be derandomized by extracting the necessary random bits from the results of the comparisons (errors).

Inversions from Sorting with Distance-Based Errors

We study the number of inversions after running the Insertion Sort or Quicksort algorithm, when errors in the comparisons occur with some probability. We investigate the case in which probabilities depend on the difference between the two numbers to be compared and only differences up to some threshold \(\tau \) are prone to errors. We give upper bounds for this model and show that for constant \(\tau \), the expected number of inversions is linear in the number of elements to be sorted. For Insertion Sort, we also yield an upper bound on the expected number of runs, i.e., the number of consecutive increasing subsequences.

Parallel Minimum Cuts in Near-linear Work and Low Depth

We present the first near-linear work and poly-logritharithmic depth algorithm for computing a minimum cut in a graph, while previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices. In a graph with n vertices and m edges, our algorithm computes the correct result with high probability in

On Computing the Total Displacement Number via Weighted Motzkin Paths

O(m łog^4 n)

Collaborative delivery with energy-constrained mobile robots

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