Sorting Buffers on HSTs

1
Sorting Buffers on HSTs

• Harald Räcke
• Matthias Englert Matthias Westermann

2
Problem Definition
Input sequence
Metric space
Output sequence
3
Applications

• Sorting Buffers are used e.g. in car
  manufacturing for sorting cars according to
  their color before they are painted.
• reducing state changes in a rendering system.
• uniform metric
  star metric
• minimizing head movements in a hard-disc
• line metric with equidistant points

4
Previous Work

• R, Sohler, Westermann ESA 2002
• uniform metric competitive ratio
• Englert, Westermann ICALP 2005
• star metric competitive ratio
• Khandekar, Pandit STACS 2005
• line metric competitive ratio

5
Our Results

• competitive ratio for HSTs
  of height
• HST all edges on the same level have the same
  length.
• The edges on the level have length
  for a constant .
• also competitive ratio for
  HSTs
• General Metrics ratios
  and .

6
Why some natural strategies fail

• points/items are kept in the buffer for too
  long. This shrinks the effective buffer-size
• Nearest Item Next
• Starting situation
• sequence b a b a b a b a b a b a b a b a ...

7
Why some natural strategies fail

• points/items are removed too early. Therefore
  clusters of points cannot be removed in one huge
  chunk.
• FIFO
• Starting situation
• sequence x aaaa bbbb x aaaa bbbb x aaaa bbbb x
  aaaa bbbb ...

8
The Algorithm for HSTs

• The algorithm is defined by the following
  continuous process.
• Every item in the buffer sends out an agent
• In time step an agent places payment
  on the first yet unpaid edge in direction
  towards the current ONLINE-position.
• When a group of agents arrives at the
  ONLINE-position, the corresponding items are
  removed from the buffer and their payment is
  removed from the edges.
• ONLINE moves to the furthest among the just
  removed items.

9
The Algorithm for HSTs
This front gets the third item now it moves
three times faster
This front also gets two items.
Now two items are working on this front. This
increases speed.
10
Analysis

• Observation
• The total payment is equal to the ONLINE-cost.
• Idea
• fix an optimal algorithm OPT
• analyze the payment that is created on an edge
  between two visits of OPT to this edge

11
Analysis

• Introduce discounts
• Whenever an ONLINE-item creates a unit of
  payment, an OPT-item creates a discount of
  on every ancestor edge.
• An ONLINE-item creates a discount of
  on every ancestor edge, as well.
• The total discount that is produced is only
  half the total payment.
• Idea
• analyze the real payment created on an edge
  (payment-discount) between two consecutive
  visits of OPT to the edge.

12
Analysis

• Amortization move payment between edges, such
  that (in the end) for every
  edge
  accepted_payment(e) f
  OPT(e)
• dont delete payment
• The amortization is done with respect to the
  actions of the ONLINE, and OPTIMAL algorithm.

13
Amortization

• every edge is assigned the following counters
• active_payment(e)
• this counter describes the payment on an edge
  asdefined by the ONLINE-algorithm.
• amortized_payment(e)
• this counter acts as a buffer. When e.g. the
  active paymenton an edge is set to 0, this
  counter may be increased.
• accepted_payment(e)in the end all other counters
  should be 0 and this countershould be
  comparable to OPT(e).
• discount(e)counts the discounts collected so far

14
Amortization

• when ONLINE goes over an edge,
  active_payment(e) is set to 0.
• for adjustment we do one of the following
• self-amortizationwe increase the counter
  amortized_payment(e) by active_payment(e). We
  set discount(e) to 0.
• amortization we increase the amortized-counter
  on some other edge by active_payment(e)
  amortized_payment(e).We set all counters at e to
  0.
• accept payment on ehere we increase the counter
  accepted_payment(e) byactive_payment(e)
  amortized_payment(e) - discount(e).We set all
  counters at e to 0.

15
Amortization

• Facts
• active_payment(e)
  clear
• amortized_payment(e)
  to prove
• Rules
• if OPTIMAL traverses an edge accept payment on
  this edge
• inequality still holds!

16
Amortization

• ONLINE only accepts payment on an edge if
• ONLINE is moving downward
• the sub-tree does not contain OPT, i.e.
  ONLINE moves away from OPT.
• active(e)amortized(e)-disount(e) lt 0
• OPT-exclusive( ) items are those items in
  that are held by OPT but not by ONLINE.
• ONLINE-exclusive( ) items are those
  held by ONLINE but not by OPT.
• Only accept payment if either the number
• of OPT-exclusive items increases by
• a -factor or if the number
• of ONLINE-exclusive items shrink by a
  -factor.

and either
or
17
Amortization

• We have to show that in all other cases we can
  amortize without amortized_payment getting
  too large.
• Remaining cases (Overview)
• away from OPT / upwards amortize to another
  edge (move the payment to another
  downward-edge)
• away from OPT / downwards when the
  exclusive items do not change too much.
• towards OPT make a self-amortization
• (ignore the payment and move it to your own
  buffer)

More to come
not possible
OK
18
Amortization

• Away from OPT / upwards
• move the payment to the buddy edge that you
  visit next on the same level (sideways
  amortization)

• how much payment does an edge receive
  because of side- ways amortization?
• cannot receive any payments due to
  sideways-amortization.
• therefore shifts at most to
• receives at most from one edge before
  ONLINE visits and removes the payment.

19
Amortization

• Away from OPT / downwards
• items you remove
• assume

These are the only items that generated
paymenton e since the last visit.
All these items generateddiscounts since the
lastvisit.
20
Amortization

• Away from OPT / downwards
• items you remove
• assume

These are the only items that generated
paymenton e since the last visit.
All these items generateddiscounts since the
lastvisit.
21
Extension to

• So far we didnt use the HST-property. The
  analysis can be adapted
• to general trees giving a competitive ratio of
• Which of the ancestor edges need the discount?
• edges, such that ONLINE is in the sub-tree dont
  need the discount.
• only create discount on edges that are at
• most edges away from the
• LCA of ONLINE and the item.
• additionally let discount grow from
• the position of the item

Edges that are close obtain discount by region
growing
22
Open Problems

• Is it possible to reduce the competitive ratio
  toon the line / in arbitrary trees.
• Can you remove the dependency on for
  arbitrarymetrics?
• Is there a constant competitive ratio?