Minimum Back-Walk-Free Latency Problem with Multiple Servers

Yaw-Ling Lin (林耀鈴 )Dept Computer Sci. & Info. Management,Providence University, Taichung, Taiwan

Yaw-Ling Lin, Providence, Taiwan 2

Minimum Latency Problem (MLP)

• Starts from s, sending goods to all other nodes.

• Traveling Salesperson Problem (TSP): Server oriented

• MLP: Client oriented• MLP is also known as

repairman problem or traveling repairman problem (TRP) s

Yaw-Ling Lin, Providence, Taiwan 3

MLP: Formal Definition

Yaw-Ling Lin, Providence, Taiwan 4

MLP vs. TSP• TSP: minimizes the salesman’s total time. Server oriented, egoistic.

– No contstant approximation algorithm for general case.– Christofides (1976): 3/2-approximation ratio for metric case; Arora (1992):

metric TSP does not have PTAS unless P=NP.– Arora (1998 JACM): PTAS on Euclidean case.

• MLP: minimizes the customers’ total time. Clients oriented, altruistic.– Alias: deliveryman problem, traveling repairman problem (TRP).– Afrati (1986): MAX-SNP-hard for metric case.– Goeman (1996): 10.78-approximation ratio for metric case (with Garg, 19

96FOCS, technique); 3.59-approximation ratio for trees.– Arora (1999 STOC): quasi-polynomial ( O(nO(log n) ) approximation scheme

for trees and Euclidean space. – Sitters (2002, IPCO): MLP on trees is NP-complete; not known for caterpi

llars.

Yaw-Ling Lin, Providence, Taiwan 5

MBLP: Back-Walk Free

Yaw-Ling Lin, Providence, Taiwan 6

An Example

Yaw-Ling Lin, Providence, Taiwan 7

Our Results

• COCOON2002, Singapore, single server MBLP( given a starting point of G )– Trees : O(n log n ) time– k-path : O(n log k) ; path is O(n) time– DAG : NP-Hard (Reduce from 3-SAT)

• This talk (CMCT2003), multiple servers MBLP– k servers on paths : O(n2) time– k servers on cycles : O(n3/k ) time– k origins on paths and cycles : O(n3 log k ) time

Yaw-Ling Lin, Providence, Taiwan 8

Properties

Yaw-Ling Lin, Providence, Taiwan 9

Properties (contd’)

Yaw-Ling Lin, Providence, Taiwan 10

Properties (contd’)

Yaw-Ling Lin, Providence, Taiwan 11

Properties (contd’)

<4,2,3,8> is right-skew; < 5, 3, 4, 1, 2, 6 > is not.<5> <3,4> <1,2,6> is decreasing right-skew partitioned.

Yaw-Ling Lin, Providence, Taiwan 12

Properties (contd’)

Yaw-Ling Lin, Providence, Taiwan 13

Path-Partition: Example

Yaw-Ling Lin, Providence, Taiwan 14

Algorithm Path-Partition

Yaw-Ling Lin, Providence, Taiwan 15

Main Result: k-MBLP on Paths

1 2 3 … n

n1 n2 nk…

Yaw-Ling Lin, Providence, Taiwan 16

k-MBLP: Recurrence Scheme

1 2 3 … nn1 n2 nk…

one-server

Yaw-Ling Lin, Providence, Taiwan 17

Base Cases Analysis

Yaw-Ling Lin, Providence, Taiwan 18

k-MBLP on Cycles

less than O(n/k) cuts

Yaw-Ling Lin, Providence, Taiwan 19

k-server Origin Problems

Yaw-Ling Lin, Providence, Taiwan 20

k-origins: Recurrence Scheme

Yaw-Ling Lin, Providence, Taiwan 21

k-origins: Complexity Analysis

Yaw-Ling Lin, Providence, Taiwan 22

k-origins on Cycles

Yaw-Ling Lin, Providence, Taiwan 23

Future Research

• MLP on caterpillars.• The binary encoding in k-origin setting

could be further exploited.• Multiple servers on trees, paths.