1. Minimizing Stall Time in Single Disk Susanne Albers, Naveen Garg, Stefano Leonardi, Carsten Witt Presented by Ruibin Xu

2. Introduction Prefetching and caching are powerful techniques for increasing performance in disk systems Prefetching: load memory blocks into the cache before the actual references (needs to evict blocks simultaneously) Caching: maintain the most frequently accessed blocks in cache

3. Introduction Both techniques have been studied extensively, but separately Now look at them in an integrated manner Focus on the offline problem

4. The problem definition Assume all blocks reside on one disk The cache size is k Serving a request takes one time unit Fetching a block takes F time units Given a request sequence σ = r 1, …, r n, how to schedule the prefetching to minimize the total stall time

5. An example k = 4 F = 5 Blocks a, b, c and d are initially in the cache The minimum stall time is 3

6. Big question Cao et. al. designed a 2-approximation algorithm. Can this problem be solved exactly in polynomial time? Yes, this paper answers this quesiton

7. The idea Use linear programming At first thought, needs to prove that the optimum solution is integral by arguing that all vertices of the corresponding polytope are integral By showing that the constraint matrix is total unimodular (ex. Bipartite matching) By combinatorial argument(ex. Matching and matroid polytopes)

8. Main novelty At second thought, the polytope corresponding to the LP to this problem has nonintegral vertices Now if we can show that any solution to the LP can be written as a convex combination of (polynomially many) integral solutions, ……

9. The roadmap 1. Construct the LP 2. Solve the LP 3. Find the convex decomposition to integral solutions

10. The LP formulation This is a 0-1 LP The length of the request sequence is n The cache size is k The fetching time is F The cache initially contains k blocks never requested in the sequence

11. The variables of the LP Consider all the intervals of the request sequence of length at most F : interval I = (i, j) of length |I|=j – i – 1, i = 0, …, n-1, j = 1, …, n, i < j

12. The variables of the LP Associate each interval I with an indicator variable X(I) where X(I) =1 indicates a prefetch starting after request i and ending before request j and X(I) =0 indicates no prefetch is performed in this interval With each interval I and distinct block a, associate variable f I,a ( e I,a ), which is 1 if block a is fetched (evicted) in interval I and 0, otherwise

13. The objective func. of the LP The prefetch occuring in interval I has a stall time F - |I| Thus the objective function is

14. The constraints of the LP There are 7 kinds of constraints A definition: an interval (a, b) is contained in an interval (c, d) if c ≤a and d ≥b, denoted by (a, b) (c, d)

15. The 1 st constraint To ensure that two prefetches are not performed simultaneously

16. The 2 nd constraint For any interval, the total amount of fetch should be exactly equal to the total amount of eviction and this value should not exceed the value of the interval

17. The 3 rd constraint A block should be in cache when it is referenced After each reference to a block, the block is in cache. It can then be evicted at most once up until the next reference to that block, and if it is, it must be also be fetched back prior to that next reference

18. The 4 th and 5 th constraint To ensure that every block is in cache at its first reference, the total fetch of a block on intervals before its first reference should be 1 and the total evict of the block on these intervals should be 0

19. The 6 th constraint A block is not evicted for more than 1 unit after its last reference

20. The last constraint On each request, the requested block is neither prefetched nor evicted And

21. Solving the LP relaxation First solve the LP relaxation. If we get an integral solution, we are done. If not, find the convex combination

22. Modify the intervals The goal: to obtain a total order of intervals An interval I 1 = (i 1, j 1 ) is properly contained in interval I 2 = (i 2, j 2 ) iff i 1 > i 2 and j 1 < j 2 We don’t want any interval is properly contained in any interval

23. Modify the intervals For each pair of nested intervals, remove one of them and add two new intervals

24. Order the intervals Now we can order the intervals by increasing starting points; If two intervals have the same start point, then they are ordered by increasing end-points

25. Properties of the optimum sol. Let C denote the cache configuration after we have performed the fetches and evicts corresponding to the first i intervals; let I be the (i+1)-st interval There exists an optimum solution for which the next two claims are satisfied

26. Properties of the optimum sol. Claim 1: In interval I, we fetch the block that is not completely in C and whose next reference is earliest Claim 2: In interval I, we evict the block which is partially or completely in C whose next reference is furthest Both claims can be proven by contradiction

27. Properties of the optimum sol. The amount of fetch of a block prescribed by claim 1 might be less than x(I). In this case, we apply the same rule to fetch another block in I The same holds for the case of evictions

28. Another view of the process of fetching/evicting Define the distance of interval I View the process of fetching/evicting as a process in time by associating the time interval [dist(I), dist(I)+x(I)) with interval I

29. Another view of the process of fetching/evicting There is a unique interval associated with each time instant Also associate a unique fetch/evict with each time instant

30. Properties of the optimum sol. From claim 1&2 and the ordering of fetches/evicts within an interval, it follows that a block a is fetched continuously till it is fully in cache But the eviction of a could be interrupted before it is completely out of cache

31. Properties of the optimum sol. Consider the fetches/evictions of a block a between two consecutive references to a Lemma 1. Every interruption in the eviction of a is for some integral time units

32. Properties of the optimum sol. A block a is partially fetched/evicted if the total extent to which a is fetched/evicted between two consecutive references is strictly less than 1 Lemma 2. If a is partially fetched/evicted, then the fetch of a begins some integral time units after the start of its eviction

33. Properties of the optimum sol. Lemma 3. If a is evicted at time t and referenced again, then there is a time t’ = t + i, for some integer i, at which a is fetched back

34. The convex decomposition Let t be in the range [0, 1) and let t i = i + t for every integer i, 0 ≤ i ≤ x(I) Claim 3. Let t 1, t 2 be two time instants such that t 2 = t 1 + i for some positive integer i, and let I 1, I 2 be the intervals associated with these time instants. Then I 1 and I 2 are disjoint.

35. The convex decomposition Lemma 4. For any time t in [0,1), the set of intervals that correspond to t i forms a feasible solution Note that each solution is obtained not for just one value of t but for a range of values, say for all t in the range [a, b]. We associate a weight b – a in the decomposition.

36. Conclusion An optimum prefetching/caching schedule for a single disk can be computed in polynomial time

37. Open problem Now the problem can be solved exactly in polynomial time by using LP, Does there exist a combinatorial, polynomial time algorithm? Yes, by using multicommodity network flows

38. The roadmap 1. Construct the LP 2. Solve the LP 3.Find the convex decomposition to integral solutions 1. Construct the multicommodity network 2. Solve the network

39. Problem No combinatorial polynomial-time algorithm for computing non-integral min-cost flow is known But we know an approximation algorithm: for any ε ≥ 0, δ ≥ 0, the algorithm computes a flow such that a fraction of at least 1 - ε of each demand in the network is satisfied and the cost of the flow is at most (1 + δ ) times the optimum

40. The network Given a request sequence of length n, construct a network with n+1 commodities Associate each request σ(i) with a commodity i, which has a source s i, a sink t i and a demand d i = 1 For each request σ(i), introduce two vertices x i and x’ i

41. An example network Sketch of the network for request sequence abcbc and F=2

42. The problem of previous network The construction allows a flow algorithm to saturate more than one of the edges that correspond to fetches executed simultaneously Needs to make sure at most one fetch operation is executed at any time

43. Solution Split the “super edge” (s i, x j ) into several parts and add one more commodity For any l, 1≤ l ≤ n-1, let [l, l+1) be the time interval starting at the service of σ(l) and ending immediately before the service of σ(l+1)

44. Solution For any fixed i and j, with 1 ≤ i ≤ n, and p i +1 ≤ j < i, introduce vertices v ij l and w ij l where l = j, …, min{j+F, i} -1 For any fixed i, with 1 ≤ i ≤ n, introduce vertices v ii i-1 and w ii i-1 How to connect? How to assign cost and capacity?

45. Solution Now add the (n+1)-st commodity Let f l be the number of prefetches whose execution overlaps with [l, l+1) Commodity n+1 has a source s n+1, a sink t n+1 and a demand d n+1

46. Solution The flow from s n+1 to t n+1 is routed through the edges (v ij l, w ij l ) and newly introduced “subsinks” t n+1 l, 1 ≤ l ≤ n-1 How to connect? How to assign cost and weight?

47. Optimal flows Any feasible integral flow of cost C in the network correspond to a feasible prefetching/caching schedule with stall time C for σ, and vice versa A non-integral flow correspond to a fractional prefetching/caching schedule

48. Apply the approximation algo. Unfortunately, the flow computed by the algorithm does not correspond to a feasible fractional prefetching/caching schedule It is possible that(1) more than one block is fetched at any time and (2)blocks are not completely in cache when requested

49. Apply the approximation algo. The solution is to choose ε and δ properly and modify the flow Choose ε=1/(4F 2 n 3 ) and δ=1/(3nF) Let Φ be the flow returned by the approximation algorithm

50. Apply the approximation algo. The flow out of each source s i, i={1,…n}, is lower bounded by 1-ε. Moreover, commodity n+1 might lack an amount of εd n+1 ≤ εFn 2 Let ρ= 1-ε- εd n+1, transform the flow Φ into a uniform flow Φ’ which directs exactly ρ units of flow from s i to t i

51. Apply the approximation algo. The flow Φ’ corresponds to a fractional solution in which all blocks have size ρ and the number of cache slots is upper bounded by k/ ρ We can interpret the fractional solution to Φ’ as a convex combination of integral ρ-solution

52. Apply the approximation algo. Let the cost of convex combination of ρ-solutions be C, we can prove that C≤OPT+1/3 By increasing the block size from ρ to 1, we obtain the integral solutions. Let the cost of convex combination of integral solutions be C’, we can prove that C’<OPT+1

53. Apply the approximation algo. Also, it can be proven that no integral component of the convex composition does hold more than k blocks in cache concurrently Therefore, the convex combination contains at least one integral solution with optimal costs.

54. Conclusion An optimal solution can be computed by a combinatorial algorithm in polynomial time The running time is O*(n 18 )