An Algorithmic View on OVSF Code Assignment

Abstract

Orthogonal Variable Spreading Factor (OVSF) codes are used in UMTS to share the radio spectrum among several connections of possibly different bandwidth requirements. The combinatorial core of the OVSF code assignment problem is to assign some nodes of a complete binary tree of height h (the code tree) to n simultaneous connections, such that no two assigned nodes (codes) are on the same root-to-leaf path. A connection that uses a 2 −d fraction of the total bandwidth requires some code at depth d in the tree, but this code assignment is allowed to change over time. Requests for connections that would exceed the total available bandwidth are rejected. We consider the one-step code assignment problem: Given an assignment, move the minimum number of codes to serve a new request. Minn and Siu propose the so-called DCA algorithm to solve the problem optimally. In contrast, we show that DCA does not always return an optimal solution, and that the problem is NP-hard. We give an exact n O(h) -time algorithm, and a polynomial-time greedy algorithm that achieves approximation ratio (h). A more practically relevant version is the online code assignment problem, where future requests are not known in advance. Our objective is to minimize the overall number of code reassignments. We present a (h)-competitive online algorithm, and show that no deterministic online algorithm can achieve a competitive ratio better than 1.5. We show that the greedy strategy (minimizing the number of reassignments in every step) is not better than (h) competitive. We give a 2-resource augmented online algorithm that achieves an amortized constant number of (re-)assignments. Finally, we show that the problem is fixed-parameter tractable.

Online OVSF Code Assignment with Resource Augmentation

“…At any time t > 0, we only know about the operations until t and have no information about any future operation C t with t > t. The problem is to find a sequence of code assignments such that the total number of reassignments is minimum. Erlebach et al [5] gave an O(h)-competitive algorithm for this problem, where h is the height of the OVSF code tree. They also proved that the lower bound on the competitive ratio of this problem is 1.5.…”

“…They also proved that the lower bound on the competitive ratio of this problem is 1.5. With resource augmentation [7], which means the online algorithm is allowed to use more bandwidth than the optimal scheme, a 4-competitive algorithm with a double-bandwidth code tree was given in [5] .…”

“…Each cell with a different color can use a different code tree for handling code requests and releases initiated at that cell. In order to achieve a constant-competitive ratio, a total of six code trees with full bandwidth will be needed using the resource augmentation approach given in [5], whereas 3.375 code trees with full bandwidth would be sufficient using the approach in Section 2. However, we will introduce here an algorithm with a competitive ratio of 5 that uses only 19/8 = 2.375 code trees of full bandwidth.…”

“…We consider the well-studied problem of Orthogonal Variable Spreading Factor (OVSF) code assignment in W-CDMA systems [5,6,9,10,12].…”

Online Bandwidth Allocation

“…Since the cost of a reallocation dominates this operation, the number of reallocations should be kept minimal. In [4] the authors considered the problem of minimizing the number of reallocations over a given schedule and showed that it is NP-hard to generate an optimal allocation schedule. In this paper, we show that the online version proposed in [4] can be solved in amortized complexity O(1) reallocations per request.…”