旅行商問題（TSP）是一個圖問題，已被廣泛應用於許多應用中，特別是在
運輸和物流方面。因為 TSP 是一個 NP 難題，所以最小化 TSP 算法的複雜
度是一個重要的問題。已經提出了許多啟發式算法來計算給定靜態圖的 TSP
路徑。但是對時間演化圖 （TEG）進行了有限的研究，其中圖可能由於更新
事件而隨時間變化，例如邊或頂點上的權重變化。這是一個更具挑戰性的問
題，因為 TSP 計算的速度必須足夠高才能趕上圖更新頻率。最近，為了應對
TEG 的挑戰，已經進行了許多研究工作。然而，它仍然是一個活躍且充滿挑
戰的研究課題。因此，在本論文中，我們通過詳細調查總結了 TEG 的最新
計算技術。我們從三個不同的研究社區收集這些技術：i）用於圖形分析的
數據挖掘社區；ii) 圖算法的理論社區；iii) 圖計算框架的計算社區。我們還
首次嘗試最小化在時間演化圖上求解 TSP 的計算時間。通過探索並行計算
能力和重用之前的計算結果，我們提出了一種分佈式增量 TSP 算法，該算
法可以在以頂點為中心的並行圖計算框架上實現，以有效地在大型變化圖上
找到 TSP 遊歷。我們的增量算法可以以最少的重新計算量保持最短的 TSP
旅程。通過我們的實驗評估，我們表明增量 TSP 算法比分佈式算法快 98%。
我們進一步改進了增量算法的結果，通過使用分而治之的範式來減小問題
的大小，並提出了分區算法 P-TSP 和 Pg-TSP。從我們的實驗結果中，我們
觀察到分區 TSP 算法 （P-TSP 和 Pg-TSP）比增量算法 （I-TSP 和 Ig-TSP）
更快，並且計算時間可以減少到至少一半更多數量的分區。我們還為 P-TSP
和 Pg-TSP 算法提出了三種劃分策略。這三種策略如下：邊尺寸屬性、頂點
尺寸屬性和 k-means 分區。其中，頂點大小屬性顯示最小計算時間。我們
還對所有增量和分區算法進行了複雜度分析。

Travelling salesman problem(TSP) is a graph problem that has been widely used in many applications, especially for transportation and logistics. Because TSP is a NP hard problem, minimizing the complexity of TSP algorithms is an important problem. Many heuristic algorithms have been proposed to compute the TSP tours of a given static graph. But limited studies have been done on time evolving graph~(TEG) where the graph can change over time due to update events, such as weight changes on edges or vertices. It is a more challenging problem because the speed of TSP computations must be high enough to catch up the graph update frequency.
Recently, many research efforts have been made with the aim to address the challenges of TEG. However it remains to be an active and challenged research topic. Therefore, in this thesis, we have summarize the state-of-art computation techniques for TEG by performing a detailed survey. We collect these techniques from three different research communities: i)The data mining community for graph analysis; ii)The theory community for graph algorithm; iii)The computation community for graph computing framework. We also make the very first attempt to minimize the computation time of solving TSP on time evolving graphs. By exploring parallel computing power and reusing previous computing results, we proposed a distributed and incremental TSP algorithm which can be implemented on vertex centric parallel graph computing frameworks to efficiently find TSP tours on large changing graphs. Our incremental algorithm can maintain shortest TSP tour with minimum amount of recomputation. Through our experimental evaluation, we have shown Incremental TSP algorithm is 98% faster than distributed algorithm. We further improve the result of incremental algorithm by using divide and conquer paradigm to reduce the problem size and proposed Partitioning algorithms P-TSP and Pg-TSP. From our experimental results, we have observed partitioning TSP algorithms~(P-TSP and Pg-TSP) are faster than the incremental algorithms~(I-TSP and Ig-TSP), and the computation time can be reduced to at least half with more number of partitions. We have also proposed three partitioning strategies for P-TSP and Pg-TSP algorithms. The three strategies are as follows: Edge size attribute, vertex size attribute and k-means partitioning. Among all of them, vertex size attribute showed the minimum computation time. We have also conducted complexity analysis for all incremental and partitioning algorithm.

1 Chapter 1: Introduction---------1

2 Chapter 2: Literature Review-----5
2.1 Graph processing-----5
2.1.1 Graph analytics-----5
2.1.2 Graph algorithms------9
2.1.3 Graphframeworks------13
2.2 Time evolving graph computation techniques-----16
2.3 TravellingSalesmanAlgorithms-----19
2.4 Summary---------19

3 Chapter 3: Distributed TSP Algorithms------21
3.1 Problem Definition-----22
3.2 D-TSP:DistributedTSP------23
3.3 Dg-TSP:DistributedgreedyTSP-----26
3.4 I-TSP:IncrementalTSPalgorithm-------29
3.5 Ig-TSP:IncrementalgreedyTSP-----33
3.6 P-TSPandPg-TSPalgorithm-------35
3.6.1 Graphpartitioningconstraints-------38

4 Chapter 4: Complexity Analysis------41

5 Chapter 5: Experimental Evaluation------44
5.1 Experimental evaluation of D-TSP, Dg-TSP, I-TSP and Ig-TSP----44
5.1.1 Experiment Design-----44
5.1.2 Discussion of results------45
5.1.3 Tuning parameter-------47
5.2 ExperimentsforP-TSPandPg-TSP-------48
5.2.1 Experiment results for partitioning strategies on P-TSP andPg-TSP----52

6 Chapter 5: Conclusion References-----58

References------60

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