O3-mini比DS R1强大

[hci]

还是用的我的老题目：

Suppose I use Rendezvous Hashing to distribute objects to different shards. At beginning I have 2 shards, once the number of objects in the shards goes over a preset threshold, I want to add a new shard. I can set this new shard's weight higher so it is more likely to get new objects. What would be a mathematically sound schema to assign weights to new shard so that the distribution of objects are still even, assuming this goes on forever?

O3-mini用24秒，产生了如下答案。我觉得比O1和R1的答案都要好。

Below is one mathematically founded idea to “bootstrap” a new shard into a rendezvous‐hashing scheme and eventually let all shards share the load evenly.

Recall that in rendezvous hashing the chance that an object goes to shard i is (roughly) proportional to its weight wᵢ relative to the total, that is,

  pᵢ = wᵢ⁄(w₁ + w₂ + … + wₙ).

Thus, if you want an even—asymptotic—distribution you want every shard’s weight to eventually be the same (say, 1). But if you add a new shard that starts “empty” (or underloaded) compared to all the “old” shards, assigning it the same weight right away would mean it only “competes” equally even though it has little data. Intuitively, you would like new objects to go preferentially to the new shard until its load gets roughly in line with the rest. One mathematically sound way to do this is to let weights be a function of the gap between the shard’s current load and the target (or average) load.

One concrete schema is as follows. Let

  L_avg = (total number of objects)⁄(number of shards)

be the target load per shard, and let Lᵢ be the current load of shard i.

Then define a dynamic weight function for shard i by

  wᵢ = 1 + f((L_avg – Lᵢ)/L_avg).

A natural choice for f is a (nonnegative) function that is large when (L_avg – Lᵢ) is large and decays to 0 as Lᵢ approaches L_avg. For example, you could pick

  wᵢ = 1 + (W₀ – 1) · exp(–λ · (Lᵢ/L_avg))      (1)

where W₀ > 1 is the initial bonus weight assigned to a “new” shard (when Lᵢ is near 0) and λ > 0 determines how fast the bonus decays as the shard’s load Lᵢ increases. (Any monotonic decreasing function that goes from W₀ to 1 would serve equally well.)

The idea is:
• When you add a shard j, its current load Lⱼ is low (say 0), so it gets weight
  wⱼ = 1 + (W₀ – 1) = W₀.
• In contrast, the “old” shards—if their loads are near L_avg—have weights close to 1.
• Thus, a new object is assigned with probability approximately
  pⱼ = W₀⁄((n_old × 1) + W₀).
For example, if you have 2 old shards and you set W₀ = 3 then p_new ≈ 3⁄(2+3)=60%—a significant share to let it “catch up.”
• As the new shard accumulates objects, its Lⱼ rises, so the exponential decay (or other function f) reduces its weight toward 1. In the long‐term, every shard will have wᵢ ≈ 1 and receive about 1/n of new objects.

This approach is mathematically sound because it makes the selection probability for each object depend on

  pᵢ = wᵢ⁄(∑ⱼ wⱼ),

so that initially underloaded (new) shards get a bonus (wᵢ > 1) and efficiently “attract” new objects until equilibrium is reached. (One may even analyze or simulate the dynamics to show that under reasonable conditions Lᵢ converges to L_avg for every shard.)

Some notes:
– The precise choice of W₀ and decay parameter λ (or an alternative decay function) may be tuned to your particular load thresholds, arrival rates, and how “bursty” your object insertions are.
– It’s also possible to “snap‐down” the bonus once the load reaches a certain fraction below L_avg. For instance, you might set
  wᵢ = { W₀, if Lᵢ ≤ L_avg – Δ; 1, if Lᵢ ≥ L_avg – Δ },
which is a simpler scheme but introduces a discontinuity that may or may not be acceptable.

In summary, a mathematically sound schema is to define weights as a dynamic function of how “underloaded” a shard is relative to the target average. A concrete example (Equation (1) above) gives you a way to smooth from an initial high bonus weight (to help new shards catch up) down to the equilibrium weight (1) that ensures equal division of load in the long run.