The k-server problem 17.Januar 2008
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1 Sanders/van Stee: Approximations- und Online-Algorithmen 1 The k-server problem 17.Januar 2008 Problem definition Examples An offline algorithm A lower bound and the k-server conjecture Several online algorithms
2 Sanders/van Stee: Approximations- und Online-Algorithmen 2 Problem definition k > 1 servers M is a metric space with metric d Servers are located at points of M Request sequence σ consists of points of M A request is served by moving a server there Cost is total distance traveled by servers Goal: minimize the total cost
3 Sanders/van Stee: Approximations- und Online-Algorithmen 3 Examples (1) Paging Uniform space (all distances are 1) Servers are slots in the cache Fault (moving a server) costs 1 Weighted paging As above, but cost of moving a page into the cache depends on the page E.g. a distributed file system Asymmetric k-server problem This space is not metric!
4 Sanders/van Stee: Approximations- und Online-Algorithmen 4 Examples (2) k-headed disk A disk with multiple read/write heads Each head can access all locations on the disk Which head should be moved for a particular request? Possible performance measure: total distance moved by all heads
5 Sanders/van Stee: Approximations- und Online-Algorithmen 5 The offline problem Can be solved using dynamic programming This is not the most efficient solution Better: reduce to mincost / maxflow problem We will construct a graph with maximum flow k Minimum cost for this flow will correspond to k-server solution
6 Sanders/van Stee: Approximations- und Online-Algorithmen 6 Construction of the graph Servers are s 1,...,s k Request sequence is r 1,...,r n Nodes are s,t,s 1,...,s k,r 1,r 1,...,r n,r n All arcs have capacity 1 Costs depend on arcs We assume all servers start in the same point, the origin O
7 Sanders/van Stee: Approximations- und Online-Algorithmen 7 The graph for 3 servers and 2 requests s 1 r 1 r 2 s s 2 t s 3 r 1 r 2 0 K d
8 Sanders/van Stee: Approximations- und Online-Algorithmen 8 The graph for 3 servers and 2 requests s 1 d(o,r ) 1 r 1 d(o,r ) 2 r 2 s s 2 d(o,r ) 2 t d(r,r ) 1 2 d(o,r ) 1 s 3 r 1 r 2 0 K d
9 Sanders/van Stee: Approximations- und Online-Algorithmen 9 The graph for 3 servers and 2 requests s 1 r 1 r 2 s s 2 K K t s 3 r 1 r 2 0 K d
10 Sanders/van Stee: Approximations- und Online-Algorithmen 10 The maximum flow Since all capacities are one, maxflow = k (consider the servers) Since all capacities are integer, we can find an integral min-cost flow of value k in time O(kn 2 ) This flow basically consists of k disjoint paths All edges (r i,r i) will be used in a min-cost solution Each path corresponds to a server visiting the requests on its path This gives an optimal schedule for the servers
11 Sanders/van Stee: Approximations- und Online-Algorithmen 11 Lower bound We show a lower bound of k for an arbitrary online algorithm ALG We use an arbitrary space with k + 1 points We compare to k different other algorithms A 1,...,A k that the adversary controls An algorithm is determined by the uncovered point (hole) Invariant: holes of ALG and k other algorithms cover the space
12 Sanders/van Stee: Approximations- und Online-Algorithmen 12 Cruel request sequence Before first request, each A i moves one server to hole of ALG to ensure invariant holds At each step, we request the hole of ALG Denote request by r, then ALG moves to r from, say, s Other algorithms: all have a server at r, exactly one (say A i ) has no server at s Now, A i moves from s to r The other k 1 algorithms do nothing
13 Sanders/van Stee: Approximations- und Online-Algorithmen 13 Cruel request sequence The stupidity of LaTeX continues to annoy me. I wish I didn t have to type this nonsense in order to get slides the right way up. ALG r A 1 A 2 Ai s r r r s s s
14 Sanders/van Stee: Approximations- und Online-Algorithmen 14 Relative costs In each step j, ALG pays some cost c j Only one of the other algorithms A 1,...,A k pays c j Summing over all algorithms and the entire sequence, we get k i=1 A i (σ) = ALG(σ) + k i=1 d(x i,x 0 ) There must be one algorithm which has a cost of at most ALG(σ)/k (plus an additive constant) This proves the lower bound
15 Sanders/van Stee: Approximations- und Online-Algorithmen 15 The k-server conjecture Any metric space allows for a deterministic, k-competitive algorithm. The work function algorithm is (2k 1)-competitive in any metric space For certain metric spaces, k-competitive algorithms are known Fundamental open question in online algorithms
16 Sanders/van Stee: Approximations- und Online-Algorithmen 16 The k-server conjecture Any metric space allows for a deterministic, k-competitive algorithm. Note: other generalizations of paging results fail! There is no k/(k h + 1)-competitive k-server algorithm for the (h, k)-server problem Not every metric space allows a randomized H k -competitive algorithm
17 Sanders/van Stee: Approximations- und Online-Algorithmen 17 The greedy algorithm Definition: serve each request by the closest server
18 Sanders/van Stee: Approximations- und Online-Algorithmen 18 The greedy algorithm Definition: serve each request by the closest server This algorithm is not competitive a b c Request sequence: c,b,a,b,a,b,a,... Greedy leaves one server at c forever The other one moves between a and b OPT moves servers to a and b and has constant cost
19 Sanders/van Stee: Approximations- und Online-Algorithmen 19 k servers on the line Algorithm Double Cover Two cases: request is between two servers, or at one side 1 2 new request move closest server servers move 2 closest servers at equal speed If two servers are at same point, choose one to move
20 Sanders/van Stee: Approximations- und Online-Algorithmen 20 Double Cover on first example Request c a b c b a b a b a b c a b c a b c a b c Eventually, servers are at a and b and stop moving
21 Sanders/van Stee: Approximations- und Online-Algorithmen 21 Analysis of Double Cover We show that DC is k-competitive We use a potential function as in the List Update problem Let min be the cost of the minimum cost matching between the servers of DC and OPT Let s i be the ith server of DC Define sum = i<j d(s i,s j ) Potential function: Φ = k min + sum
22 Sanders/van Stee: Approximations- und Online-Algorithmen 22 The potential function Φ = k min + sum, so it is bounded from below We show: 1. If OPT moves a distance d, Φ increases by at most kd 2. If DC moves a distance d, Φ decreases by at least d Since Φ 0 at all times, this shows DC is k-competitive Property 1 holds since sum is unchanged by move of adversary min cannot increase by more than d
23 Sanders/van Stee: Approximations- und Online-Algorithmen 23 Change of Φ when DC moves DC moves only 1 server over a distance d: it moves away from all other servers sum increases by (k 1)d there exists a minimum cost matching where this server is matched to this request (one OPT server is there) Therefore, min decreases by at least d Overall decrease of Φ is at least k d (k 1)d = d 1 move closest server
24 Sanders/van Stee: Approximations- und Online-Algorithmen 24 Change of Φ when DC moves DC moves two servers, s 1 and s 2, by a distance d: one of them is matched to the request in some minimum cost matching min is decreased by at least d by this move other server moves at most d away from its match min does not increase overall 2 OPT servers d d
25 Sanders/van Stee: Approximations- und Online-Algorithmen 25 Change of Φ when DC moves DC moves two servers, s 1 and s 2, by a distance d: what is the change of sum? total distance from s 1 and s 2 to any other online server is unchanged distance between s 1 and s 2 decreases by 2d Overall decrease of sum (and therefore Φ) is at least 2d d d 2 OPT servers
26 Sanders/van Stee: Approximations- und Online-Algorithmen 26 k servers on trees Algorithm Double Cover can be extended for trees It still has a competitive ratio of k Definition of DC-TREE: At all times, all the servers neighboring the request are moving in a constant speed towards the request On a line, DC-TREE is identical to DC new request servers 1 move closest server 2 move 2 closest servers at equal speed
27 Sanders/van Stee: Approximations- und Online-Algorithmen 27 DC-TREE DC-TREE may move all k servers simultaneously and this text is here so that it fits with the next slide r
28 Sanders/van Stee: Approximations- und Online-Algorithmen 28 DC-TREE While moving towards a request, some servers may get cut off and stop moving r
29 Sanders/van Stee: Approximations- und Online-Algorithmen 29 Upper bound for DC-TREE We use the same potential function Φ = k min + sum A move by OPT still increases Φ by at most kd We break the action of DC-TREE to serve a single request into phases In each phase, the subset of servers that moves is fixed Need to show: Φ decreases at least by total distance traveled by DC-TREE We consider separately the change of min and sum in a phase
30 Sanders/van Stee: Approximations- und Online-Algorithmen 30 The change of min Denote the number of neighbours in a phase by m One of these is matched to the request in a minimum cost matching Moving that server by d decreases min by d Moving the m 1 other servers by d increases min by at most (m 1)d min increases by at most (m 2)d
31 Sanders/van Stee: Approximations- und Online-Algorithmen 31 The change of sum: non-moving servers Consider a server s which is not moving (no neighbour of the request) r Exactly one server is moving away from s, m 1 others are moving towards s Change in sum caused by this server is (m 2)d s
32 Sanders/van Stee: Approximations- und Online-Algorithmen 32 The change of sum: non-moving servers We need to sum over the k m non-moving servers sum decreases by (k m)(m 2)d
33 Sanders/van Stee: Approximations- und Online-Algorithmen 33 The change of sum: moving servers Each pair of moving servers gets closer together by 2d Summing over m(m 1)/2 pairs, this gives a decrease in sum of dm(m 1)
34 Sanders/van Stee: Approximations- und Online-Algorithmen 34 The change of Φ min increases by at most (m 2)d Due to non-moving servers, sum decreases by (k m)(m 2)d Due to moving servers, sum decreases by dm(m 1) In total, Φ = k min + sum decreases by at least (k m)(m 2)d + dm(m 1) k(m 2)d = dm This is exactly the total distance that DC-TREE moves
35 Sanders/van Stee: Approximations- und Online-Algorithmen 35 Application: arbitrary graph G take a spanning tree T, apply DC-TREE on it Let n be the number of nodes of G An edge of length d in G has a detour on T of length at most (n 1)d Thus OPT-TREE(σ) (n 1)OPT(σ) Since DC-TREE is k-competitive on trees, we have DC-TREE(σ) k OPT-TREE(σ) We have a (n 1)k-competitive algorithm
36 Sanders/van Stee: Approximations- und Online-Algorithmen 36 Application: paging Suppose there are N slow memory pages Create a star graph with N edges of length 1/2 The central node is labeled v The other nodes are the page nodes
37 Sanders/van Stee: Approximations- und Online-Algorithmen 37 DC-TREE for paging (1) Servers start on k page nodes On first request, all servers move to v One server continues to requested page On subsequent requests, other servers move away from v Once all servers have left, next request causes all servers to return to v One server continues to request, etc.
38 Sanders/van Stee: Approximations- und Online-Algorithmen 38 DC-TREE for paging (2) This algorithm is equivalent to FLUSH-WHEN-FULL Moving to v is equivalent to clearing the cache This gives an alternative proof that FWF is k-competitive
39 Sanders/van Stee: Approximations- und Online-Algorithmen 39 Euclidean spaces DC is k-competitive for the line This is the one-dimensional Euclidean space Can we extend this to the higher dimensions? Even for the plane, no efficient algorithm with good competitive ratio is known Efficient = computational cost per request does not depend on length of input sequence
40 Sanders/van Stee: Approximations- und Online-Algorithmen 40 The Work Function Algorithm Tries to mimic OPT Keeps track of optimal offline cost so far Tries to have a configuration similar to OPT Is (2k 1)-competitive for any metric space
41 Sanders/van Stee: Approximations- und Online-Algorithmen 41 Work functions Configuration = set of locations of servers This is a multiset (two servers may be at same location) For a configuration C, the work function w(c) is the minimum cost to reach C (from the starting configuration) Suppose sequence so far is σ, new request is r How do we compute w σr (C), given w σ (C)?
42 Sanders/van Stee: Approximations- und Online-Algorithmen 42 Calculation of work function If r C, then w σr (C) = w σ (C) Otherwise, we need to move one server from some other configuration B The difference between B and C is one point (server) We need to minimize the cost to get to B while serving σ i, and then move to r Thus, w σr (C) = min x C (w σ(c x + r) + d(x,r))
43 Sanders/van Stee: Approximations- und Online-Algorithmen 43 Definition of WFA Let C be the current configuration Let r be the new request We serve r with server s C which satisfies Notes: s = arg min(w(c x + r) + d(x,r)) x C Minimizing only d(x,r) is what the greedy algorithm does Minimizing w(c x + r) mimics OPT so far (retrospective greedy)
44 Sanders/van Stee: Approximations- und Online-Algorithmen 44 Idea behind WFA From C, we can move to k different configurations to serve r (we can move any of k servers) We move to the best one that is not too far away In effect, the algorithm is trying to find the optimal servers, without paying too much We do not use any properties of the metric space
45 Sanders/van Stee: Approximations- und Online-Algorithmen 45 Performance of WFA WFA is (2k 1)-competitive The proof uses a (complicated) potential function For some special metric spaces, WFA is known to be k-competitive E.g., the line, any metric space with at most k + 2 points The popular conjecture is that WFA is k-competitive in any metric space
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