The Frequent Items Problem in Online Streaming under Various Performance Measures

In this paper, we strengthen the competitive analysis results obtained for a fundamental online streaming problem, the Frequent Items Problem. Additionally, we contribute with a more detailed analysis of this problem, using alternative performance measures, supplementing the insight gained from competitive analysis. The results also contribute to the general study of performance measures for online algorithms. It has long been known that competitive analysis suffers from drawbacks in certain situations, and many alternative measures have been proposed. However, more systematic comparative studies of performance measures have been initiated recently, and we continue this work, using competitive analysis, relative interval analysis, and relative worst order analysis on the Frequent Items Problem.Comment: IMADA-preprint-c

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

We consider a setting where we are given a graph \mathcal {g}=(\mathcal {r},e)\mathcal {g}=(\mathcal {r},e), where \mathcal {r}=\{r_1,\ldots ,r_n\}\mathcal {r}=\{r_1,\ldots ,r_n\} is a set of polygonal regions in the plane. Placing a point p_ip_i inside each region r_ir_i turns gg into an edge-weighted graph g_{\varvec{p}}g_{\varvec{p}}, {\varvec{p}}=\{p_1,\ldots ,p_n\}{\varvec{p}}=\{p_1,\ldots ,p_n\}, where the cost of (r_i,r_j)\in e(r_i,r_j)\in e is the distance between p_ip_i and p_jp_j. The shortest path problem with neighborhoods asks, for given r_sr_s and r_tr_t, to find a placement \varvec{p}\varvec{p} such that the cost of a resulting shortest stst-path in \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}} is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. The minimum spanning tree problem with neighborhoods asks to find a placement \varvec{p}\varvec{p} such that the cost of a resulting minimum spanning tree is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. We study these problems in the l_1l_1 metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is \mathsf {apx}\mathsf {apx}-hard, even if the neighborhood regions are segments

Query-competitive algorithms for cheapest set problems under uncertainty

Considering the model of computing under uncertainty where element weights are uncertain but can be obtained at a cost by query operations, we study the problem of identifying a cheapest (minimum-weight) set among a given collection of feasible sets using a minimum number of queries of element weights. For the general case we present an algorithm that makes at most d·OPT+d queries, where d is the maximum cardinality of any given set and OPT is the optimal number of queries needed to identify a cheapest set. For the minimum multi-cut problem in trees with d terminal pairs, we give an algorithm that makes at most d·OPT+1 queries. For the problem of computing a minimum-weight base of a given matroid, we give an algorithm that makes at most 2·OPT queries, generalizing a known result for the minimum spanning tree problem. For each of our algorithms we give matching lower bounds