Time Complexity Analysis of the Binary Tree Roll Algorithm

Abstract

This paper presents the time complexity analysis of the Binary Tree Roll algorithm. The time complexity is analyzed theoretically and the results are then confi rmed empirically. The theoretical analysis consists of fi nding recurrence relations for the time complexity, and solving them using various methods. The empirical analysis consists of exhaustively testing all trees with given numbers of nodes and counting the minimum and maximum steps necessary to complete the roll algorithm. The time complexity is shown, both theoretically and empirically, to be linear in the best case and quadratic in the worst case, whereas its average case is shown to be dominantly linear for trees with a relatively small number of nodes and dominantly quadratic otherwise.

Space Complexity Analysis of the Binary Tree Roll Algorithm

“…The motivation for this paper was the fact that the binary tree roll algorithm, in either its CCW() or CW() variant, has so far been analyzed for time complexity [7] but not for space complexity. The latter is the goal of this paper and it will be done as follows, focusing on the CCW() variant.…”