Algorithmica最新文献

{"title":"Line Intersection Searching Amid Unit Balls in 3-Space","authors":"Pankaj K. Agarwal,&nbsp;Esther Ezra","doi":"10.1007/s00453-024-01284-7","DOIUrl":"10.1007/s00453-024-01284-7","url":null,"abstract":"<div><p>Let <span>(mathscr {B})</span> be a set of <i>n</i> unit balls in <span>({mathbb {R}}^3)</span>. We present a linear-size data structure for storing <span>(mathscr {B})</span> that can determine in <span>(O^*(sqrt{n}))</span> time whether a query line intersects any ball of <span>(mathscr {B})</span> and report all <i>k</i> such balls in additional <i>O</i>(<i>k</i>) time. The data structure can be constructed in <span>(O(nlog n))</span> time. (The <span>(O^*(cdot ))</span> notation hides subpolynomial factors, e.g., of the form <span>(O(n^{{varepsilon }}))</span>, for arbitrarily small <span>({varepsilon }&gt; 0)</span>, and their coefficients which depend on <span>({varepsilon })</span>.) We also consider the dual problem: Let <span>(mathscr {L})</span> be a set of <i>n</i> lines in <span>({mathbb {R}}^3)</span>. We preprocess <span>(mathscr {L})</span>, in <span>(O^*(n^2))</span> time, into a data structure of size <span>(O^*(n^2))</span> that can determine in <span>(O(log {n}))</span> time whether a query unit ball intersects any line of <span>(mathscr {L})</span>, or report all <i>k</i> such lines in additional <i>O</i>(<i>k</i>) time.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 2","pages":"223 - 241"},"PeriodicalIF":0.9,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Compact Genetic Algorithm Struggles on Cliff Functions","authors":"Frank Neumann,&nbsp;Dirk Sudholt,&nbsp;Carsten Witt","doi":"10.1007/s00453-024-01281-w","DOIUrl":"10.1007/s00453-024-01281-w","url":null,"abstract":"<div><p>Estimation of distribution algorithms (EDAs) are general-purpose optimizers that maintain a probability distribution over a given search space. This probability distribution is updated through sampling from the distribution and a reinforcement learning process which rewards solution components that have shown to be part of good quality samples. The compact genetic algorithm (cGA) is a non-elitist EDA able to deal with difficult multimodal fitness landscapes that are hard to solve by elitist algorithms. We investigate the cGA on the <span>Cliff</span> function for which it was shown recently that non-elitist evolutionary algorithms and artificial immune systems optimize it in expected polynomial time. We point out that the cGA faces major difficulties when solving the <span>Cliff</span> function and investigate its dynamics both experimentally and theoretically. Our experimental results indicate that the cGA requires exponential time for all values of the update strength 1/<i>K</i>. We show theoretically that, under sensible assumptions, there is a negative drift when sampling around the location of the cliff. Experiments further suggest that there is a phase transition for <i>K</i> where the expected optimization time drops from <span>(n^{Theta (n)})</span> to <span>(2^{Theta (n)})</span>.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 4","pages":"507 - 536"},"PeriodicalIF":0.9,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01281-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143668328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Partition Strategies for the Maker–Breaker Domination Game","authors":"Guillaume Bagan,&nbsp;Eric Duchêne,&nbsp;Valentin Gledel,&nbsp;Tuomo Lehtilä,&nbsp;Aline Parreau","doi":"10.1007/s00453-024-01280-x","DOIUrl":"10.1007/s00453-024-01280-x","url":null,"abstract":"<div><p>The Maker–Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in <span>(n^{k+3})</span> for interval graphs with at most <i>k</i> nested intervals.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 2","pages":"191 - 222"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Algorithms for Online b-Matching with Variable Vertex Capacities","authors":"Susanne Albers,&nbsp;Sebastian Schubert","doi":"10.1007/s00453-024-01282-9","DOIUrl":"10.1007/s00453-024-01282-9","url":null,"abstract":"<div><p>We study the <i>b</i>-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph <span>(G=(Sdot{cup }R,E))</span>. Every vertex <span>(sin S)</span> is a server with a capacity <span>(b_s)</span>, indicating the number of possible matching partners. The vertices <span>(rin R)</span> are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios. As for deterministic algorithms, we give a greedy algorithm <span>RelativeBalance</span> and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated <span>Ranking</span> algorithm by Karp, Vazirani and Vazirani. We prove that the original <span>Ranking</span> strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of <span>(1-1/e)</span>, independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server <i>s</i> by <span>(b_s)</span> vertices of unit capacity and to then run <span>Ranking</span> on this graph with <span>(sum _{sin S} b_s)</span> vertices on the left-hand side. Additionally, we extend this result to the vertex-weighted <i>b</i>-matching problem. Technically, we formulate a new configuration LP for the <i>b</i>-matching problem and conduct a primal-dual analysis.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 2","pages":"167 - 190"},"PeriodicalIF":0.9,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01282-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure","authors":"Hans L. Bodlaender,&nbsp;Carla Groenland,&nbsp;Hugo Jacob,&nbsp;Lars Jaffke,&nbsp;Paloma T. Lima","doi":"10.1007/s00453-024-01274-9","DOIUrl":"10.1007/s00453-024-01274-9","url":null,"abstract":"<div><p>In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing <i>W</i>[1]-hardness proofs for these problems, since XNLP-hardness implies <i>W</i>[<i>t</i>]-hardness for all <i>t</i>. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are <span>Independent Set</span>, <span>Dominating Set</span>, <span>Odd Cycle Transversal</span>, <span>(</span><i>q</i><span>-)Coloring</span>, <span>Max Cut</span>, <span>Maximum Regular Induced Subgraph</span>, <span>Feedback Vertex Set</span>, <span>Capacitated (Red-Blue) Dominating Set</span>, <span>Capacitated Vertex Cover</span> and <span>Bipartite Bandwidth</span>.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 4","pages":"465 - 506"},"PeriodicalIF":0.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01274-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143668054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Better Hardness Results for the Minimum Spanning Tree Congestion Problem","authors":"Huong Luu,&nbsp;Marek Chrobak","doi":"10.1007/s00453-024-01278-5","DOIUrl":"10.1007/s00453-024-01278-5","url":null,"abstract":"<div><p>In the spanning tree congestion problem, given a connected graph <i>G</i>, the objective is to compute a spanning tree <i>T</i> in <i>G</i> that minimizes its maximum edge congestion, where the congestion of an edge <i>e</i> of <i>T</i> is the number of edges in <i>G</i> for which the unique path in <i>T</i> between their endpoints traverses <i>e</i>. The problem is known to be <span>(mathbb{N}mathbb{P})</span>-hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratio <i>o</i>(<i>n</i>). In the decision version of this problem, denoted <span>({varvec{K}-textsf {STC}})</span>, we need to determine if <i>G</i> has a spanning tree with congestion at most <i>K</i>. It is known that <span>({varvec{K}-textsf {STC}})</span> is <span>(mathbb{N}mathbb{P})</span>-complete for <span>(Kge 8)</span>, and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand, <span>({varvec{3}-textsf {STC}})</span> can be solved in polynomial time, with the complexity status of this problem for <span>(Kin { left{ 4,5,6,7 right} })</span> remaining an open problem. We substantially improve the earlier hardness results by proving that <span>({varvec{K}-textsf {STC}})</span> is <span>(mathbb{N}mathbb{P})</span>-complete for <span>(Kge 5)</span>. This leaves only the case <span>(K=4)</span> open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider <span>({varvec{K}-textsf {STC}})</span> restricted to graphs of radius 2, and we prove that this variant is <span>(mathbb{N}mathbb{P})</span>-complete for all <span>(Kge 6)</span>. \u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 1","pages":"148 - 165"},"PeriodicalIF":0.9,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01278-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Euclidean Maximum Matchings in the Plane—Local to Global","authors":"Ahmad Biniaz,&nbsp;Anil Maheshwari,&nbsp;Michiel Smid","doi":"10.1007/s00453-024-01279-4","DOIUrl":"10.1007/s00453-024-01279-4","url":null,"abstract":"<div><p>Let <i>M</i> be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that <i>M</i> is <i>globally maximum</i> if it is a maximum-length matching on all points. We say that <i>M</i> is <i>k</i>-<i>local maximum</i> if for any subset <span>(M'={a_1b_1,dots ,a_kb_k})</span> of <i>k</i> edges of <i>M</i> it holds that <span>(M')</span> is a maximum-length matching on points <span>({a_1,b_1,dots ,a_k,b_k})</span>. We show that local maximum matchings are good approximations of global ones. Let <span>(mu _k)</span> be the infimum ratio of the length of any <i>k</i>-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that <span>(mu _kgeqslant frac{k-1}{k})</span> for any <span>(kgeqslant 2)</span>. We show the following improved bounds for <span>(kin {2,3})</span>: <span>(sqrt{3/7}leqslant mu _2&lt; 0.93 )</span> and <span>(sqrt{3}/2leqslant mu _3&lt; 0.98)</span>. We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for <span>(mu _2)</span> we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor <span>(2/sqrt{3})</span>, then the resulting disks have a common intersection.\u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 1","pages":"132 - 147"},"PeriodicalIF":0.9,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Online Paging with Heterogeneous Cache Slots","authors":"Marek Chrobak,&nbsp;Samuel Haney,&nbsp;Mehraneh Liaee,&nbsp;Debmalya Panigrahi,&nbsp;Rajmohan Rajaraman,&nbsp;Ravi Sundaram,&nbsp;Neal E. Young","doi":"10.1007/s00453-024-01270-z","DOIUrl":"10.1007/s00453-024-01270-z","url":null,"abstract":"<div><p>It is natural to generalize the online <span>(k)</span>-Server problem by allowing each request to specify not only a point <i>p</i>, but also a subset <i>S</i> of servers that may serve it. To date, only a few special cases of this problem have been studied. The objective of the work presented in this paper has been to more systematically explore this generalization in the case of uniform and star metrics. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page <i>p</i>, but also a subset <i>S</i> of cache slots, and is satisfied by having a copy of <i>p</i> in some slot in <i>S</i>. We call this problem <i>Slot-Heterogenous Paging</i>. In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore we parameterize the problem by specifying a family <span>({mathcal {S}}subseteq 2^{[k]})</span> of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size <i>k</i> and family <span>({mathcal {S}})</span>:</p><ul>\u0000 <li>\u0000 <p>If all request sets are allowed (<span>({mathcal {S}}=2^{[k]}setminus {emptyset })</span>), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (<span>({mathcal {S}}={[k]})</span>).</p>\u0000 </li>\u0000 <li>\u0000 <p>As a function of <span>(|{mathcal {S}}|)</span> and <i>k</i>, the optimal deterministic ratio is polynomial: at most <span>(O(k^2|{mathcal {S}}|))</span> and at least <span>(Omega (sqrt{|{mathcal {S}}|}))</span>.</p>\u0000 </li>\u0000 <li>\u0000 <p>For any laminar family <span>({mathcal {S}})</span> of height <i>h</i>, the optimal ratios are <i>O</i>(<i>hk</i>) (deterministic) and <span>(O(h^2log k))</span> (randomized).</p>\u0000 </li>\u0000 <li>\u0000 <p>The special case of laminar <span>({mathcal {S}})</span> that we call <i>All-or-One Paging</i> extends standard Paging by allowing each request to specify a specific slot to put the requested page in. The optimal deterministic ratio for <i>weighted</i> All-or-One Paging is <span>(Theta (k))</span>. Offline All-or-One Paging is <span>(mathbb{N}mathbb{P})</span>-hard.</p>\u0000 </li>\u0000 </ul><p> Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set <span>(P)</span> of <i>pages</i>, and is satisfied by fetching any page from <span>(P)</span> into the cache. The optimal ratios for the latter problem (with laminar family of height <i>h</i>) are at most <i>hk</i> (deterministic) and <span>(hH_k)</span> (randomized).</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 1","pages":"89 - 131"},"PeriodicalIF":0.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01270-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Anti-factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)","authors":"Dániel Marx,&nbsp;Govind S. Sankar,&nbsp;Philipp Schepper","doi":"10.1007/s00453-024-01265-w","DOIUrl":"10.1007/s00453-024-01265-w","url":null,"abstract":"<div><p>In the general <span>AntiFactor</span> problem, a graph <i>G</i> and, for every vertex <i>v</i> of <i>G</i>, a set <span>(X_vsubseteq {mathbb {N}})</span> of forbidden degrees is given. The task is to find a set <i>S</i> of edges such that the degree of <i>v</i> in <i>S</i> is <i>not</i> in the set <span>(X_v)</span>. Standard techniques (dynamic programming plus fast convolution) can be used to show that if <i>M</i> is the largest forbidden degree, then the problem can be solved in time <span>((M+2)^{{operatorname {tw}}}cdot n^{{mathcal {O}}(1)})</span> if a tree decomposition of width <span>({operatorname {tw}})</span> is given. However, significantly faster algorithms are possible if the sets <span>(X_v)</span> are sparse: our main algorithmic result shows that if every vertex has at most <span>(x)</span> forbidden degrees (we call this special case <span>AntiFactor</span>_x), then the problem can be solved in time <span>((x+1)^{{mathcal {O}}({operatorname {tw}})}cdot n^{{mathcal {O}}(1)})</span>. That is, <span>AntiFactor</span>_x is fixed-parameter tractable parameterized by treewidth <span>({operatorname {tw}})</span> and the maximum number <span>(x)</span> of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #<span>AntiFactor</span>₁ is already #<span>W</span> <span>([1])</span>-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set <span>(X)</span>, we denote by <span>(X)</span>-<span>AntiFactor</span> the special case where every vertex <i>v</i> has the same set <span>(X_v=X)</span> of forbidden degrees. We show the following lower bound for every fixed set <span>(X)</span>: if there is an <span>(epsilon &gt;0)</span> such that #<span>(X)</span>-<span>AntiFactor</span> can be solved in time <span>((max X+2-epsilon )^{{operatorname {tw}}}cdot n^{{mathcal {O}}(1)})</span> given a tree decomposition of width <span>({operatorname {tw}})</span>, then the counting strong exponential-time hypothesis (#SETH) fails.\u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 1","pages":"22 - 88"},"PeriodicalIF":0.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01265-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Energy Constrained Depth First Search","authors":"Shantanu Das,&nbsp;Dariusz Dereniowski,&nbsp;Przemysław Uznański","doi":"10.1007/s00453-024-01275-8","DOIUrl":"10.1007/s00453-024-01275-8","url":null,"abstract":"<div><p>Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such a route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer <i>B</i> (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree <i>T</i> using the minimum number of routes, each starting and ending at the root and each being of length at most <i>B</i>. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal <i>R</i> of the tree <i>T</i>, we cover <i>R</i> with routes <span>(R_1,ldots ,R_l)</span>, each of length at most <i>B</i> such that: <span>(R_i)</span> starts at the root, reaches directly the farthest point of <i>R</i> visited by <span>(R_{i-1})</span>, then <span>(R_i)</span> continues along the path <i>R</i> as far as possible, and finally <span>(R_i)</span> returns to the root. We call the above algorithm <i>piecemeal-DFS</i> and we prove that it achieves the asymptotically minimal number of routes <i>l</i>, regardless of the choice of <i>R</i>. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that <i>R</i> can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree <i>T</i> is not known in advance. Each route <span>(R_i)</span> can be constructed without any knowledge of the yet unvisited part of <i>T</i>. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 12","pages":"3759 - 3782"},"PeriodicalIF":0.9,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01275-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142636684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}