Competitive Paging Algorithms Amos Fiat Richard Karp Michael

Competitive Paging Algorithms Amos Fiat, Richard Karp, Michael Luby, Lyle Mc. Geoch, Daniel Sleator, Neal Young presented by Seth Phillips

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

Introduction – Paging Problem System with k pages of fast memory (cache/RAM) n Has n-k pages of slow memory (RAM/virtual) n Requesting a page not in fast memory n Eject a page to make room (page fault) n

Online paging algorithm Decision made without knowledge of future requests n Performance analyzed against off-line algorithm – complete knowledge n Deterministic algorithms are k-competitive n

Randomized Algorithms Analyzed as a sum cost of the algorithm n Cost on a sequence of input averaged over all the random choices that the algorithm makes while processing the sequence n

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

K-server Problem Let G be an n-vertex graph with positive edge lengths obeying the triangle inequality n Let k mobile servers occupy vertices of G n Given a sequence of requests, each of which specifies a vertex, decide how to move the servers in response to each request n

K-server continued Requests must be satisfied in order of their occurrence in the sequence n Cost of handling a sequence is equal to the total distance moved by the servers n It is conjectured that there exists a kcompetitive k-server algorithm for any graph n

Uniform k-server problem Cost of moving a server from any vertex to any other vertex is 1 n Isomorphic to the paging problem: any vertex with a server is in fast memory and the vertices of the graph represent the address space n

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

Marking Algorithm Randomized algorithm for uniform k-server problem on graph with n vertices n Servers are initially on vertices 1 through k n Algorithm maintains a set of marked vertices n Initially these are the vertices covered by the servers n

Marking Each time a vertex is requested, it is marked n If k+1 vertices are marked, all marks except the most recently requested vertex are erased n

Serving If vertex is covered, then no servers move n If vertex is not covered, then a server is chosen uniformly at random from the unmarked vertices n That server is moved to cover n

Competitiveness 2*H_k competitive (H_k on the order of ln(k)) n Algorithm implicitly divides request sequence into phases. n First phase begins with r(i), where I is the smallest integer such that r(i) is not in the set {1 through k}. n

Comp. continued In general the phase starting with r(i) ends with r(j) n j is the smallest integer such that the set {r(i), r(i+1), . . . , r(j+1)} has cardinality k+1 n

Comp. continued Implicitly the first request of every phase is made to an unmarked vertex n A vertex is clean if it was not requested in the previous phase and has not yet been requested in this phase – we’ll say there are l of these n A vertex is stale if it was requested in the previous phase and has not yet been requested in this phase n

Adversary Amortized Cost At least l/2 because: n Let d be the number of servers that do not coincide with marking’s at the beginning of the phase n Let d’ be this quantity at the end of the phase n Cost(A) >= l – d because l requests have to be met, mitigated by a possible d different server locations n

Adversary Cost Continued C(Adv) >= d’ because: n The vertices of S are those covered by Marking at the end of the phase, so d’ servers are not in S n Since Adversary is lazy (does not unnecessarily move servers) at least d’ of A’s servers were outside of S for the entire phase n

Adversary Cost Continued C(Adv) >= max(l – d, d’) >= ½(l – d + d’) n D and d’ naturally telescope so simply C(Adv) >= l/2 n

Marking Expected Cost l requests to clean vertices – each costs 1 n k – l requests to stale vertices – each based on the probability that there is no server there n Highest cost is when the l requests to clean vertices come first (allows the most stale vertices to be reassigned) n That leaves the cost of the k-l stales to be: n l/k + l/(k-1) + l/(k-2) +. . . + l/(l+1) = l*(H_k – H_l) n

Final Cost Competitiveness l*H_k – l*H_l + l <= l*H_k n Since the cost of the adversary is l/2: n The Marking algorithm is 2*H_k competitive n For the n-1 server problem it is H_n-1 competitive, but for time constraints I’ll skip this n

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

EATR Stands for End After Twice Requested n This is an algorithm specific to the uniform 2 server problem n Servers initially on vertices 1 + 2 n Stale redefined to not clean and not the most recently requested vertex n When a stale vertex is requested, the servers are placed on the two most recently requested vertices n

EATR Analysis Let l be the number of clean vertices requested during a phase n The number of stale vertices before the request that terminates the phase is l+1 n The probably of a server on each of these is 1/(l+1) n The expected cost of each phase is l + l/(l+1) n

Adversary Analysis The best possible cost incurred by any algorith for each phase is at least l n The competitive factor is therefore: (l + l/(l+1))/l = 1 + 1/(l+1) <= 3/2 n

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

Lower Bound There is no c-competitive randomized algorithm for the uniform (n-1) server problem on n vertices with c < H_n-1 n (Time constraints) The article proves this, which also makes the marking algorithm in a class of best possible algorithms for this problem n

Overview Introduction n Server Problems n Marking Algorithm n EATR n Lower Bound n Competitive Against Other Algorithms n

Algorithms Competitive Against Several Others Adopt a viewpoint where each algorithm is tailored for a specific choice of k and n. n The ordered pair (k, n) is called the type of algorithm n Let A be adeterministic and B be a deterministic on-line algorithms of the same type and c be a positive constant n If for every sequence r of requests C_A(r) <= c*C_B(r) + alpha then A is c-competitive with B n

ACASO continued Let c* = (c(1), c(2), . . . C(m)) be a sequence of postive real numbers n c* is realizable if, for every (k, n) and every sequence B(1), B(2). . . B(m) there exists an algorithm A of type (k, n) such then A is c(i) competitive with B(i) n c* realizable iff: Sum from 1 to m of (1/c(i)) <= 1 n