考慮一 n 個 vertices 的 tree T，每個 vertex 上都有一個非負整數的 weight，而每一條 edge 上都有一個正整數的 cost。一個 T 的 partition 為在 T 上刪除一些 edges 後所形成的 subtrees。令 l 和 u 為兩個非負整數且 l <= u。一個 [l, u]-partition 為一每個 subtree 的 weight 都在 [l, u] 中的 partition。一個 p-[l, u]-partition 為一有 p 個 subtrees 的 [l, u]-partition。一個 partition 的 cost 定義為被刪除的 edge 的 cost 總和。本論為的重點為下列兩個問題： (1) 如何求出一個 p-[l, u]-partition 以及 (2) 如何求出一個 minimum-cost [l, u]-partition。對於問題 (1)，本論文提出一 O(n p^{4} loglog p / log^{2} p) 時間的演算法。對於問題 (2)，本論文證明它為 NP-hard 並提出一 O(n u^{2} loglog u / log^{2} u) 時間的 pseudo-polynomial 演算法。上述兩個演算法藉由套用目前最好的 (min, +)-convolution 演算法來加速原本的演算法，分別改進了之前 O(n p^{4}) 及 O(n u^{2}) 的結果。本論文也研究問題 (1) 及問題 (2) 的延伸問題： 如何求出一個 minimum-cost p-[l, u]-partition。對於這個延伸問題，本論文提出一 O(n u^{2} p^{2} loglog p / log^{2} p) 時間的演算法。

Consider a tree T with n vertices, in which each vertex is associated with a nonnegative integer weight and each edge is associated with a positive integer cost. A partition of T is a set of subtrees induced by removing some edges. Let l <= u be two nonnegative integers. An [l, u]-partition is a partition such that the total weight of each subtree is in [l, u]. A p-[l, u]-partition is an [l, u]-partition with p subtrees. The cost of a partition is the total cost of the removed edges. The focus of this thesis is the following two problems: (1) the problem of finding a p-[l, u]-partition and (2) the problem of finding a minimum-cost [l, u]-partition. For (1), an O(n p^{4} loglog p / log^{2} p)-time algorithm is presented. For (2), its NP-hardness is proved. In addition, an O(n u^{2} loglog u / log^{2} u)-time pseudo-polynomial algorithm is presented. The presented algorithms for (1) and (2) improve, respectively, the previous upper bounds from O(n p^{4}) and O(n u^{2}). The improvement is achieved by an efficient application of the current best algorithm for computing the (min, +)-convolution of two vectors to reduce the running time of the previous algorithms. This thesis also considers the following nature extension of (1) and (2): finding a minimum-cost p-[l, u]-partition. For this extended problem, an O(n u^{2} p^{2} loglog p / log^{2} p)-time algorithm is presented.

Abstract i
Contents iii
List of Figures v
List of Tables vi
Chapter 1 Introduction 1
1.1 Previous Results and Related Work 3
1.1.1 Previous Results 4
1.1.2 Related Work 6
1.2 Summary of Results 12
1.3 Organization of the Thesis 14
Chapter 2 The p-[l, u]-partition Problem on Trees 15
2.1 Notation 15
2.2 Ito et al.'s Algorithm 17
2.2.1 An O(np4u2)-time Algorithm 17
2.2.2 An O(np4)-time Algorithm 21
2.3 An Improved Algorithm for the p-[l, u]-partition Problem on Trees 26
2.3.1 The (min, +)-convolution 27
2.3.2 An Improved Algorithm 29
Chapter 3 The Minimum-cost (Maximum-cost) [l, u]-partition Problem on Trees 34
3.1 The Minimum-cost [l, u]-partition Problem on Trees 34
3.1.1 The NP-hardness of the Minimum-cost [l, u]-partition Problem on Trees 35
3.1.2 Lukes's Algorithm 37
3.1.3 An Improved Algorithm for the Minimum-cost [l, u]-partition Problem on Trees 40
3.2 The Maximum-cost [l, u]-partition Problem on Trees 42
3.2.1 The NP-hardness of the Maximum-cost [l, u]-partition Problem on Trees 42
3.2.2 A Pseudo-polynomial Time Algorithm 46
Chapter 4 The Minimum-cost (Maximum-cost) p-[l, u]-partition Problem on Trees 48
4.1 The Minimum-cost p-[l, u]-partition Problem on Trees 48
4.1.1 An O(np2u2)-time Algorithm 48
4.1.2 An Improved Algorithm 52
4.2 The Maximum-cost p-[l, u]-partition Problem on Trees 53
Chapter 5 Conclusions and Future Work 56
References 60

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