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Two Results on LPT: A Near-Linear Time Algorithm and Parcel Delivery Using Drones

Authors L. Sunil Chandran , Rishikesh Gajjala , Shravan Mehra, Saladi Rahul

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Author Details

L. Sunil Chandran
  • Indian Institute of Science, Bengaluru, India
Rishikesh Gajjala
  • Indian Institute of Science, Bengaluru, India
Shravan Mehra
  • Indian Institute of Science, Bengaluru, India
  • University of Birmingham, UK
Saladi Rahul
  • Indian Institute of Science, Bengaluru, India

Funding

  • Mehra, Shravan: This work was supported by SERB Core Research Grant (CRG/2022/ 006770): "Bridging Quantum Physics with Theoretical Computer Science and Graph Theory’’.
  • Rahul, Saladi: This work was supported by the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-230001).

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L. Sunil Chandran, Rishikesh Gajjala, Shravan Mehra, and Saladi Rahul. Two Results on LPT: A Near-Linear Time Algorithm and Parcel Delivery Using Drones. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.17

Abstract

The focus of this paper is to increase our understanding of the Longest Processing Time First (LPT) heuristic. LPT is a classical heuristic for the fundamental problem of uniform machine scheduling. For different machine speeds, LPT was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). Since then, extensive work has been done to improve the approximation factor of the LPT heuristic. However, all known implementations of the LPT heuristic take O(mn) time, where m is the number of machines and n is the number of jobs. In this work, we come up with the first near-linear time implementation for LPT. Specifically, the running time is O((n+m)(log²m + log n)). Somewhat surprisingly, the result is obtained by mapping the problem to dynamic maintenance of lower envelope of lines, which has been well studied in the computational geometry community.
Our second contribution is to analyze the performance of LPT for the Drones Warehouse Problem (DWP), which is a natural generalization of the uniform machine scheduling problem motivated by drone-based parcel delivery from a warehouse. In this problem, a warehouse has multiple drones and wants to deliver parcels to several customers. Each drone picks a parcel from the warehouse, delivers it, and returns to the warehouse (where it can also get charged). The speeds and battery lives of the drones could be different, and due to the limited battery life, each drone has a bounded range in which it can deliver parcels. The goal is to assign parcels to the drones so that the time taken to deliver all the parcels is minimized. We prove that the natural approach of solving this problem via the LPT heuristic has an approximation factor of ϕ, where ϕ ≈ 1.62 is the golden ratio.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Scheduling
  • Approximation algorithms
  • Fine-grained complexity

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