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Explore cache location problems, cache types, models, and efficient algorithms to minimize flow delay in networks.
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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 • 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 en-route caches (TERCs). • A computationally efficient dynamic programming algorithm is present for the single server case.
Introduction • 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 load-balancing. • Danzig et al. had observed the advantage of placing caches inside the backbone rather than at its edge.
Transparent En-route Caches • 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 • 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) • 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 • The shortest path routing is used. • The network is represented by a undirected graph G = (V, E): • 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: • fs,c the flow from server vs to client vc • Ps,c the hit ratio of the flow
The formal model (Cont’d) • 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 • This model does not capture hierarchical structures.
The k-TERC location problem • The formal definition of the TERC k-cache location problem is exactly as the general k-cache location problem, except that the minimization in the objective function is over the set
Theorem • 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:
Theorem (Cont’d) The solution for the problem with F’={fs,cps,c} and a hit ratio of one is given by
Assumption • Base on the Theorem, we assume that all flows have the same hit ratio which we denote by p.
Single web server case • 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 • 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).
The optimal dynamic-programming algorithm • 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 , where k is the maximum number of caches to place. • For each ,where h is the height of the tree, we compute the quantity .
The optimal dynamic-programming algorithm • is the cost of the subtree rooted at i with optimal located caches, where the next cache up the tree is at distance l from i. • is the sum of the demands in the subtree rooted at i that do not pass through a cache in the optimal solution of .
The optimal dynamic-programming algorithm • If no cache is to be put at node i • If we put a cache at node i
The optimal dynamic-programming algorithm • While running the dynamic program we should also compute the appropriate , 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).
