Shengyu Zhang

Distributed rate allocation for inelastic flows

A common assumption behind most of the recent research on network rate allocation is that traffic flows are elastic, which means that their utility functions are concave and continuous and that there is no hard limit on the rate allocated to each flow. These critical assumptions lead to the tractability of the analytic models for rate allocation based on network utility maximization, but also limit the applicability of the resulting rate allocation protocols. This paper focuses on inelastic flows and removes these restrictive and often invalid assumptions. First, we consider nonconcave utility functions, which turn utility maximization into difficult, nonconvex optimization problems. We present conditions under which the standard price-based distributed algorithm can still converge to the globally optimal rate allocation despite nonconcavity of utility functions. In particular, continuity of price-based rate allocation at all the optimal prices is a sufficient condition for global convergence of rate allocation by the standard algorithm, and continuity at at least one optimal price is a necessary condition. We then show how to provision link capacity to guarantee convergence of the standard distributed algorithm. Second, we model real-time flow utilities as discontinuous functions. We show how link capacity can be provisioned to allow admission of all real-time flows, then propose a price-based admission control heuristics when such link capacity provisioning is impossible, and finally develop an optimal distributed algorithm to allocate rates between elastic and real-time flows.

Any AND-OR Formula of Size

Consider the problem of evaluating an AND-OR formula on an

N

N

Can Be Evaluated in Time

-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time

N^{1/2+o(1)}

N^{1/2+o(1)}

on a Quantum Computer

. In particular, approximately balanced formulas can be evaluated in

ANY AND-OR FORMULA OF SIZE N CAN BE EVALUATED IN TIME N1/2+o(1) ON A QUANTUM COMPUTER

O(\sqrt{N})

On the power of Ambainis lower bounds

queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

Distributed rate allocation for inelastic flows: optimization frameworks, optimality conditions, and optimal algorithms

Consider the problem of evaluating an AND-OR formula on an N-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time N 1/2+o(1) . In particular, approximately balanced formulas can be evaluated in O(√N) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

The communication complexity of the Hamming distance problem

The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Albs. We first use known Albs to derive Ω(n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous Ω(n) one. We then show that all the three known Ambainis lower bounds have a limitation √N min{C0(f), C1(f)}, where C0(f) and C1(f) are the 0- and 1-certificate complexities, respectively. This implies that for many problems such as TRIANGLE, k-CLIQUE, BIPARTITENESS and BIPARTITE/GRAPH MATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all the Ambainis lower bounds are not tight. For total functions, this upper bound for Albs can be further improved to min{√C0(f)C1(f), √NċCI(f)}, where CI(f) is the size of max intersection of a 0- and a 1-certificate set. Again this implies that Albs cannot improve the best known lower bound for some specific problems such as AND-OR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Albs and give a new Alb style lower bound method, which may be easier to use for some problems.