The Cache Location Problem
IEEE/ACM Transactions on Networking,
Vol. 8, No. 5, October 2000
P. Krishnan, Danny Raz, Member, IEEE, and
Yuval Shavitt, Member, IEEE
Abstract
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The goal is to minimize the overall flow or the
average delay by placing a given number of
caches in the network.
The location problems are formulated both
for general caches and for transparent enroute caches (TERCs).
A computationally efficient dynamic
programming algorithm is present for the
single server case.
Introduction
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The popular locations for caches are at the
edge of networks in the form of browser and
proxy caches.
Significant research has gone into optimizing
cache performance, co-operation among
several caches, and cache hierarchies. Web
servers are also replicated to achieve loadbalancing.
Danzig et al. had observed the advantage of
placing caches inside the backbone rather
than at its edge.
Transparent En-route Caches
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When using TERCs, caches are only located
along routes from clients to servers.
An en-route cache intercepts any request that
passes through it, and either satisfies the
request or forwards the request toward the
server along the regular routing path.
TERCs are easier to manage than replicated
web servers since they are oblivious both to
the end-user and the server.
Model and definitions
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Considering a general wide area network,
where the internal nodes are routers and the
external nodes are either servers, clients, or
gateways to different subnets.
A client can request a web page from any of
the servers, and the server vs sends this page
to the client vc on the shortest path from the
server to the client.
When caches are present, a client can
request the page from a cache vk rather than
from the server.
Model and definitions(Cont’d)
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Simplifying “full dependency” assumption: If
a page will be found in any cache, it will be
found in the first cache on the way to the
server.
Each client flow is associated with a single
number pf that is the cachability of this flow.
In other words, pf is the flow hit ratio.
The full dependency assumption implies that
all the flows have the same hit ratio p, the hit
ratio at any node in the network is also p.
The formal model
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The shortest path routing is used.
The network is represented by a undirected
graph G = (V, E):
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d(e)
the length of edge e
d(vi, vj) the sum of the link distances along the
route between nodes vi and vj.
The request pattern is modeled by the
demand set F:
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fs,c
Ps,c
the flow from server vs to client vc
the hit ratio of the flow
The formal model (Cont’d)
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K is the set of at most k nodes where the
caches are to be placed.
The cost cs,c of demand fs,c using a cache in
location vk is
Cs ,c  f s ,c   ps ,c  d vc , vk   1  ps ,c  d vc , vk   d vk , vs 
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This model does not capture hierarchical
structures.
The general k-cache location
problem
The k-TERC location problem
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The formal definition of the TERC k-cache
location problem is exactly as the general kcache location problem, except that the
minimization in the objective function is over
the set v  K  v  path v , v
k 
 s  

c
s

Theorem
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The solution of the problem with demands
F={fs,c} and flow hit ratios P={ps,c} is
equivalent to solving the problem for
F’={fs,cps,c} with hit ratio of one.
Proof:

s ,c
min f s ,c   ps ,c  d vc , vk   1  ps ,c  d vc , vk   d vk , vs 
vk K  s ,c 
  min
s ,c
vk K  s ,c 
f s ,c  d vc , vs   f s ,c  ps ,c  d vk , vs 
where K s, c   K  vs   pathvc , vs 
Theorem (Cont’d)
The solution for the problem with F’={fs,cps,c}
and a hit ratio of one is given by

s ,c
min
vk K  s ,c 
f s ,c ps ,c  d vc , vk 
  min f s ,c ps ,c  d vc , vs   d vk , vs 
s ,c
vk K  s ,c 
  min f s ,c ps ,c  d vc , vs   f s ,c ps ,c  d vk , vs 
s ,c
vk K  s ,c 
Assumption
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Base on the Theorem, we assume that
all flows have the same hit ratio which
we denote by p.
line
1 server
Poly
Bounded
Degree tree
Poly
m servers
Poly
NP
Tree
Poly
General
graph
NP
NP
NP
Single web server case
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Even the case when we have a single
server is NP-hard for general networks.
This case can be solved on a tree graph.
Fortunately, if the shortest path routing
algorithm implied by the Internet is
stable, the routes to various clients as
viewed by any single server should be a
tree graph.
Simple greedy algorithm
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The intuitive greedy algorithm places caches
on the tree iteratively in a greedy fashion.
It checks each node of the tree to determine
where to place the first cache, and chooses
the node that minimizes the costs.
It assigns the first cache to this node, and
looks for an appropriate location for the next
cache.
The complexity of the greedy algorithm is
O(nk).
Worst case
The optimal dynamicprogramming algorithm
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The general tree is converted into a binary
tree by introducing at most n dummy nodes.
Sort all the nodes in reverse breadth first
order, i.e, all descendants of a node are
number before the node itself.
For each
node
i having children iL and iR, for
~
~
each k ,0  k  k , where k is the maximum
number of caches to place.
For each l ,0  l  h ,where h is the height of
the tree, we compute the quantity C  i, k~ , l  .

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The optimal dynamicprogramming algorithm
 ~ 
 C  i, k , l 
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is the
cost of the subtree rooted
~
at i with k optimal located caches,
where the next cache up the tree is at
distance l from i.
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F  i, k , l  is the sum of the demands in the

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subtree rooted at i that do not pass
through a cache in the optimal solution
of C  i, k , l  .
~
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~

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The optimal dynamicprogramming algorithm
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If no cache is to be put at node i
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If we put a cache at node i
The optimal dynamicprogramming algorithm
The optimal dynamicprogramming algorithm
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While running the dynamic program we ~

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should also compute the appropriate F  i, k , l  ,
and keep track of the location of the caches
in these solutions.
The amount of data we have to keep is
O(nhk).
The overall time complexity is bounded by
O(nhk2).
Greedy versus optimal
Comparison of several
placement strategies