Fast and Cost-Effective Online Load-Balancing in - Computing ...

KONSTANTINOU ET AL.: FAST AND COST-EFFECTIVE ONLINE LOAD-BALANCING IN DISTRIBUTED RANGE-QUERIABLE SYSTEMS 1353
Fig. 4. Worst case of initial setup and converged balanced network for the NIX and MIG cases.
for: item exchanges between nodes (C itx ), overlay maintenance
during migrations (C ovm ), and locating underloaded basically the cost for a node insertion or deletion (see
item exchanges then is C itx ¼ ðN 1ÞM N
N 1
¼ Oð M NÞ which is
nodes, where in each migration M N
keys are offloaded from a number of NIX operations. Node A is located between
N
P 1 to the helper. Finally (fourth picture of Fig. 4), a total of nodes that their load is near their thres value (left side of
N 1
i¼1 M N ¼ðN 1Þ M N keys are transferred. The cost for Fig. 6). In this situation, a chain of NIX operations would
peers during probing (C prb ).
In Theorems 1 and 2, we use the aggregate method of
amortized analysis to calculate the average cost of each
balancing operation and completion time of NIX and MIG
Theorem 3 of Karger’s work [1]). The probing cost C prb is
OðlogNÞ since it involves contacting logN neighbors.
Moreover, in most DHT-like networks, overlay maintenance
costs C ovm ¼ OðlogNÞ messages. Therefore, the
in the worst case of an initial setup (i.e., worst upper bound total MIG cost is C itx þ C ovm þ C prb ¼ Oð M N
of amortized cost). We utilize the notations of Section 2.
Migrations take a constant number of steps as, unlike
Theorem 1. In the worst case, the running time of NIX is OðNÞ
and the amortized cost per balancing operation is OðMÞ.
NIX operations, they are executed in parallel: therefore,
we consider MIG running time to be Oð1Þ (although
Proof. In the first picture of Fig. 4, we present an initial setup
overlay maintenance usually takes OðlogNÞ time, this can
happen lazily after the key transfer phase).
tu
of node and item combination that leads NIX to its worst
behavior in terms of completion time and item exchanges.
To gain insight into the behavior of the two algorithms
Buckets represent nodes and balls depict items. Bars
in a more general case, let us consider a typical “balls into
above items represent the unit item load l j ¼ 1. For
bins” setup, with N items being uniformly distributed
simplicity, we consider that for each node, thres i ¼ 1. At
among N nodes (we only consider N out of M items, since
the beginning, N 1 contains all M objects, of which only the
these items affect node loads). The fraction of underloaded
leftmost N are requested (i.e., they have l j ¼ 1;j2ð0;NŠ)
and the rest M-N are not queried (i.e., l j ¼ 0;
nodes, i.e., nodes with a load less or equal to 1, is
j 2ðM N;MŠ). All other nodes are empty. Since
calculated by estimating the probability of a node to hold
L 1 ¼ N > thres 1 , N 1 will perform a NIX operation with
either one or no popular item. Utilizing the equation that
its neighbor N 2 at t 0 and it will transfer to it a total of M 1 calculates the probability of a particular bin to have exactly
keys, keeping only the leftmost key, so that L 1 ¼ l 1 ¼ k balls we have: P½N j is underloadedŠ ¼ P 1
k¼0 P ½L j ¼ kŠ ¼
P 1
1