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

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

TL;DR

This work comes up with the first near-linear time implementation for LPT, and it is proved that the natural approach of solving this problem via the LPT heuristic has an approximation factor of $\phi$, where $\phi \approx 1.62$ is the golden ratio.

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. Computing, 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 $O((n+m)(\log^2{m}+\log{n}))$. 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 $φ\approx 1.62$ is the golden ratio.

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

TL;DR

This work comes up with the first near-linear time implementation for LPT, and it is proved that the natural approach of solving this problem via the LPT heuristic has an approximation factor of , where is the golden ratio.

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. Computing, 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 time, where is the number of machines and 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 . 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 , where is the golden ratio.
Paper Structure (24 sections, 7 theorems, 10 equations, 2 figures, 3 algorithms)

This paper contains 24 sections, 7 theorems, 10 equations, 2 figures, 3 algorithms.

Table of Contents

  1. LPT heuristic for uniform scheduling
  2. First result: A near-linear time implementation of LPT
  3. Second result: LPT for the drones warehouse problem (DWP)
  4. The drones warehouse problem (DWP)
  5. Our results and techniques
  6. Related work on machine scheduling
  7. Uniform machines scheduling problem (USP)
  8. ISP $\subset$ USP $\subset$ DWP
  9. DWP $\subset$ USP-C $\subset$ UnrelSP.
  10. Other algorithms and optimization measures.
  11. Near linear time implementation
  12. Longest processing time first (LPT)
  13. Dynamic Lower Envelope
  14. $\phi$-approximation for the Drone Warehouse Problem
  15. Algorithm
  16. ...and 9 more sections

Key Result

Theorem 2.1

There is an $O((n+m)(\log^2{m}+\log{n}))$ time implementation of the LPT heuristic.

Figures (2)

  • Figure 4.1: Landscape of the approximation ratio of the LPT heuristic for machine scheduling problems and our drone warehouse problem (DWP). The relation $A \subset B$ in the figure implies that $A$ is a special case of $B$. Therefore, the approximation factor increases from left to right in the figure. The interval $[a,b]$ means that the approximation ratio of the LPT heuristic is at least $a$ and at most $b$. As there is no total order among jobs in UnrelSP, LPT is not applicable for UnrelSP.
  • Figure 5.1: Dynamic lower envelope of lines $L_1: y=-x+4$, $L_2: y = x$, $L_3: y = 2x$, $L_4: y = 1$

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 3.1
  • Lemma 6.1
  • Definition 6.2
  • Definition 6.3
  • Definition 6.4
  • Lemma 6.5
  • Lemma 6.6
  • Lemma 6.7
  • Lemma 6.8