Pub Date : 2025-03-10DOI: 10.1016/j.ipl.2025.106574
Takashi Ishizuka
One of the most famous TFNP subclasses is PPP, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint [1] has introduced a TFNP problem related to the pigeonhole principle over a quotient set, called Quotient Pigeon, and shown that the problem Quotient Pigeon is not only PPP-hard but also PLS-hard. In this paper, we formulate other computational problems related to the pigeonhole principle over a quotient set via an explicit representation of the equivalence classes. Our new formulation introduces a non-trivial -complete problem for every . Furthermore, we consider the computational complexity of a computational problem related to Kőnig's lemma, which is a weaker variant of the problem formulated by Pasarkar et al. [2]. We show that our weaker variant is PPAD-hard and is in .
{"title":"On the complexity of some restricted variants of Quotient Pigeon and a weak variant of Kőnig","authors":"Takashi Ishizuka","doi":"10.1016/j.ipl.2025.106574","DOIUrl":"10.1016/j.ipl.2025.106574","url":null,"abstract":"<div><div>One of the most famous <span>TFNP</span> subclasses is <span>PPP</span>, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint <span><span>[1]</span></span> has introduced a <span>TFNP</span> problem related to the pigeonhole principle over a quotient set, called <span>Quotient Pigeon</span>, and shown that the problem <span>Quotient Pigeon</span> is not only <span>PPP</span>-hard but also <span>PLS</span>-hard. In this paper, we formulate other computational problems related to the pigeonhole principle over a quotient set via an explicit representation of the equivalence classes. Our new formulation introduces a non-trivial <span><math><mtext>PPP</mtext><mo>∩</mo><msub><mrow><mtext>PPA</mtext></mrow><mrow><mi>k</mi></mrow></msub></math></span>-complete problem for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. Furthermore, we consider the computational complexity of a computational problem related to Kőnig's lemma, which is a weaker variant of the problem formulated by Pasarkar et al. <span><span>[2]</span></span>. We show that our weaker variant is <span>PPAD</span>-hard and is in <span><math><mtext>PPP</mtext><mo>∩</mo><mtext>PPA</mtext></math></span>.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106574"},"PeriodicalIF":0.7,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601479","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}
Pub Date : 2025-03-03DOI: 10.1016/j.ipl.2025.106573
Chao Yang , Zhujun Zhang
The friends-and-strangers problem is defined on two graphs X and Y of the same order n, where X is the location graph on a set of locations and Y is the friendship graph on a set of people . A configuration is a bijection from to , representing an arrangement of n people at n locations. The friends-and-stranger problem investigates the connectivity of two arbitrary configurations by a sequence of swaps of two people that are neighbors (i.e., adjacent in X) and friends (i.e., adjacent in Y). In this paper, we show that the friends-and-strangers problem with subcubic planar location graphs X is PSPACE-complete by a reduction from the Ncl (non-deterministic constraint logic) problem.
{"title":"Friends-and-strangers is PSPACE-complete","authors":"Chao Yang , Zhujun Zhang","doi":"10.1016/j.ipl.2025.106573","DOIUrl":"10.1016/j.ipl.2025.106573","url":null,"abstract":"<div><div>The friends-and-strangers problem is defined on two graphs <em>X</em> and <em>Y</em> of the same order <em>n</em>, where <em>X</em> is the location graph on a set of locations <span><math><mi>V</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <em>Y</em> is the friendship graph on a set of people <span><math><mi>V</mi><mo>(</mo><mi>Y</mi><mo>)</mo></math></span>. A configuration is a bijection from <span><math><mi>V</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> to <span><math><mi>V</mi><mo>(</mo><mi>Y</mi><mo>)</mo></math></span>, representing an arrangement of <em>n</em> people at <em>n</em> locations. The friends-and-stranger problem investigates the connectivity of two arbitrary configurations by a sequence of swaps of two people that are neighbors (i.e., adjacent in <em>X</em>) and friends (i.e., adjacent in <em>Y</em>). In this paper, we show that the friends-and-strangers problem with subcubic planar location graphs <em>X</em> is <span>PSPACE</span>-complete by a reduction from the <span>Ncl</span> (non-deterministic constraint logic) problem.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106573"},"PeriodicalIF":0.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534766","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}
Pub Date : 2025-02-28DOI: 10.1016/j.ipl.2025.106572
Jordan Dempsey , Leo van Iersel , Mark Jones , Norbert Zeh
Maximum agreement forests have been used as a measure of dissimilarity of two or more phylogenetic trees on a given set of taxa. An agreement forest is a set of trees that can be obtained from each of the input trees by deleting edges and suppressing degree-2 vertices. A maximum agreement forest is such a forest with the minimum number of components. We present a simple 4-approximation algorithm for computing a maximum agreement forest of multiple unrooted binary trees. This algorithm applies LP rounding to an extension of a recent ILP formulation of the maximum agreement forest problem on two trees by Van Wersch et al. [13]. We achieve the same approximation ratio as the algorithm by Chen et al. [3], but our algorithm is extremely simple. We also prove that no algorithm based on the ILP formulation by Van Wersch et al. can achieve an approximation ratio of , for any , even on two trees. To this end, we prove that the integrality gap of the ILP approaches 4 as the size of the two input trees grows.
{"title":"A simple 4-approximation algorithm for maximum agreement forests on multiple unrooted binary trees","authors":"Jordan Dempsey , Leo van Iersel , Mark Jones , Norbert Zeh","doi":"10.1016/j.ipl.2025.106572","DOIUrl":"10.1016/j.ipl.2025.106572","url":null,"abstract":"<div><div>Maximum agreement forests have been used as a measure of dissimilarity of two or more phylogenetic trees on a given set of taxa. An agreement forest is a set of trees that can be obtained from each of the input trees by deleting edges and suppressing degree-2 vertices. A maximum agreement forest is such a forest with the minimum number of components. We present a simple 4-approximation algorithm for computing a maximum agreement forest of multiple unrooted binary trees. This algorithm applies LP rounding to an extension of a recent ILP formulation of the maximum agreement forest problem on two trees by Van Wersch et al. <span><span>[13]</span></span>. We achieve the same approximation ratio as the algorithm by Chen et al. <span><span>[3]</span></span>, but our algorithm is extremely simple. We also prove that no algorithm based on the ILP formulation by Van Wersch et al. can achieve an approximation ratio of <span><math><mn>4</mn><mo>−</mo><mi>ε</mi></math></span>, for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, even on two trees. To this end, we prove that the integrality gap of the ILP approaches 4 as the size of the two input trees grows.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106572"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534765","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}
Pub Date : 2025-02-15DOI: 10.1016/j.ipl.2025.106571
Jan Kleinekathöfer, Alireza Mahzoon, Rolf Drechsler
Decision Diagrams (DDs) are among the most popular representations for Boolean functions. They are widely used in the synthesis and verification of digital circuits. The size (i.e., number of nodes) and computation time (required time for performing operations) are two important parameters that determine the efficiency of a DD in different applications. It has been proven that some DDs can represent specific functions in polynomial space or perform certain operations in polynomial time. For example, Binary Decision Diagrams (BDDs) are capable of representing a wide variety of functions (e.g. integer addition) in polynomial space with respect to the input size. However, there are also some functions (e.g., integer multiplication) for which the exponential lower-bounds have been proven for the BDD sizes.
In this paper, we investigate the space complexity of representing an integer addition, where one of the operands is shifted to the right by an arbitrary value. We call this function the shifted addition. This function is widely used in many digital circuits, e.g., floating point adders. We prove that the size of the BDD representing a shifted addition has exponential space complexity with respect to the input size. It is an important step towards clarifying the reasons behind the failure of BDD-based verification and synthesis when they are applied to the circuits containing shifted addition, e.g., floating point adders.
{"title":"Lower bound proof for the size of BDDs representing a shifted addition","authors":"Jan Kleinekathöfer, Alireza Mahzoon, Rolf Drechsler","doi":"10.1016/j.ipl.2025.106571","DOIUrl":"10.1016/j.ipl.2025.106571","url":null,"abstract":"<div><div><em>Decision Diagrams</em> (DDs) are among the most popular representations for Boolean functions. They are widely used in the synthesis and verification of digital circuits. The size (i.e., number of nodes) and computation time (required time for performing operations) are two important parameters that determine the efficiency of a DD in different applications. It has been proven that some DDs can represent specific functions in polynomial space or perform certain operations in polynomial time. For example, <em>Binary Decision Diagrams</em> (BDDs) are capable of representing a wide variety of functions (e.g. integer addition) in polynomial space with respect to the input size. However, there are also some functions (e.g., integer multiplication) for which the exponential lower-bounds have been proven for the BDD sizes.</div><div>In this paper, we investigate the space complexity of representing an integer addition, where one of the operands is shifted to the right by an arbitrary value. We call this function the shifted addition. This function is widely used in many digital circuits, e.g., floating point adders. We prove that the size of the BDD representing a shifted addition has exponential space complexity with respect to the input size. It is an important step towards clarifying the reasons behind the failure of BDD-based verification and synthesis when they are applied to the circuits containing shifted addition, e.g., floating point adders.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106571"},"PeriodicalIF":0.7,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143427877","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}
Pub Date : 2025-02-14DOI: 10.1016/j.ipl.2025.106570
Dekel Tsur
In the Cliques or Trees Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a graph in which every connected component is either a clique or a tree. In this paper we give an -time deterministic algorithm, an -time randomized algorithm, and a kernel with vertices for Cliques or Trees Vertex Deletion.
{"title":"Faster algorithms and a smaller kernel for Cliques or Trees Vertex Deletion","authors":"Dekel Tsur","doi":"10.1016/j.ipl.2025.106570","DOIUrl":"10.1016/j.ipl.2025.106570","url":null,"abstract":"<div><div>In the <span>Cliques or Trees Vertex Deletion</span> problem, the input is a graph <em>G</em> and an integer <em>k</em>, and the goal is to decide whether there is a set of at most <em>k</em> vertices whose removal from <em>G</em> result in a graph in which every connected component is either a clique or a tree. In this paper we give an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>3.46</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time deterministic algorithm, an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>3.103</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time randomized algorithm, and a kernel with <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> vertices for <span>Cliques or Trees Vertex Deletion</span>.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106570"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420534","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}
Pub Date : 2025-02-11DOI: 10.1016/j.ipl.2025.106569
Daniel Eichhorn, Sven O. Krumke
The Online Delay Management on a Single Train Line (ODMP) deals with the question at which station a train should wait for delayed passengers, instead of forcing them to take the next train. Waiting at a station increases the delay of all passengers that are already on board, and the goal is to minimize the total passenger delay. An online algorithm learns about the number of delayed passengers at a station only when reaching this station. We study the ODMP with an additional prediction on the future input data, which an online algorithm can utilize. Two desired qualities for online algorithms with prediction are called consistency and robustness, denoting the competitive ratio in case of best and worst prediction respectively. We present a family of algorithms, which uses a hyperparameter measuring the “doubt” about the given prediction. This allows to achieve -consistency and -robustness. Moreover, we provide a lower bound for the trade-off between consistency and robustness for two variously detailed prediction models, showing that our algorithm achieves an asymptotically optimal trade-off for small values of λ.
{"title":"Online delay management on a single train line with predictions","authors":"Daniel Eichhorn, Sven O. Krumke","doi":"10.1016/j.ipl.2025.106569","DOIUrl":"10.1016/j.ipl.2025.106569","url":null,"abstract":"<div><div>The Online Delay Management on a Single Train Line (ODMP) deals with the question at which station a train should wait for delayed passengers, instead of forcing them to take the next train. Waiting at a station increases the delay of all passengers that are already on board, and the goal is to minimize the total passenger delay. An online algorithm learns about the number of delayed passengers at a station only when reaching this station. We study the ODMP with an additional prediction on the future input data, which an online algorithm can utilize. Two desired qualities for online algorithms with prediction are called consistency and robustness, denoting the competitive ratio in case of best and worst prediction respectively. We present a family of algorithms, which uses a hyperparameter <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> measuring the “doubt” about the given prediction. This allows to achieve <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>λ</mi><mo>)</mo></math></span>-consistency and <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>/</mo><mi>λ</mi><mo>)</mo></math></span>-robustness. Moreover, we provide a lower bound for the trade-off between consistency and robustness for two variously detailed prediction models, showing that our algorithm achieves an asymptotically optimal trade-off for small values of <em>λ</em>.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"190 ","pages":"Article 106569"},"PeriodicalIF":0.7,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420533","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}
Pub Date : 2025-01-21DOI: 10.1016/j.ipl.2025.106560
Arnab Bhattacharyya , Sutanu Gayen , Kuldeep S. Meel , Dimitrios Myrisiotis , A. Pavan , N.V. Vinodchandran
We show that computing the total variation distance between two product distributions is -complete. This is in stark contrast with other distance measures such as Kullback–Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.
{"title":"Total variation distance for product distributions is #P-complete","authors":"Arnab Bhattacharyya , Sutanu Gayen , Kuldeep S. Meel , Dimitrios Myrisiotis , A. Pavan , N.V. Vinodchandran","doi":"10.1016/j.ipl.2025.106560","DOIUrl":"10.1016/j.ipl.2025.106560","url":null,"abstract":"<div><div>We show that computing the total variation distance between two product distributions is <span><math><mi>#</mi><mi>P</mi></math></span>-complete. This is in stark contrast with other distance measures such as Kullback–Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106560"},"PeriodicalIF":0.7,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099054","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}
This work examines the Conditional Approval Framework for elections involving multiple interdependent issues, specifically focusing on the Conditional Minisum Approval Voting Rule. We first conduct a detailed analysis of the computational complexity of this rule, demonstrating that no approach can significantly outperform the brute-force algorithm under common computational complexity assumptions and various natural input restrictions. In response, we propose two practical restrictions (the first in the literature) that make the problem computationally tractable and show that these restrictions are essentially tight. Overall, this work provides a clear picture of the tractability landscape of the problem, contributing to a comprehensive understanding of the complications introduced by conditional ballots and indicating that conditional approval voting can be applied in practice, albeit under specific conditions.
{"title":"On the Tractability Landscape of the Conditional Minisum Approval Voting Rule","authors":"Georgios Amanatidis , Michael Lampis , Evangelos Markakis , Georgios Papasotiropoulos","doi":"10.1016/j.ipl.2025.106561","DOIUrl":"10.1016/j.ipl.2025.106561","url":null,"abstract":"<div><div>This work examines the Conditional Approval Framework for elections involving multiple interdependent issues, specifically focusing on the Conditional Minisum Approval Voting Rule. We first conduct a detailed analysis of the computational complexity of this rule, demonstrating that no approach can significantly outperform the brute-force algorithm under common computational complexity assumptions and various natural input restrictions. In response, we propose two practical restrictions (the first in the literature) that make the problem computationally tractable and show that these restrictions are essentially tight. Overall, this work provides a clear picture of the tractability landscape of the problem, contributing to a comprehensive understanding of the complications introduced by conditional ballots and indicating that conditional approval voting can be applied in practice, albeit under specific conditions.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106561"},"PeriodicalIF":0.7,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099053","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}
Pub Date : 2025-01-16DOI: 10.1016/j.ipl.2025.106559
Alexandros A. Voudouris
We consider a distributed voting problem with a set of agents that are partitioned into disjoint groups and a set of obnoxious alternatives. Agents and alternatives are represented by points in a metric space. The goal is to compute the alternative that maximizes the total distance from all agents using a two-step mechanism which, given some information about the distances between agents and alternatives, first chooses a representative alternative for each group of agents, and then declares one of them as the overall winner. Due to the restricted nature of the mechanism and the potentially limited information it has to make its decision, it might not be always possible to choose the optimal alternative. We show tight bounds on the distortion of different mechanisms depending on the amount of the information they have access to; in particular, we study full-information and ordinal mechanisms.
{"title":"Metric distortion of obnoxious distributed voting","authors":"Alexandros A. Voudouris","doi":"10.1016/j.ipl.2025.106559","DOIUrl":"10.1016/j.ipl.2025.106559","url":null,"abstract":"<div><div>We consider a distributed voting problem with a set of agents that are partitioned into disjoint groups and a set of <em>obnoxious</em> alternatives. Agents and alternatives are represented by points in a metric space. The goal is to compute the alternative that <em>maximizes</em> the total distance from all agents using a two-step mechanism which, given some information about the distances between agents and alternatives, first chooses a representative alternative for each group of agents, and then declares one of them as the overall winner. Due to the restricted nature of the mechanism and the potentially limited information it has to make its decision, it might not be always possible to choose the optimal alternative. We show tight bounds on the <em>distortion</em> of different mechanisms depending on the amount of the information they have access to; in particular, we study full-information and ordinal mechanisms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106559"},"PeriodicalIF":0.7,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143156875","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}
Pub Date : 2025-01-14DOI: 10.1016/j.ipl.2025.106558
Michael T. Goodrich
The Quickhull algorithm is a simple algorithm for constructing the convex hull of a set of n points. Quickhull is usually described for points in the plane, in which case it is defined as a divide-and-conquer algorithm, where one has a pair of points such that p and r are on the convex hull, and one then finds the point, q, farthest from the line , which must also be on the convex hull, and then uses the triangle to divide the remaining points and recursively solve the resulting subproblems. It is well-known that Quickhull has a worst-case running time of , but it runs much faster than this for some input distributions. In a highly cited paper, Barber, Dobkin, and Huhdanpaa conjecture that the Quickhull algorithm runs in worst-case time, where h is the size of the convex hull, when the input points have precision . In this paper, we give an explicit lower-bound construction that shows that, in general, the worst-case running time of the Quickhull algorithm is . Our lower bound proof also provides a counter-example to the Quickhull precision conjecture of Barber et al., in that we give an explicit construction of a set, S, of n points with precision such that h is but the worst-case running time of Quickhull on S is , not .
{"title":"A lower bound for the Quickhull convex hull algorithm that disproves the Quickhull precision conjecture","authors":"Michael T. Goodrich","doi":"10.1016/j.ipl.2025.106558","DOIUrl":"10.1016/j.ipl.2025.106558","url":null,"abstract":"<div><div>The <em><strong>Quickhull</strong></em> algorithm is a simple algorithm for constructing the convex hull of a set of <em>n</em> points. Quickhull is usually described for points in the plane, in which case it is defined as a divide-and-conquer algorithm, where one has a pair of points <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> such that <em>p</em> and <em>r</em> are on the convex hull, and one then finds the point, <em>q</em>, farthest from the line <span><math><mover><mrow><mi>p</mi><mi>r</mi></mrow><mo>‾</mo></mover></math></span>, which must also be on the convex hull, and then uses the triangle <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> to divide the remaining points and recursively solve the resulting subproblems. It is well-known that Quickhull has a worst-case running time of <span><math><mi>Θ</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, but it runs much faster than this for some input distributions. In a highly cited paper, Barber, Dobkin, and Huhdanpaa conjecture that the Quickhull algorithm runs in worst-case <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>h</mi><mo>)</mo></math></span> time, where <em>h</em> is the size of the convex hull, when the input points have precision <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>. In this paper, we give an explicit lower-bound construction that shows that, in general, the worst-case running time of the Quickhull algorithm is <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mi>h</mi><mo>)</mo></math></span>. Our lower bound proof also provides a counter-example to the Quickhull precision conjecture of Barber et al., in that we give an explicit construction of a set, <em>S</em>, of <em>n</em> points with precision <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> such that <em>h</em> is <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> but the worst-case running time of Quickhull on <em>S</em> is <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mi>h</mi><mo>)</mo></math></span>, not <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>h</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106558"},"PeriodicalIF":0.7,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099051","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}
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