By Qifan Y.

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**Extra info for A 2. 79 competitive online algorithm for two processor real-time systems with uniform value density**

**Example text**

It follows that orthoplex has 2d vertices. Hypercube: √1d × (±1, ±1, · · · , ±1) give the coordinates of the vertices. It follows that the hypercube has 2d vertices. Spherical LSH for Approximate Nearest Neighbor Search on Hypersphere 33 Let us consider how to obtain the nearest vertex eﬃciently. (3), it is computationally easier to solve hA (p) = argmaxi (A˜ vi · p). (6) vi | i = 1, · · · , N } in advance, a d + 1 dot-product If we calculate {vi = A˜ calculation would suﬃce to return hA (p) for the simplex.

B supports emptiness and one-reporting queries in O(log n/ log log n) time. The data structure for the two-dimensional dynamic range counting problem is almost identical with the data structure of Theorem 1. The only diﬀerence is that we store in every node v a data structure Sv of Lemma 2 that supports range counting queries on a narrow grid in O(log n/ log log n) time. Theorem 2. There is a linear space data structure C for orthogonal range counting queries with O((log n/ log log n)2 ) query time and O(log9/2 n/(log log n)2 ) update time.

Eini , then we store ei1 , δi2 = ei2 − ei1 , . . δini = eini − eini −1 in Gi . Each diﬀerence δij is gamma coded[7], so that δij is stored with O(log δij ) bits. We choose the size of each group Gi in such a way that all encoded elements in Gi require at most 8 log m − 4 bits and at least 2 log m − 1 bits. It can be shown (s. [5]) that all groups Gi require O(m) bits; hence, there are O(m/ log m) groups. In [5] it is also shown that since Gi is stored in O(1) words, we can insert and delete elements into Gi and search in Gi in O(1) time using table look-up.

### A 2. 79 competitive online algorithm for two processor real-time systems with uniform value density by Qifan Y.

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