Hierarchical Well-Separated Trees (HST)

1
Hierarchical Well-Separated Trees (HST)

• Edges distances are uniform across a level of
  the tree
• Stretch s factor by which distances decrease
  from root to leaf
• Distortion factor by which distance between 2
  points increases when HST is used to traverse
  instead of direct distance
• Upper bound is O(s logs n)

Diagram from Fakcharoenphol, Rao Talwar 2003
2
Pure Randomized vs. Fractional Algorithms

• Fractional view keep track only of marginal
  distributions of some quantities
• Lossy compared to pure randomized
• Which marginals to track?
• Claim for some algorithms, fractional view can
  be converted back to randomized algorithm with
  little loss

3
Fractional View of K-server Problem

• For node j, let T(j) leaves of the subtree of T
  rooted in j
• At time step t, for leaf i, pit probability of
  having a server at i
• If there is a request at i on time t, pit should
  be 1
• Expected number of servers across T(j) kt(j)
  Si?T(j) pit
• Movement cost to get servers at j Sj?T W(j)
  kt(j) kt-1(j)

j
T(j)
i
Parts of diagram from Bansal 2011
4
The Allocation Problem

• Decide how to (re-)distribute k servers among d
  locations (each location of uniform distance from
  a center and may request arbitrary no. of
  servers)
• Each location i has a request denoted as ht(0),
  ht(1), ht(k)
• ht(j) cost of serving request using j servers
• Ex. request at i0 is 8, 2, 1, 0, 0
  (monotonic decrease)
• Total cost hit cost movement cost

I can work with 1, but Id like 3!
Parts of diagram from Bansal 2011
5
Fractional View of Allocation Problem

• Let xi,jt (non-negative) probability of having
  j servers at location i at time t
• Sum of probabilities Sj xi,jt 1
• No. of servers used must not exceed no. available
• Si Sj j xi,jt k
• Hit cost incurred Sj ht(j) xi,jt
• Movement cost incurred Si Sj (Sjltj xi,jt
  Sjltj xi,jt-1)
• Note fractional A.P. too weak to obtain
  randomized A.P. algorithm
• But we dont really care about A.P., we care
  about K-server problem!

6
From Allocation to K-Server

• Theorem 2 It suffices to have a (1?,
  ?(?))-competitive fractional AP algorithm on
  uniform metric to get a k-server algorithm that
  is O(ßl)-competitive algorithm (Coté et al. 2008)
• Theorem 1 Bansal et al.s k-server algorithm has
  a competitive ratio of Õ(log2 k log3 n)

7
The Main Algorithm

• Embed the n points into a distribution m over
  s-HSTs with stretch s Q(log n log(k log n))
• (No time to discuss, this step is essentially
  from the paper of Fakcharoenphol, Rao Talwar
  2003)
• According to distribution m, pick a random HST T
• Extra step Transform the HST to a weighted HST
  (Well briefly touch on this)

Diagram from Bansal 2011
8
The Main Algorithm

• Solve the (fractional) allocation problem on Ts
  root node immediate children, then recursively
  solve the same problem on each child
• Intuitive application of Theorem 2
• d immediate children of a given node
• At root node k all k servers
• At internal node i k resulting (re)allocation
  of servers from is parent

Allocation instances
Diagram from Bansal 2011
9
Detour Weighted HST

• Degenerate case of normal HST
• Depth l O(n) (can happen if n points are on a
  line with geometrically increasing distances)

10
Detour Weighted HST

• Solution allow lengths of edges to be
  non-uniform
• Allow distortion from leaf-to-leaf to be at most
  2s/(s1)
• Depth l O(log n)
• Consequence Uniform A.P. becomes weighted-star
  A.P.

11
Proving the Main Algorithm

• Theorem 1 Bansal et al.s k-server algorithm has
  a competitive ratio of Õ(log2 k log3 n)
• Idea of proof How does competitive ratio and
  distortion evolve as we transform
• Fractional allocation algorithm
• ?
• Fractional k-server algorithm on HST
• ?
• Randomized k-server algorithm on HST

12
Supplemental Theorems

• Theorem 3 For e gt 0, there exists a fractional
  A.P. algorithm on a weighted-star metric that is
  (1e, O(log(k/e)))-competitive (Refinement of
  theorem 2, to be discussed by Tanvirul)
• Theorem 4 If T is a weighted s-HST with depth l,
  if Theorem 3 holds, then there is a fractional
  k-server algorithm that is O(l
  log(kl))-competitive as long as s W(l log(kl))
• Theorem 5 If T is a s-HST with sgt5, then any
  fractional k-server algorithm on T converts to a
  randomized k-server algorithm on T that is about
  as competitive (only O(1) loss)
• Theorem 6 If T is a s-HST with n leaves and any
  depth, it can transform to a weighted s-HST with
  identical leaves but with depth O(log n) and
  leaf-to-leaf distance distorted only by at most
  2s/(s1)

13
Proof of Theorem 1

• Embed the n points into a distribution m over
  s-HSTs with stretch s Q(log n log(k log n))
• Distortion at O(s logs n)
• Resulting HSTs may have depth l up to O(n)
• According to distribution m, pick a random HST T
  and transform to a weighted HST
• From Theorem 6, depth l reduced to O(log n)
• Stretch s is now Q(l log (kl)))

14
Proof of Theorem 1

• Solve the (fractional) allocation problem on Ts
  root node immediate children, then recursively
  solve the allocation problem on children
• This is explicitly Theorem 2 refined by Theorem 3
  ?
• Stretch s Q(l log (kl))), so Theorem 4 is
  applicable!
• Transform to a fractional k-server algorithm with
  competitiveness
• O(l log (kl))) O(log n log (k log n))
• Applying Theorem 5, we get similar
  competitiveness for the randomized k-server
  algorithm

15
Proof of Theorem 1

• Expected distortion to optimal solution OptM,
  given the cost of the solution on T, cT
• EmcT O(s logs n) OptM
• AlgM AlgT
• O(log n log (k log n)) cT
• EmAlgM O(log n log (k log n)) EmcT
• O(log n log (k log n)) O(s
  logs n) OptM

16
Proof of Theorem 1

• EmAlgM O(log n log (k log n)) O(s logs n)
  OptM
• This implies a competitive ratio of
• O(log n log (k log n)) O(s logs n)
• O(log n log (k log n)) O(s (log n / log s))
• Olog3 n (log (k log n))2 / log log n
• O(log2 k log3 n log log n)
• Õ(log2 k log3 n)