Skip to main content
Springer Nature Link
Log in

Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

  • Conference paper
Automata, Languages, and Programming (ICALP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8572))

Included in the following conference series:

  • 2847 Accesses

Abstract

We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(nlogn + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((nlogn + k)loglogn) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Pătraşcu [SoCG’11] that attains the same time complexity but in expectation.

The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d ≥ 4.

A key tool behind Chan et al.’s previous randomized algorithm is shallow cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA’14] obtained a deterministic O(nlogn)-time algorithm to construct such cuttings. We first present an improved deterministic construction algorithm that runs in O(nloglogn) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

') var buybox = document.querySelector("[data-id=id_"+ timestamp +"]").parentNode var buyingOptions = buybox.querySelectorAll(".buying-option") ;[].slice.call(buyingOptions).forEach(initCollapsibles) // springerPlus roll out 10% starts here var springerPlusGroup = setLocalStorageSpringerPlus(); var rollOutSpringerPlus = springerPlusGroup === "B" function setLocalStorageSpringerPlus() { var selectUserKey = "springerPlusRollOut"; var springerPlusGroup = "X"; if (!window.localStorage) return springerPlusGroup; try { var selectUserValue = window.localStorage.getItem(selectUserKey) springerPlusGroup = selectUserValue || randomDistributionSpringerPlus(selectUserKey) } catch (err) { console.log(err) } return springerPlusGroup; } function randomDistributionSpringerPlus(selectUserKey) { var randomGroup = Math.random() < 0.7 ? "A" : "B" window.localStorage.setItem(selectUserKey, randomGroup) return randomGroup } if (rollOutSpringerPlus) { revealSpringerPlus(); } function revealSpringerPlus() { if(buybox) { document.querySelectorAll(".c-springer-plus").forEach(function(node) { node.style.display = "block" }) } } //springerPlus ends here var buyboxMaxSingleColumnWidth = 480 function initCollapsibles(subscription, index) { var toggle = subscription.querySelector(".buying-option-price") subscription.classList.remove("expanded") var form = subscription.querySelector(".buying-option-form") var priceInfo = subscription.querySelector(".price-info") var buyingOption = toggle.parentElement if (toggle && form && priceInfo) { toggle.setAttribute("role", "button") toggle.setAttribute("tabindex", "0") toggle.addEventListener("click", function (event) { var expandedBuyingOptions = buybox.querySelectorAll(".buying-option.expanded") var buyboxWidth = buybox.offsetWidth ;[].slice.call(expandedBuyingOptions).forEach(function(option) { if (buyboxWidth <= buyboxMaxSingleColumnWidth && option != buyingOption) { hideBuyingOption(option) } }) var expanded = toggle.getAttribute("aria-expanded") === "true" || false toggle.setAttribute("aria-expanded", !expanded) form.hidden = expanded if (!expanded) { buyingOption.classList.add("expanded") } else { buyingOption.classList.remove("expanded") } priceInfo.hidden = expanded }, false) } } function hideBuyingOption(buyingOption) { var toggle = buyingOption.querySelector(".buying-option-price") var form = buyingOption.querySelector(".buying-option-form") var priceInfo = buyingOption.querySelector(".price-info") toggle.setAttribute("aria-expanded", false) form.hidden = true buyingOption.classList.remove("expanded") priceInfo.hidden = true } function initKeyControls() { document.addEventListener("keydown", function (event) { if (document.activeElement.classList.contains("buying-option-price") && (event.code === "Space" || event.code === "Enter")) { if (document.activeElement) { event.preventDefault() document.activeElement.click() } } }, false) } function initialStateOpen() { var buyboxWidth = buybox.offsetWidth ;[].slice.call(buybox.querySelectorAll(".buying-option")).forEach(function (option, index) { var toggle = option.querySelector(".buying-option-price") var form = option.querySelector(".buying-option-form") var priceInfo = option.querySelector(".price-info") if (buyboxWidth > buyboxMaxSingleColumnWidth) { toggle.click() } else { if (index === 0) { toggle.click() } else { toggle.setAttribute("aria-expanded", "false") form.hidden = "hidden" priceInfo.hidden = "hidden" } } }) } initialStateOpen() if (window.buyboxInitialised) return window.buyboxInitialised = true initKeyControls() })()

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Afshani, P.: On dominance reporting in 3D. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 41–51. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Afshani, P., Tsakalidis, K.: Optimal deterministic shallow cuttings for 3D dominance ranges. In: Proc. of the 25th An. SODA, pp. 1389–1398. ACM-SIAM (2014)

    Google Scholar 

  3. Bentley, J.L., Wood, D.: An optimal worst case algorithm for reporting intersections of rectangles. IEEE Trans. Computers 29(7), 571–577 (1980)

    Article  MathSciNet  Google Scholar 

  4. Chan, T.M.: All-pairs shortest paths with real weights in O(n 3/log n) time. Algorithmica 50(2), 236–243 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chan, T.M.: Persistent predecessor search and orthogonal point location on the word RAM. ACM Transactions on Algorithms 9(3), 22 (2013)

    Article  MathSciNet  Google Scholar 

  6. Chan, T.M., Larsen, K.G., Pătraşcu, M.: Orthogonal range searching on the RAM, revisited. In: Proc. of the 27th SoCG, pp. 1–10. ACM (2011)

    Google Scholar 

  7. Chan, T.M., Pătraşcu, M.: Transdichotomous results in computational geometry, I: Point location in sublogarithmic time. SIAM J. Comp. 39(2), 703–729 (2009)

    Article  MATH  Google Scholar 

  8. Chan, T.M., Pătraşcu, M.: Counting inversions, offline orthogonal range counting, and related problems. In: Proc. of the 21st An. SODA, pp. 161–173. ACM-SIAM (2010)

    Google Scholar 

  9. Chazelle, B., Guibas, L.J.: Fractional cascading: I. A data structuring technique. Algorithmica 1(2), 133–162 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comp. Syst. Sci. 48(3), 533–551 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Goodrich, M.T.: Planar separators and parallel polygon triangulation. J. Comp. Syst. Sci. 51(3), 374–389 (1995)

    Article  MathSciNet  Google Scholar 

  12. Gupta, P., Janardan, R., Smid, M., DasGupta, B.: The rectangle enclosure and point-dominance problems revisited. I. J. C. Geom. & Appl. 7(5), 437–455 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Han, Y.: Deterministic sorting in O(nloglogn) time and linear space. J. Algorithms 50(1), 96–105 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lagogiannis, G., Makris, C., Tsakalidis, A.K.: A new algorithm for rectangle enclosure reporting. Inf. Process. Lett. 72(5-6), 177–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lee, D.T., Preparata, F.P.: An improved algorithm for the rectangle enclosure problem. J. Algorithms 3(3), 218–224 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comp. 9(3), 615–627 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Makris, C., Tsakalidis, K.: An improved algorithm for static 3D dominance reporting in the pointer machine. In: Chao, K.-M., Hsu, T.-s., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 568–577. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Matoušek, J.: Reporting points in halfspaces. Comp. Geom. 2, 169–186 (1992)

    Article  MATH  Google Scholar 

  19. Preparata, F.P., Shamos, M.I.: Computational Geometry - An Introduction. Springer (1985)

    Google Scholar 

  20. Ramos, E.A.: On range reporting, ray shooting and k-level construction. In: Proc. of the 15th SoCG, pp. 390–399. ACM (1999)

    Google Scholar 

  21. Vaishnavi, V., Wood, D.: Data structures for the rectangle containment and enclosure problems. Comp. Graphics and Image Processing 13(4), 372–384 (1980)

    Article  Google Scholar 

  22. van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Mathematical Systems Theory 10(1), 99–127 (1976)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MADALGO, Department of Computer Science, Aarhus University, Denmark

    Peyman Afshani

  2. David R. Cheriton School of Computer Science, University of Waterloo, Canada

    Timothy M. Chan

  3. Computer Science and Engineering Department, CUHK, Hong Kong

    Konstantinos Tsakalidis

Authors
  1. Peyman Afshani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timothy M. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Konstantinos Tsakalidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstrasse 3, 85748, München, Germany

    Javier Esparza

  2. LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Pierre Fraigniaud

  3. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Thore Husfeldt

  4. Department Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Elias Koutsoupias

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Afshani, P., Chan, T.M., Tsakalidis, K. (2014). Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-43948-7_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43947-0

  • Online ISBN: 978-3-662-43948-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation