1. Why is bin packing interesting?
Simplest version of industrial packing problems
Cutting stock allocation problem
CD allocation problem (floppy disk allocation problem)
Layouts on VLSI chips (2D version)
Assigning commercials to station breaks (easier)
Packing trucks within a given weight limit
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.
Classic combinatorial optimization problem
Research dates back to the 1940s
In the mid 80’s labeled “The problem that wouldn’t go away”
Subject of research on approximations
before approximations were cool

2. First Fit Decreasing
Sort the items in decreasing order by size
Open one bin
for each item Si
if there is an open bin where Si will fit
then place Si into the leftmost such bin
else open a new bin
place Si into that new bin .2
.25
.33 .4 .3
Absolute performance ratio of FFD is 3/2, and asymptotic performance ratio is 11/9
.75
.67 .5 .4 .2
FFD < 11/9 OPT (i.e …)
Running Time: O(n log n)
[Johnson, Demers, Ullman, Garey, Graham, 1974]
Loooooooooong proof (originally 100+ pages)

3. More on First Fit Decreasing
Why does FFD have such an improved performance ratio?
Pack the trouble items first (big items)
Pack the easier items later (small items)
Can we do better than FFD?
Modified First Fit Decreasing
[Johnson & Garey, 1985]
Like FFD, but with special handling for items in (1/6, 1/3]
Competitive ratio: 71/60 = …
Running Time: O(n log n)

4. Summary Method Competitive Ratio Any Fit 2 Next Fit 2 Worst Fit 2
First Fit 1.7
Best Fit 1.7
FFD … *
Modified FFD … *
Fully polynomial time
approximation scheme YES *
* = asymptotic competitive ratios (vs absolute competitive ratios)
For all L, A(L) <= Ratio(A)*OPT(L) + KA(L) where KA(L) = o(OPT(L))
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

5. Changing the Rules – Fully Dynamic Bin Packing
NOT given the items all at once
Instead: given items one at a time – INSERTs (DELETES)
Upon an INSERT/DELETE, update the packing
NO apriori limitations on this update – the contents of the bins may be changed at will
Maintain a good packing at all times
O(log n) time per INSERT/DELETE (to update)
So, n INSERTs in O(n log n) time

6. Distinct from Off-line, On-Line, Dynamic BP
Off-line bin packing – given all items at once, process & pack
(FFD, MFFD, PTAS)
On-line bin packing – given items one at a time (INSERTS), and must pack immediately, but cannot move already packed items
On-line Algorithms
Any Fit, Next Fit 2
First Fit, Best Fit 1.7
Harmonic [Seiden, 2001]
Lower bound [van Vliet, 1992]

7. Distinct from Off-line, On-Line, Dynamic BP
Variant of On-line bin packing [Gambosi, etal, 1990]– INSERTs
only, each item may be moved a constant number of times
Algorithm A: Competitive ratio: 3/2
Running time: O(n)
Algorithm B: Competitive ratio: 4/3
Running time: O(n log n)
Dynamic bin packing [Coffman, Garey & Johnson, 1983] – INSERT & DELETE items, but items cannot be moved once packed
2.770 < Dynamic First Fit < 2.898
Lower bound: 2.5

8. Return to Fully Dynamic Bin Packing
NOT given the items all at once
Instead: given items one at a time – INSERTS
Upon an INSERT, update the packing
NO apriori limitations on this update – the contents of the bins may be changed at will
Maintain a good packing at all times
O(log n) time per INSERT (to update)
So, n INSERTs in O(n log n) time

9. A first attempt: Fully Dynamic FFD?
Recall FFD Preprocess by sorting items
First Fit pack the items
Making FFD fully dynamic
After each INSERT
Update the sorting in O(log n) – OK!
Update the packing
May need to completely redo the packing
 O(n log n) per INSERT – that’s bad!

10. Another off-line algo: First Fit Grouped
Groups of items:
BIG – contains items of size (1/2, 1]
LARGE – contains items of size (1/3, 1/2]
Small – contains items of size (1/4, 1/3]
tiny – contains items of size (0, 1/4]
First Fit Grouped (FFG) Algorithm:
Place each item into its group
Pack the items in BIG using first fit
Pack the items in LARGE using first fit
Pack the items in Small using first fit
Pack the items in tiny using First Fit
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

11. FFG – An example
BIG LARGE Small tiny
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.
[Johnson, 1974]
Running Time: O(n log n)
Competitive Ratio of FFG: 4/3
First Fit: FFD: 11/9

12. Fully Dynamic First Fit Grouped
Item I puts on glasses & what does it see?
if I is tiny , then I sees everything
if I is Small, then I sees only the BIG, LARGE and Small items
if I is LARGE , then I sees only the BIG and LARGE items
if I is BIG , then I sees only the BIG items
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.
These are “myopic” glasses!

13. What items “see” Tiny items see:
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

14. Fully Dynamic First Fit Grouped
To process INSERT of item I (assume partial packing exists)
I puts on (myopic) glasses
Item I packs itself in the packing it sees
(smaller items are evicted from the bin I goes into)
Refill the bin that I goes into using:
Smaller evicted items
Smaller items in “inferior” bins (ie largest bin item is in a smaller group than I)
If an inferior bin is disturbed, evict all remaining items
Repack evicted items using recursion
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

15. An example of FDFFG Existing packing:
That item First Fit packs itself into the second bin:
Unpacked items:
.35, .35, .30
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.
Refill the second bin with the .30 item:
Unpacked items:
.35, .35

16. FDFFG Example - continued
It packs into 3rd bin, as does other .35 and then .30 item:
Unpacked items:
.30, .25
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

17. FDFFG Example – still continued …
The .30 item sees this packing:
That .30 item packs into 4th bin & two .25 items refill that bin:
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.
The remaining two .25 items pack themselves into a 5th bin

18. Performance of FDFFG Key elements of FDFFG
Competitive Ratio: 4/3 (same as FFG)
Running Time: O(log n) per INSERT
Key elements of FDFFG
Evicted items are smaller than the inserted item
“bundle” tiny items and move as a unit*
Thus: Relatively few items/bundles will be evicted
(hence, repacked)
* Is this cheating? (no – will say why later)
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

19. O(n log n) for sequence of n INSERT/DELETEs
Can we do better than 4/3?
Yes: Mostly Myopic Packing (MMP)
Key: Avoid bad cases of FDFFG via special handling
MMP Competitive Ratio: 5/4 = 1.25
Looong proof
MMP running time: O(log n) per INSERT or DELETE
O(n log n) for sequence of n INSERT/DELETEs
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

20. Mostly Myopic Packing Same general approach as FDFFG
Much more complicated than FDFFG
Introduce: miniscule items: (0, 1/5]
tiny items: (1/5, 1/4]
Carry over: myopic glasses
bundle miniscule (not tiny) - size (1/10,1/5]
Introduce: coalitions!
Prior to putting on glasses, L & S items try to form LLS-coalitions
Constant number of items/bundles “touched” per INSERT
Each “touch” costs O(log n)
Thus, O(log n) per INSERT (DELETE)
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

21. The 5/4 Competitive Ratio (the 2% sketch)
Analyze the structure of the MMP and OPT packings
Focus on 8 types of “front” bins: B-bins, LLS, LSTT
Lower bound the LLS-bins in MMP relative to OPT
MMPLLS ≥ 1/4(OPTLLS – (L7+S7)) + OPTLSTT – max(L8,S8) - 1
Lower bound the L, S and T-items in B-bins in MMP
Allows upper bound on number of items in “back” bins in MMP
relative to the number in OPT
The packings
In OPT: B OPTLLS + OPTLSTT OPTBACK
In MMP: B MMPLLS MMPBACK
put bounds together to yield the 5/4 bound
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

22. What about that possible “cheating”?
Theorem: If A is a fully dynamic bin packing algorithm
moving at most a fixed (i.e. constant) number of
items per INSERT/DELETE
Then the competitive ratio of A is at least 4/3
Implication: “bundling” (or similar), is necessary to do better than 4/3
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

23. Partially dynamic bin packing
INSERTs only, packing can be rearranged
For any  > 0 there are algorithms with:
Competitive ratio 1 + 
Running time O(log n) amortized per INSERT (PTAS)
Running time O(log2 n) amortized per INSERT (FPTAS)
What about uniform running times for INSERT only?
MMP is the best we know: competitive ratio 5/4
running time O(log n) per INSERT
The problem we consider is topology control in ad hoc networks. So, what is an ad hoc network? An ad hoc network is a collection of nodes[click] that communicate with one other over [click] wireless medium.
When the destination node can not be reached directly by the source, the nodes in the middle can act as routers to forward the transmission.
Such networks require no fixed infrastructure, so can be quick deployed.
it has application in battle field communication, disaster recovery, sensors networks, etc.

24. Related papers
Z. Ivkovich and E.L. Lloyd, "Fully dynamic algorithms for bin packing: Being mostly myopic helps," SIAM Journal on Computing, 28(1998),
Z. Ivkovich and E.L. Lloyd, "Partially dynamic bin packing can be solved within 1+e in (amortized) polylogarithmic time", Information Processing Letters 63(1997),
Z. Ivkovich and E.L. Lloyd, "A fundamental restriction on fully dynamic maintenance of bin packing," Information Processing Letters, 59(1996),