Ion Mandoiu

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Approximation Algorithms in VLSI Design and
Wireless Networks

• Ion Mandoiu
• UC San Diego

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What Are Approximation Algorithms?

• Polynomial time algorithms for an optimization
  problem guaranteed to return solutions close to
  optimum
• Closeness measure multiplicative error factor
• ?-approximation algorithm solution cost within
  factor of ? of best possible
• Why approximate?
• Exact algorithms too slow for the application!
• Most practical optimizations are NP-hard ?
  unlikely to admit any poly-time exact algorithms
• Requirements for practical applicability
• Very low degree polynomial runtime, often
  sub-quadratic
• E.g., Moores law in VLSI rules out over time any
  O(nc) flat placement algorithm for cgt1
• Practical improvements over current best
  heuristics

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Some Problems I Have Worked on

• VLSI Physical Design
• Single net routing (rectilinear Steiner trees,
  connectivity augmentation)
• Congestion/timing driven global routing (MCF)
• Buffer insertion (tree partitioning)
• Clock routing (zero-skew trees)
• Scan chain synthesis (ATSP)
• Ad Hoc Wireless Networks
• Broadcasting (minimum forwarding set)
• Power range assignment (symmetric connectivity)
• Other
• Network design (element connectivity, bounded
  edge-length Steiner trees)
• Routing (QoS Steiner trees)
• Auctions (XOR auctions)
• DNA Arrays (border-length minimization)

• VLSI Physical Design
• Single net routing (rectilinear Steiner trees,
  connectivity augmentation)
• Congestion/timing driven global routing (MCF)
• Buffer insertion (tree partitioning)
• Clock routing (zero-skew trees)
  ? this talk
• Scan chain synthesis (ATSP)
• Ad Hoc Wireless Networks
• Broadcasting (minimum forwarding set)
  ? this talk
• Power range assignment (symmetric connectivity)
• Other
• Network design (element connectivity, bounded
  edge-length Steiner trees)
• Routing (QoS Steiner trees)
• Auctions (XOR auctions)
• DNA Arrays (border-length minimization)

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Outline

• Zero-Skew Steiner Trees
• Problem definition
• Lower bound
• Spanning tree stretching
• Rooted-Kruskal algorithm
• Open problems
• Minimum Forwarding Set
• Broadcasting in ad hoc mobile networks
• Flooding vs. forwarding set approach
• Reduction to single quadrant
• Skyline based algorithm
• Combinatorial refinement algorithm
• Open problems

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Zero-Skew Trees
Zero-Skew Tree rooted tree in which all
root-to-leaf paths have the same length
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The Zero-Skew Tree Problem
Given set of terminals in rectilinear
plane Find zero-skew tree with minimum total
length

• Previous results CKKRST99
• NP-hard for general metric spaces
• Factor 2e 5.44 approximation

• This talk
• Factor 4 approximation for general metric spaces
• Factor 3 approximation for rectilinear plane

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ZST Lower-Bound
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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Constructive Lower-Bound
Computing N(r) is NP-hard, but
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Constructive Lower-Bound
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Constructive Lower-Bound
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Stretching Rooted Spanning Trees

• ZST root spanning tree root

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Stretching Rooted Spanning Trees
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Stretching Rooted Spanning Trees
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Zero-Skew Spanning Tree Problem
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How good are the MST and Min-Star?
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The Rooted-Kruskal Algorithm

• While ? 2 roots remain

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The Rooted-Kruskal Algorithm
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How good is Rooted-Kruskal?
Lemma delay(T) ? length(T)
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How good is Rooted-Kruskal?
Lemma length(T) ? 2 OPT
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Factor 4 Approximation
Algorithm Rooted-Kruskal Stretching

• Length after stretching length(T) delay(T)
• delay(T) ? length(T)
• length(T) ? 2 OPT

? ZST length ? 4 OPT
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Stretching Using Steiner Points
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Factor 3 Approximation
Algorithm Rooted-Kruskal Improved Stretching

• Length after stretching length(T) ½ delay(T)
• delay(T) ? length(T)
• length(T) ? 2 OPT

? ZST length ? 3 OPT
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Practical Considerations

• For a fixed topology, minimum length ZST can be
  found in linear time using the Deferred Merge
  Embedding (DME) algorithm Eda91, BK92, CHH92
• Practical algo Rooted-Kruskal Stretching DME

Theorem Both stretching algorithms lead to the
same ZST topology when applied to the
Rooted-Kruskal tree
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Running Time

• Stretching O(N logN)
• Rooted-Kruskal O(N logN) using the dynamic
  closest-pair data structure of B98
• DME O(N) Eda91, BK92, CHH92

? O(N logN) overall
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Extension to Other Metric Spaces
Everything works as in rectilinear plane, except

• No equivalent of DME known for other spaces

• The space must be metrically convex to apply
  second stretching algorithm

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Bounded-Skew Trees
b-bounded-skew tree difference between length of
any two root-to-leaf paths is at most b
Bounded-Skew Tree Problem given a set of
terminals and bound bgt0, find a b-bounded-skew
tree with minimum total length

• Previous approximation guarantees CKKRST 99
• factor 16.11 for arbitrary metrics
• factor 12.53 for rectilinear plane

Our results factor 14, resp. 9 approximation
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BST construction idea lower bound
Two stage BST construction

• Cover terminals by disjoint b-bounded-skew trees
• Connect roots via a zero-skew tree

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Constructing the tree cover
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BST Approximation
Algorithm Output tree cover ? approximate ZST on
W
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BST Approximation
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Summary of Results on ZST/BST
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Open Problems

• Complexity of ZST problem in rectilinear plane
• Complexity of finding the spanning tree with
  minimum lengthdelay?
• Zero-skew Steiner ratio supremum, over all
  sets of terminals, of the ratio between minimum
  ZST length and minimum spanning tree lengthdelay
• What is the ratio for rectilinear plane?
• What is the ratio for arbitrary spaces? ( ?4,
  ?3)
• Planar ZST / BST

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Outline

• Zero-Skew Steiner Trees
• Problem definition
• Lower bound
• Spanning tree stretching
• Rooted-Kruskal algorithm
• Open problems
• Minimum Forwarding Set
• Broadcasting in ad hoc mobile networks
• Flooding vs. forwarding set approach
• Reduction to single quadrant
• Skyline based algorithm
• Combinatorial refinement algorithm
• Open problems

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Ad Hoc Wireless Networks

• Features
• Lack of a centralized entity
• All the communication carried over the wireless
  medium
• Limited wireless bandwidth
• Limited battery power
• Multi-hop routing
• Assumptions for this talk
• Omnidirectional antennas
• All nodes have equal transmission range
• ?Unit disk graph model

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Broadcast in Ad Hoc Wireless Networks

• Broadcast relaying a message to all (reachable)
  nodes in the network
• Fundamental networking operation
• Very frequently used in ad hoc wireless networks
• Host paging
• Sending alarm signals
• Route discovery
• Wired network implementation uses flooding
• Every node retransmits the message to all its
  immediate neighbors upon receiving the first copy
  of the broadcast message

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Flooding In Ad Hoc Wireless Networks
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Drawbacks of Flooding

• Many redundant transmissions
• Reduce battery life
• Heavy contention
• Retransmitting nodes are close to each other
• High collision rate
• Highly correlated retransmissions
• RTS/CTS dialogue inapplicable
• ? Broadcast storm Ni et al., Mobicom99

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Forwarding Set Approach

• Avoid broadcast storm by keeping track of
  2-neighborhood via beaconing and selectively
  forwarding broadcast messages Quayyum et al.
  HICSS02 Sinha et al. INFOCOM01
• Forwarding set subset of 1-hop neighbors that
  cover all 2-hop neighbors

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Minimum Forwarding Set Problem

• Location oblivious formulation Quayyum et
  al.Sinha et al.
• Given local topology of 2-hop neighborhood
• Find minimum size forwarding set
• Easily reducible to the set cover problem
  ?Well-known greedy algorithm
  gives O(log m) approximation
• Approximation factor is tight

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Does Location Info Help?

• Location info (GPS based) becoming easily
  available
• Location-aware problem formulation
• Given locations of source, 1-hop neighbors, and
  2-hop neighbors
• Find minimum size set of 1-hop neighbors that
  cover all 2-hop neighbors
• Special case of the NP-hard Minimum Disk Cover
  problem
• ?O(1) approximation using bounded VC-dimension
  Bronnimann and Goodrich 1994
• Our results
• 6-approximation in O(n log n) time
• 3-approximation in O(n log2n) time

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Reduction to Single Quadrant

• Lemma All 2-hop neighbors in Q1 are covered
  by OPT1 ?OPT2?OPT4 where OPTiOPT ? Qi

• High Level Algorithm
• Compute forwarding set Fi for 2-hop neighbors in
  quadrant Qi, i1,,4
• Output union of Fis

Theorem If each Fi is within a factor of ?
of optimum forwarding set for 2-hop neighbors in
Qi, then output has size ? 3?OPT
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2-approximation For Single Quadrant
Use only 1-hop neighbors with unit disks
participating in the skyline for the quadrant!
Key observation skyline disks covering any 2-hop
neighbor form an interval
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2-approximation For Single Quadrant

• Compute skyline
• Find skyline disk interval for each 2-hop
  neighbor
• Find minimum set of skyline disks hitting each
  interval
• Sort intervals by right end
• Add the right end disk of first interval, remove
  all intervals hit by it
• Repeat until no intervals left

• O(n log n) runtime
• Approximation ratio of 2 for single quadrant
• - Any disk fully covered by ? 2 skyline disks

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Exact Algorithm for Single Quadrant

• O(n2) runtime
• O(n log2 n) with more involved algorithm

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Open Problems

• What is the complexity of Minimum Forwarding Set
  in the plane?
• Improve factor 3 for Minimum Forwarding Set
• without using reduction to quadrangle
• Improve factor for (unit radius) Minimum Disk
  Cover problem
• O(1) approximation by Bronnimann and Goodrich
• 102-approximation algorithm as follows
• 6-approximation for covering points in a triangle
  (skyline algorithm LP rounding)
• Any optimum disk can cover points in at most 17
  triangles in the standard tiling

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Thank You for Your Attention!