Ph.D. Thesis Proposal Data Caching in Ad Hoc and Sensor Networks

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Ph.D. Thesis ProposalData Caching in Ad Hoc
and Sensor Networks

• Bin Tang
• Computer Science Department
• Stony Brook University

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Summary of My Work

• Data Caching
• Update cost constraint
• Optimal algorithm for tree approximation
  algorithm for general graph.
• Memory constraint with multiple data items
• Approximation algorithm for general graph
• number constraint w/h read/write/storage cost
• Optimal algorithm for tree
• Localized distributed implementations.
• Compare with existing work

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Motivation

• Ad hoc and sensor networks are resource
  constrained
• Limited bandwidth, battery energy, and memory
• Caching can save access (communication) cost, and
  thus, bandwidth and energy
• Under update cost, memory, number constraint

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Rooted in

• Facility location problem set up facilities in a
  network to minimize total access cost and setting
  up cost
• K-median problem set up k facilities to minimize
  total access cost

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1. Cache Placement in Sensor Networks Under
Update Cost Constraint
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Problem Statement

• Sensor Network Model
• A data item stored at a server node.
• Updated at a certain frequency.
• Other nodes access the data item at a certain
  frequency.
• Problem Statement
• Select nodes to cache the data item to
• Goal Minimize total access cost
• Constraint Total update cost.

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Why update cost constraint?

• Nodes close to the server bear most of the
  update cost.

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Problem Formulation

• Given
• Network graph G(V,E).
• A data item stored at a server node
• Update frequency
• Access frequency for each other node
• Update cost constraint ?
• Goal
• Select cache nodes to minimize the total access
  cost
• Total update cost is less than ?

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Total Access/Update Cost

• Total Access Cost
• ? i ? V (hop length between i and its nearest
  cache x access frequency of i)
• Total Update cost cost of the optimal Steiner
  tree over server and all caches

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Algorithm Design Outline

• Tree Networks
• Optimal dynamic programming algorithm.
• General Networks
• Multiple-unicast update model -- Approximation
  algorithm.
• Steiner-tree update model Heuristic and
  Distributed.

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Tree Networks
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Subtree notation
r

• Server r
• Consider a subtree Tv.
• Let path (v,x) on its leftmost branch be all
  caches.
• Let C_v be the optimal access cost in Tv using
  additional update cost d
• Next Recursive equation for C_v

v
x
Tv
Tr
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Dynamic Programming Algorithm for Tvunder update
cost constraint d

• Let u leftmost deepest node in the optimal set
  of caches in Tv
• Path(v,u) can be all caches (update cost doesnt
  increase)
• For a fixed u,
• C_v
• Constant optimal access cost in Rv,u for
  constraint (d d_u)
• Here, d_u is the cost to update u (using
  path(v,x)).

Tv Lv,u Tu Rv,u
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DP recursive equation for Tv

• C_v minu ? Tv (access cost in Lv,u using
  path(v,x) or path(v,u)
• access cost in Tu using u
  optimal cost in Rv,u with constraint d d_u)
• Here, d_u is the cost in updating u (using
  path(v,x)).
• Note that Rv,u has a path (v, parent(u)) of
  caches on its leftmost branch.

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Time complexity

• Time complexity O(n4n3 ?)
• Analysis
• Precomputation takes O(n4)
• Lv,u with cache path (v,x) O(n4), for all v,u,x
• Tu O(n2), for all u
• Recursive equation takes O(n3 ?)
• n2? entries for each pair of (v,x) and all
  values of ?
• Each entry takes O(n) n possible u

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General Graph Network

• Two Update Cost Models
• Multiple-Unicast
• Optimal Steiner Tree

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Multiple-Unicast Update Model

• Update cost Sum of shortest path lengths from
  server to each cache node
• Benefit of node A Decrease in total access cost
  due to selection of A as a cache
• Benefit per unit update cost.

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Greedy Algorithm

• Iteratively Select the node with the highest
  benefit per unit update cost, until the update
  cost is exhausted
• Theorem Greedy solutions benefit is at least
  63 of the optimal benefit.

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Steiner-Tree Update Cost Model

• Steiner-tree update cost Cost of 2-approximation
  Steiner tree over cache nodes
• Incremental Steiner update cost of node A
  Increase in Steiner-tree update cost due to A
  becoming a cache
• Greedy-Steiner Algorithm
• Iteratively, select the node with the highest
  benefit per unit above-defined update cost.

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Distributed Greedy-Steiner Algorithm

• Each non-cache node estimates its benefit per
  unit update cost
• If the estimate is maximum among all its
  non-cache neighbors, then it decides to cache
• Algorithm
• In each rounds, each node decides to cache based
  on above.
• The server gathers new cache node information,
  and computes the total update cost
• The remaining update cost is broadcast to the
  network, and the new round begins

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Performance Evaluation

• (i) network-related -- number of nodes and
  transmission radius,
• (ii) application-related -- number of
  clients.
• Random network of 2,000 to 5,000 nodes in a 30 x
  30 region.

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Compared Caching Schemes

• Centralized Greedy
• Centralized Greedy-Steiner
• Distributed Greedy-Steiner
• Dynamic Programming on Shortest Path Tree of
  Clients
• Dynamic Programming on Steiner Tree over Clients
  and Server

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Varying Network Size Transmission radius 2,
percentage of clients 50, update cost 25 of
the Steiner tree cost
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Varying Transmission Radius - Network size
4000, percentage of clients 50, update cost
25 of the Steiner tree cost
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Varying number of clients Transmission Radiu
2, update cost 50 of the Steiner tree cost,
network size 3000
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To Recap

• Data caching problem under update cost
  constraint.
• Optimal algorithm for tree an approximation
  algorithm for general graph.
• Efficient distributed implementations.
• More general cache placement problem
• (a) under memory constraint (b) multiple data
  items.

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2. Data Caching under Memory Constraint
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Problem Addressed

• In a general ad hoc network with limited memory
  at each node, where to cache data items, such
  that the total access (communication) cost is
  minimized?

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Problem Formulation

• Given
• Network graph G(V,E)
• Multiple data items
• Access frequencies (for each node and data item)
• Memory constraint at each node
• Select data items to cache at each node under
  memory constraint
• Minimize total access cost ?nodes ?data items
• (distance from node to the nearest cache
  for that data item) x (access frequency)

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Related Work

• Related to facility-location problem and K-median
  problem No memory constraint
• Baev and Rajaraman
• 20.5-approximation algorithm for uniform-size
  data item
• For non-uniform size, no polynomial-time
  approximation unless P NP
• We circumvent the intractability by approximating
  benefit instead of access cost

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Related Work - continued

• Two major empirical works on distributed caching
• Hara infocom99
• Yin and Cao Infocom 04 (we compare our work
  with theirs)
• Our work is the first to present a distributed
  caching scheme based on an approximation algorithm

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Algorithms

• Centralized Greedy Algorithm (CGA)
• Delivers a solution whose benefit is at least
  1/2 of the optimal benefit
• Distributed Greedy Algorithm (DGA)
• Purely localized

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Centralized Greedy Algorithm (CGA)

• Benefit of caching a data item at a node
• the reduction of total access cost
• i.e., (total access cost before caching)
• (total access cost after caching)

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Centralized Greedy Algorithm (CGA)

• CGA iteratively selects the most beneficial
• (data item, node to cache at) pair.
• I.e., we pick (at each stage) the pair that has
  the maximum benefit.
• Theorem CGA is (1/2)approximate for uniform
  data item.
• ¼-approximate for non-uniform size data item

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CGA Approximation Proof Sketch

• G modified G, where each node
• has twice memory of that in G
• caches data items selected by CGA and optimal
• B(Optimal in G)
• lt B(Greedy Optimal in G)
• B(Greedy) B(Optimal) w.r.t Greedy
• lt B(Greedy) B(Greedy) Due to greedy choice
• 2 x B(Greedy)

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Distributed Greedy Algorithm (DGA)

• Each node caches the most beneficial data items,
  where the benefit is based on local traffic
  only.
• Local Traffic includes
• Its own data requests
• Data requests to its data items
• Data requests forwarding to others

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DGA Nearest Cache Table

• Why do we need it?
• Forward requests to the nearest cache
• Local Benefit calculation
• What is it?
• Each nodes keeps the ID of nearest cache for each
  data item
• Entries of the form (data item, the nearest
  cache)
• Above is on top of routing table.
• Maintenance next slide

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Maintenance of Nearest-cache Table

• When node i caches data Dj
• broadcast (i, Dj) to neighbors
• Notify server, which keeps a list of caches
• On recv (i, Dj)
• if i is nearer than current nearest-cache of Dj,
  update and forward

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Maintenance of Nearest-cache Table -II

• i deletes Dj
• get list of caches Cj from server of Dj
• broadcast (i, Dj, Cj) to neighbors
• On recv (i, Dj, Cj)
• if i is current nearest-cache for Dj, update
  using Cj and forward

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Maintenance of Nearest-cache Table -III

• More details pertaining to
• Mobility
• Second-nearest cache entries (needed for benefit
  calculation for cache deletions)
• Benefit thresholds

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Performance Evaluation

• CGA vs. DGA Comparison
• DGA vs. HybridCache Comparison

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CGA vs. DGA

• Summary of simulation results
• DGA performs quite close to CGA, for wide range
  of parameter values

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Varying Number of Data Items and Memory Capacity
Transmission radius 5, number of nodes 500
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DGA vs. Yin and Caos work.

• Yin and Caoinfocom04
• CacheData caches passing-by data item
• CachePath caches path to the nearest cache
• HybridCache caches data if size is small
  enough, otherwise caches the path to the data
• Only work of a purely distributed cache placement
  algorithm with memory constraint

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DGA vs. HybridCache

• Simulation setup
• Ns2, routing protocol is DSDV
• Random waypoint model, 100 nodes move at a speed
  within (0,20m/s), 2000m x 500m area
• Tr250m, bandwidth2Mbps
• Performance metrics
• Average query delay
• Query success ratio
• Total number of messages

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• Server Model
• 1000 data items, divided into two servers.
• Data item size 100, 1500 bytes
• Data access models
• Random Each node accesses 200 data items
  randomly from the 1000 data items
• Spatial (details skipped)
• Naïve caching algorithm caches any passing-by
  data, uses LRU for cache replacement

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Varying query generate time on random access
pattern
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Summary of Simulation Results

• Both HybridCache and DGA outperform Naïve
  approach
• DGA outperforms HybridCache in all metrics
• Especially for frequent queries and small cache
  size
• For high mobility, DGA has slightly worse average
  delay, but much better query success ratio

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To Recap

• Data caching problem for multiple items under
  memory constraint
• Centralized approximation algorithm
• Localized distributed implementation
• No update or storage cost are considered
  (otherwise, no performance guarantee)
• Can we consider and minimize the total cost of
  read/write/storage ?

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3. Data Caching Under Number Constraint
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Problem Formulation

• Given
• Network graph G(V,E).
• A data item to be stored in the network
• Access (read) frequency for each node
• Write frequency for each node
• Caching (storage) cost for each node
• Number of allowable caching node P
• Goal
• Select cache nodes to minimize the total cost
• Under number constraint

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Total Cost

• Total read cost total write cost total
  storage cost
• ? i ? V (hop length between i and its
  nearest cache x access frequency of i)
• ? i ? V (cost of optimal steiner tree over i
  and all caches x write frequency of i)
• ? i ? cache nodes (storage cost at i)

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Related Work

• K-median problem (access and storage cost)
• Tamir attains the best time complexity in tree
• We generalize it with write cost in both tree (
  O(n2P3) ) and general graph
• Kalpakis et al. solves the same problem, with
  time complexity O(n6P3)

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Tree Topology
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Tamirs DP Algorithm on tree Tr

• Transform arbitrary tree into full binary tree
• Each non-leaf node v has two children v1, v2
• For each v in binary tree, compute and sort the
  distance from v to all nodes
• leaves to root dynamic programming algorithm

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Our DP Algorithm

• Ideal For each node v in Tr
• the cost of sub-tree Tv
• access cost of nodes in Tv
• storage cost of caching nodes in Tv
• write cost of all the writer nodes in Tr due to
  edges in Tv

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DP Algorithm - Definitions

• G(v, q, r) optimal cost for subtree Tv, exact q
  caches in Tv, closest to v is at most r hops away
• F(v, q, r) optimal cost for Tv, exact q caches
  in Tv some cache nodes outside of Tv, closest to
  v is r hops away
• F(v, r) optimal cost for Tv, no cache in Tv
  some cache nodes outside of Tv, closest to v is r
  hops away

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Recursive DP Equations p cache nodes allowed

• 1. G(v, q, 0) -- v is cache node
• storage cost at v
• the cost of Tv1, Tv2
• the write cost on vv1, vv2
• 2. G(v, qltp, rgt0) there is some cache node
  outside of Tv
• min G(v, q, r-1), // there is cache in Tv
  r-1 hops from v
• cost in closest cache to v is r hops away

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Recursive DP Equations - continued

• 3. G(v, qP, rgt0) no cache node outside of Tv
• min G(v, q, r-1),
• the cost of closest cache is r hops
  away
• 4. F(v, q, r) there is cache node outside of Tv
• min G(v, q, r-1),
• the cost of closest cache to v is r hops
  away

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• Minimum total cost of original tree Tr
• min 1pP G(r, p, L, L is the hops of r
  to the farthest node in Tr
• Time Complexity O(n2P3)
• For each p, vary q from 1 to q
• For each (v, q), vary closest cache node to v (n
  possibilities) and spit q in to Tv1, Tv2 (q such
  possibilities)

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Conclusion

• We design optimal, near optimal and heuristics
  for data caching under different constraint in ad
  hoc and sensor networks
• We show our algorithms can be implemented in
  distributed way

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Questions?