Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Y. Okamoto
{"title":"Algorithmic Theory of Qubit Routing","authors":"Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Y. Okamoto","doi":"10.48550/arXiv.2305.02059","DOIUrl":"https://doi.org/10.48550/arXiv.2305.02059","url":null,"abstract":"The qubit routing problem, also known as the swap minimization problem, is a (classical) combinatorial optimization problem that arises in the design of compilers of quantum programs. We study the qubit routing problem from the viewpoint of theoretical computer science, while most of the existing studies investigated the practical aspects. We concentrate on the linear nearest neighbor (LNN) architectures of quantum computers, in which the graph topology is a path. Our results are three-fold. (1) We prove that the qubit routing problem is NP-hard. (2) We give a fixed-parameter algorithm when the number of two-qubit gates is a parameter. (3) We give a polynomial-time algorithm when each qubit is involved in at most one two-qubit gate.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"72 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131704446","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}
Davide Bilò, Keerti Choudhary, S. Cohen, T. Friedrich, Simon Krogmann, Martin Schirneck
{"title":"Compact Distance Oracles with Large Sensitivity and Low Stretch","authors":"Davide Bilò, Keerti Choudhary, S. Cohen, T. Friedrich, Simon Krogmann, Martin Schirneck","doi":"10.48550/arXiv.2304.14184","DOIUrl":"https://doi.org/10.48550/arXiv.2304.14184","url":null,"abstract":"An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $sigma geq 1$ is a data structure that preprocesses an input graph $G$. When queried with the triple $(s,t,F)$, where $s, t in V$ and $F subseteq E$ contains at most $f$ edges of $G$, the oracle returns an estimate $widehat{d}_{G-F}(s,t)$ of the distance $d_{G-F}(s,t)$ between $s$ and $t$ in the graph $G-F$ such that $d_{G-F}(s,t) leq widehat{d}_{G-F}(s,t) leq sigma d_{G-F}(s,t)$. For any positive integer $k ge 2$ and any $0<alpha<1$, we present an $f$-DSO with sensitivity $f = o(log n/loglog n)$, stretch $2k-1$, space $O(n^{1+frac{1}{k}+alpha+o(1)})$, and an $widetilde{O}(n^{1+frac{1}{k} - frac{alpha}{k(f+1)}})$ query time. Prior to our work, there were only three known $f$-DSOs with subquadratic space. The first one by Chechik et al. [Algorithmica 2012] has a stretch of $(8k-2)(f+1)$, depending on $f$. Another approach is storing an $f$-edge fault-tolerant $(2k-1)$-spanner of $G$. The bottleneck is the large query time due to the size of any such spanner, which is $Omega(n^{1+1/k})$ under the ErdH{o}s girth conjecture. Bil`o et al. [STOC 2023] gave a solution with stretch $3+varepsilon$, query time $O(n^{alpha})$ but space $O(n^{2-frac{alpha}{f+1}})$, approaching the quadratic barrier for large sensitivity. In the realm of subquadratic space, our $f$-DSOs are the first ones that guarantee, at the same time, large sensitivity, low stretch, and non-trivial query time. To obtain our results, we use the approximate distance oracles of Thorup and Zwick [JACM 2005], and the derandomization of the $f$-DSO of Weimann and Yuster [TALG 2013], that was recently given by Karthik and Parter [SODA 2021].","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126393200","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}
A. K. Abu-Affash, Paz Carmi, Ori Luwisch, Joseph S. B. Mitchell
{"title":"Geometric Spanning Trees Minimizing the Wiener Index","authors":"A. K. Abu-Affash, Paz Carmi, Ori Luwisch, Joseph S. B. Mitchell","doi":"10.48550/arXiv.2303.01096","DOIUrl":"https://doi.org/10.48550/arXiv.2303.01096","url":null,"abstract":"The Wiener index of a network, introduced by the chemist Harry Wiener, is the sum of distances between all pairs of nodes in the network. This index, originally used in chemical graph representations of the non-hydrogen atoms of a molecule, is considered to be a fundamental and useful network descriptor. We study the problem of constructing geometric networks on point sets in Euclidean space that minimize the Wiener index: given a set $P$ of $n$ points in $mathbb{R}^d$, the goal is to construct a network, spanning $P$ and satisfying certain constraints, that minimizes the Wiener index among the allowable class of spanning networks. In this work, we focus mainly on spanning networks that are trees and we focus on problems in the plane ($d=2$). We show that any spanning tree that minimizes the Wiener index has non-crossing edges in the plane. Then, we use this fact to devise an $O(n^4)$-time algorithm that constructs a spanning tree of minimum Wiener index for points in convex position. We also prove that the problem of computing a spanning tree on $P$ whose Wiener index is at most $W$, while having total (Euclidean) weight at most $B$, is NP-hard. Computing a tree that minimizes the Wiener index has been studied in the area of communication networks, where it is known as the optimum communication spanning tree problem.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122853673","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}
J. Boyar, Lene M. Favrholdt, Shahin Kamali, Kim S. Larsen
{"title":"Online Interval Scheduling with Predictions","authors":"J. Boyar, Lene M. Favrholdt, Shahin Kamali, Kim S. Larsen","doi":"10.48550/arXiv.2302.13701","DOIUrl":"https://doi.org/10.48550/arXiv.2302.13701","url":null,"abstract":"In online interval scheduling, the input is an online sequence of intervals, and the goal is to accept a maximum number of non-overlapping intervals. In the more general disjoint path allocation problem, the input is a sequence of requests, each involving a pair of vertices of a known graph, and the goal is to accept a maximum number of requests forming edge-disjoint paths between accepted pairs. These problems have been studied under extreme settings without information about the input or with error-free advice. We study an intermediate setting with a potentially erroneous prediction that specifies the set of intervals/requests forming the input sequence. For both problems, we provide tight upper and lower bounds on the competitive ratios of online algorithms as a function of the prediction error. For disjoint path allocation, our results rule out the possibility of obtaining a better competitive ratio than that of a simple algorithm that fully trusts predictions, whereas, for interval scheduling, we develop a superior algorithm. We also present asymptotically tight trade-offs between consistency (competitive ratio with error-free predictions) and robustness (competitive ratio with adversarial predictions) of interval scheduling algorithms. Finally, we provide experimental results on real-world scheduling workloads that confirm our theoretical analysis.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130626501","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}
{"title":"Revisiting Graph Persistence for Updates and Efficiency","authors":"T. Dey, T. Hou","doi":"10.48550/arXiv.2302.12796","DOIUrl":"https://doi.org/10.48550/arXiv.2302.12796","url":null,"abstract":"It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we revisit graph persistence and propose efficient algorithms especially for local updates on filtrations, similar to what is done in ordinary persistence for computing the vineyard. We show that, for a filtration of length $m$, (i) switches (transpositions) in ordinary graph persistence can be done in $O(log m)$ time; (ii) zigzag persistence on graphs can be computed in $O(mlog m)$ time, which improves a recent $O(mlog^4n)$ time algorithm assuming $n$, the size of the union of all graphs in the filtration, satisfies $ninOmega({m^varepsilon})$ for any fixed $0<varepsilon<1$; (iii) open-closed, closed-open, and closed-closed bars in dimension $0$ for graph zigzag persistence can be updated in $O(log m)$ time, whereas the open-open bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in $O(sqrt{m},log m)$ time.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125058544","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}
Magnus Berg, J. Boyar, Lene M. Favrholdt, Kim S. Larsen
{"title":"Online Minimum Spanning Trees with Weight Predictions","authors":"Magnus Berg, J. Boyar, Lene M. Favrholdt, Kim S. Larsen","doi":"10.48550/arXiv.2302.12029","DOIUrl":"https://doi.org/10.48550/arXiv.2302.12029","url":null,"abstract":"We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an irrevocable decision must be made regarding whether or not the edge should be included into the spanning tree. In order to assess the quality of our algorithms, we define an appropriate error measure and analyze the performance of the algorithms as a function of the error. We prove that, according to competitive analysis, the simplest algorithm, Follow-the-Predictions, is optimal. However, intuitively, one should be able to do better, and we present a greedy variant of Follow-the-Predictions. In analyzing that algorithm, we believe we present the first random order analysis of a non-trivial online algorithm with predictions, by which we obtain an algorithmic separation. This may be useful for distinguishing between algorithms for other problems when Follow-the-Predictions is optimal according to competitive analysis.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128055258","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}
{"title":"Quick Minimization of Tardy Processing Time on a Single Machine","authors":"B. Schieber, Pranav Sitaraman","doi":"10.48550/arXiv.2301.05460","DOIUrl":"https://doi.org/10.48550/arXiv.2301.05460","url":null,"abstract":"We consider the problem of minimizing the total processing time of tardy jobs on a single machine. This is a classical scheduling problem, first considered by [Lawler and Moore 1969], that also generalizes the Subset Sum problem. Recently, it was shown that this problem can be solved efficiently by computing $(max,min)$-skewed-convolutions. The running time of the resulting algorithm is equivalent, up to logarithmic factors, to the time it takes to compute a $(max,min)$-skewed-convolution of two vectors of integers whose sum is $O(P)$, where $P$ is the sum of the jobs' processing times. We further improve the running time of the minimum tardy processing time computation by introducing a job ``bundling'' technique and achieve a $tilde{O}left(P^{2-1/alpha}right)$ running time, where $tilde{O}left(P^alpharight)$ is the running time of a $(max,min)$-skewed-convolution of vectors of size $P$. This results in a $tilde{O}left(P^{7/5}right)$ time algorithm for tardy processing time minimization, an improvement over the previously known $tilde{O}left(P^{5/3}right)$ time algorithm.","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114994101","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}
Chengyuan Deng, Jie Ying Gao, Jalaj Upadhyay, Chen Wang
{"title":"Differentially Private Range Query on Shortest Paths","authors":"Chengyuan Deng, Jie Ying Gao, Jalaj Upadhyay, Chen Wang","doi":"10.1007/978-3-031-38906-1_23","DOIUrl":"https://doi.org/10.1007/978-3-031-38906-1_23","url":null,"abstract":"","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117343086","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}
{"title":"Density Approximation for Moving Groups","authors":"Max van Mulken, B. Speckmann, Kevin Verbeek","doi":"10.1007/978-3-031-38906-1_45","DOIUrl":"https://doi.org/10.1007/978-3-031-38906-1_45","url":null,"abstract":"","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128825246","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}
{"title":"Dominator Coloring and CD Coloring in Almost Cluster Graphs","authors":"Aritra Banik, Prahlad Narasimhan Kasthurirangan, Venkatesh Raman","doi":"10.1007/978-3-031-38906-1_8","DOIUrl":"https://doi.org/10.1007/978-3-031-38906-1_8","url":null,"abstract":"","PeriodicalId":380945,"journal":{"name":"Workshop on Algorithms and Data Structures","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124347756","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}
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