Workshop on Algorithms and Data Structures最新文献

Faster Algorithms for Cycle Hitting Problems on Disk Graphs 磁盘图上循环命中问题的更快算法

Workshop on Algorithms and Data Structures Pub Date : 2023-11-07 DOI: 10.1007/978-3-031-38906-1_3

Shinwoo An, Kyungjin Cho, Eunjin Oh

引用次数: 0

Zip-zip Trees: Making Zip Trees More Balanced, Biased, Compact, or Persistent Zip- Zip树:使Zip树更加平衡、偏向、紧凑或持久

Workshop on Algorithms and Data Structures Pub Date : 2023-07-14 DOI: 10.48550/arXiv.2307.07660

Ofek Gila, M. Goodrich, R. Tarjan

{"title":"Zip-zip Trees: Making Zip Trees More Balanced, Biased, Compact, or Persistent","authors":"Ofek Gila, M. Goodrich, R. Tarjan","doi":"10.48550/arXiv.2307.07660","DOIUrl":"https://doi.org/10.48550/arXiv.2307.07660","url":null,"abstract":"We define simple variants of zip trees, called zip-zip trees, which provide several advantages over zip trees, including overcoming a bias that favors smaller keys over larger ones. We analyze zip-zip trees theoretically and empirically, showing, e.g., that the expected depth of a node in an $n$-node zip-zip tree is at most $1.3863log n-1+o(1)$, which matches the expected depth of treaps and binary search trees built by uniformly random insertions. Unlike these other data structures, however, zip-zip trees achieve their bounds using only $O(loglog n)$ bits of metadata per node, w.h.p., as compared to the $Theta(log n)$ bits per node required by treaps. In fact, we even describe a ``just-in-time'' zip-zip tree variant, which needs just an expected $O(1)$ number of bits of metadata per node. Moreover, we can define zip-zip trees to be strongly history independent, whereas treaps are generally only weakly history independent. We also introduce emph{biased zip-zip trees}, which have an explicit bias based on key weights, so the expected depth of a key, $k$, with weight, $w_k$, is $O(log (W/w_k))$, where $W$ is the weight of all keys in the weighted zip-zip tree. Finally, we show that one can easily make zip-zip trees partially persistent with only $O(n)$ space overhead w.h.p.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123528715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Observation Routes and External Watchman Routes 观察路线和外部看守路线

Workshop on Algorithms and Data Structures Pub Date : 2023-06-20 DOI: 10.48550/arXiv.2306.11522

A. Dumitrescu, Csaba D. T'oth

{"title":"Observation Routes and External Watchman Routes","authors":"A. Dumitrescu, Csaba D. T'oth","doi":"10.48550/arXiv.2306.11522","DOIUrl":"https://doi.org/10.48550/arXiv.2306.11522","url":null,"abstract":"We introduce the Observation Route Problem ($textsf{ORP}$) defined as follows: Given a set of $n$ pairwise disjoint compact regions in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) some point in each region from some point of the tour. The observer does emph{not} need to see the entire boundary of an object. The tour is emph{not} allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods ($textsf{TSPN}$) and the External Watchman Route Problem ($textsf{EWRP}$). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results: (I) Given a family of $n$ disjoint convex bodies in the plane, computing a shortest observation route does not admit a $(clog n)$-approximation unless $textsf{P} = textsf{NP}$ for an absolute constant $c>0$. (This holds for both limited and unlimited vision.) (II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is $textsf{NP}$-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.) (III) Given a family of $n$ disjoint fat convex polygons, an observation tour whose length is at most $O(log{n})$ times the optimal can be computed in polynomial time. (This holds for limited vision.) (IV) For every $n geq 5$, there exists a convex polygon with $n$ sides and all angles obtuse such that its perimeter is emph{not} a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126294863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Finding Diameter-Reducing Shortcuts in Trees 在树中寻找减小直径的捷径

Workshop on Algorithms and Data Structures Pub Date : 2023-05-27 DOI: 10.48550/arXiv.2305.17385

Davide Bilò, Luciano Gualà, S. Leucci, Luca Pepè Sciarria

{"title":"Finding Diameter-Reducing Shortcuts in Trees","authors":"Davide Bilò, Luciano Gualà, S. Leucci, Luca Pepè Sciarria","doi":"10.48550/arXiv.2305.17385","DOIUrl":"https://doi.org/10.48550/arXiv.2305.17385","url":null,"abstract":"In the emph{$k$-Diameter-Optimally Augmenting Tree Problem} we are given a tree $T$ of $n$ vertices as input. The tree is embedded in an unknown emph{metric} space and we have unlimited access to an oracle that, given two distinct vertices $u$ and $v$ of $T$, can answer queries reporting the cost of the edge $(u,v)$ in constant time. We want to augment $T$ with $k$ shortcuts in order to minimize the diameter of the resulting graph. For $k=1$, $O(n log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bil`o, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than $10/9$-approximate solution for trees for $kgeq 3$. For any constant $varepsilon>0$, we instead design a linear-time $(1+varepsilon)$-approximation algorithm for paths and $k = o(sqrt{log n})$, thus establishing a dichotomy between paths and trees for $kgeq 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time $4$-approximation algorithm for trees, and to compute the diameter of graphs with $n + k - 1$ edges in time $O(n k log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117059563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Sublinear-Space Streaming Algorithms for Estimating Graph Parameters on Sparse Graphs 稀疏图参数估计的亚线性空间流算法

Workshop on Algorithms and Data Structures Pub Date : 2023-05-26 DOI: 10.48550/arXiv.2305.16815

Xiuge Chen, R. Chitnis, Patrick Eades, Anthony Wirth

引用次数: 0

Linear Layouts of Bipartite Planar Graphs 二部平面图的线性布局

Workshop on Algorithms and Data Structures Pub Date : 2023-05-25 DOI: 10.48550/arXiv.2305.16087

Henry Förster, Michael Kaufmann, Laura Merker, S. Pupyrev, Chrysanthi N. Raftopoulou

{"title":"Linear Layouts of Bipartite Planar Graphs","authors":"Henry Förster, Michael Kaufmann, Laura Merker, S. Pupyrev, Chrysanthi N. Raftopoulou","doi":"10.48550/arXiv.2305.16087","DOIUrl":"https://doi.org/10.48550/arXiv.2305.16087","url":null,"abstract":"A linear layout of a graph $ G $ consists of a linear order $prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v prec x prec y prec w $ ($ v prec x prec w prec y $) may not be in the same queue (stack). The best known lower and upper bounds for the number of queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42 [Bekos et al., Algorithmica 2022], respectively. While queue layouts of special classes of planar graphs have received increased attention following the breakthrough result of [Dujmovi'c et al., J. ACM 2020], the meaningful class of bipartite planar graphs has remained elusive so far, explicitly asked for by Bekos et al. In this paper we investigate bipartite planar graphs and give an improved upper bound of 28 by refining existing techniques. In contrast, we show that two queues or one queue together with one stack do not suffice; the latter answers an open question by Pupyrev [GD 2018]. We further investigate subclasses of bipartite planar graphs and give improved upper bounds; in particular we construct 5-queue layouts for 2-degenerate quadrangulations.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127532724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Lower Bounds for Non-Adaptive Shortest Path Relaxation

Workshop on Algorithms and Data Structures Pub Date : 2023-05-16 DOI: 10.48550/arXiv.2305.09230

D. Eppstein

引用次数: 0

Geometric Hitting Set for Line-Constrained Disks and Related Problems 线约束圆盘的几何命中集及相关问题

Workshop on Algorithms and Data Structures Pub Date : 2023-05-15 DOI: 10.48550/arXiv.2305.09045

Gang Liu, Haitao Wang

{"title":"Geometric Hitting Set for Line-Constrained Disks and Related Problems","authors":"Gang Liu, Haitao Wang","doi":"10.48550/arXiv.2305.09045","DOIUrl":"https://doi.org/10.48550/arXiv.2305.09045","url":null,"abstract":"Given a set $P$ of $n$ weighted points and a set $S$ of $m$ disks in the plane, the hitting set problem is to compute a subset $P'$ of points of $P$ such that each disk contains at least one point of $P'$ and the total weight of all points of $P'$ is minimized. The problem is known to be NP-hard. In this paper, we consider a line-constrained version of the problem in which all disks are centered on a line $ell$. We present an $O((m+n)log(m+n)+kappa log m)$ time algorithm for the problem, where $kappa$ is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to $O((n + m)log(m + n))$. In addition, we solve the problem in $O((m + n)log(m + n))$ time in the $L_{infty}$ and $L_1$ metrics, in which a disk is a square and a diamond, respectively. Our techniques can also be used to solve other geometric hitting set problems. For example, given in the plane a set $P$ of $n$ weighted points and a set $S$ of $n$ half-planes, we solve in $O(n^4log n)$ time the problem of finding a minimum weight hitting set of $P$ for $S$. This improves the previous best algorithm of $O(n^6)$ time by nearly a quadratic factor.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"436 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132590024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Reconfiguration of Time-Respecting Arborescences 时变树形的重构

Workshop on Algorithms and Data Structures Pub Date : 2023-05-12 DOI: 10.48550/arXiv.2305.07262

Takehiro Ito, Yuni Iwamasa, Naoyuki Kamiyama, Yasuaki Kobayashi, Yusuke Kobayashi, Shun-ichi Maezawa, Akira Suzuki

{"title":"Reconfiguration of Time-Respecting Arborescences","authors":"Takehiro Ito, Yuni Iwamasa, Naoyuki Kamiyama, Yasuaki Kobayashi, Yusuke Kobayashi, Shun-ichi Maezawa, Akira Suzuki","doi":"10.48550/arXiv.2305.07262","DOIUrl":"https://doi.org/10.48550/arXiv.2305.07262","url":null,"abstract":"An arborescence, which is a directed analogue of a spanning tree in an undirected graph, is one of the most fundamental combinatorial objects in a digraph. In this paper, we study arborescences in digraphs from the viewpoint of combinatorial reconfiguration, which is the field where we study reachability between two configurations of some combinatorial objects via some specified operations. Especially, we consider reconfiguration problems for time-respecting arborescences, which were introduced by Kempe, Kleinberg, and Kumar. We first prove that if the roots of the initial and target time-respecting arborescences are the same, then the target arborescence is always reachable from the initial one and we can find a shortest reconfiguration sequence in polynomial time. Furthermore, we show if the roots are not the same, then the target arborescence may not be reachable from the initial one. On the other hand, we show that we can determine whether the target arborescence is reachable form the initial one in polynomial time. Finally, we prove that it is NP-hard to find a shortest reconfiguration sequence in the case where the roots are not the same. Our results show an interesting contrast to the previous results for (ordinary) arborescences reconfiguration problems.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122063551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Block Crossings in One-Sided Tanglegrams 单侧缠结图中的块交叉

Workshop on Algorithms and Data Structures Pub Date : 2023-05-08 DOI: 10.48550/arXiv.2305.04682

Alexander Dobler, M. Nöllenburg

引用次数: 0