Locality helps sleep scheduling Jukka Suomela Helsinki Institute - - PowerPoint PPT Presentation

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Oct 03, 2023 148 likes •304 views

Locality helps sleep scheduling Jukka Suomela Helsinki Institute for Information Technology HIIT Department of Computer Science, University of Helsinki, Finland jukka.suomela@cs.helsinki.fi http://www.cs.helsinki.fi/jukka.suomela/ Workshop on

SLIDE 1

Locality helps sleep scheduling

Jukka Suomela

Helsinki Institute for Information Technology HIIT Department of Computer Science, University of Helsinki, Finland jukka.suomela@cs.helsinki.fi http://www.cs.helsinki.fi/jukka.suomela/

Workshop on World-Sensor-Web (WSW 2006) 31 October 2006

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SLIDE 2

Sleep scheduling problem

◮ Given: a sensor network ◮ Assumption: there may be some redundant nodes ◮ Objective: find a sleep schedule that

maximises the lifetime of the network

◮ Constraints:

◮ Energy-constrained nodes ◮ At any point of time,

a node can sleep only if it is redundant

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SLIDE 3

Redundancy graphs

◮ Focus: pairwise redundancy of the nodes ◮ Example: if v1 is active then v2 may be asleep

and vice versa (e.g., sensors are close to each other)

v3 v6 v4 v5 v7 v2 v1

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SLIDE 4

Redundancy graphs

◮ Redundancy relations can be represented as a graph

v3 v6 v4 v5 v7 v2 v1

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SLIDE 5

Redundancy graphs

◮ A valid set of active (nonsleeping) nodes

= a dominating set in the redundancy graph

◮ Two examples; red = active, white = asleep:

v3 v6 v4 v5 v7 v2 v1 v3 v6 v4 v5 v7 v2 v1

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SLIDE 6

Redundancy graphs

◮ Sleep schedule = a time interval for each dominating set ◮ No single node is active for more than 1 unit of time ◮ An example of a sleep schedule of length 2:

v3 v6 v4 v5 v7 v2 v1

Active for 1 time unit

v3 v6 v4 v5 v7 v2 v1

Active for 1 time unit

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SLIDE 7

Domatic partition

◮ One possible solution: find a domatic partition ◮ That is, partition the nodes into disjoint dominating sets

(maximum number of such sets = domatic number)

◮ Assign 1 time unit to each such set ◮ Length of sleep schedule = number of such sets

v3 v6 v4 v5 v7 v2 v1

Green nodes are a dominating set and red nodes are another disjoint dominating set

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SLIDE 8

Domatic partition

◮ However, domatic partition does not necessarily give

an optimal sleep schedule

◮ Example: ring of 5 nodes (neighbours pairwise redundant) ◮ Domatic number is 2 ◮ We obtain a sleep schedule of length 2:

1 1

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SLIDE 9

Fractional domatic partition

◮ We have to allow fractional solutions ◮ This is an LP relaxation of domatic partition

(fractional domatic partition)

◮ An optimal sleep schedule of length 5/2:

1/2 1/2 1/2 1/2 1/2

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SLIDE 10

Fractional domatic partition

◮ Sleep scheduling with pairwise redundancy =

fractional domatic partition of the redundancy graph

◮ There is a lot of research on finding domatic partitions

(and more general set cover packings, set K-cover), but little research on the fractional versions

◮ Unfortunately, both domatic partition and

fractional domatic partition in general graphs are hard to approximate within factor (1 − ǫ) ln |V |

◮ Solution: observe that realistic redundancy graphs

are not arbitrary graphs

◮ We focus on what we call (d, N)-local graphs

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SLIDE 11

Local graphs

◮ Nodes are points in a d-dimensional space

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SLIDE 12

Local graphs

◮ Nodes are points in a d-dimensional space ◮ No more than N nodes in any unit disk

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SLIDE 13

Local graphs

◮ Nodes are points in a d-dimensional space ◮ No more than N nodes in any unit disk ◮ No edges longer than 1 unit

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SLIDE 14

Fractional domatic partition in local graphs

Main result:

◮ Polynomial-time approximation scheme (PTAS)

for fractional domatic partition in local graphs

◮ That is, for any ǫ > 0, there is

a polynomial-time (1 + ǫ)-approximation algorithm

Techniques:

◮ Garg-K¨

nemann LP approximation scheme:

◮ Problem reduced to minimising weighted dominating set

◮ Divide-and-conquer technique based on modular grids:

◮ Multiple partitions of the plane ◮ Solve weighted dominating set optimally in each cell ◮ Nodes near borders of the cells may do extra work ◮ However, at least one of the partitions is good:

there is not too much weight near the borders

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SLIDE 15

Summary

◮ Focus on pairwise redundancy ◮ Sleep scheduling in sensor networks

= fractional domatic partition in redundancy graphs

◮ Hard to solve or approximate in general ◮ Focus on (d, N)-local graphs ◮ Polynomial-time approximation scheme (PTAS)

for fractional domatic partition in local graphs

◮ Assumptions on locality help with sleep scheduling

Jukka Suomela Helsinki Institute for Information Technology HIIT Department of Computer Science, University of Helsinki, Finland jukka.suomela@cs.helsinki.fi http://www.cs.helsinki.fi/jukka.suomela/ 15 / 15