Boundary labeling connects each point site in a rectangular map to a label on the sides of the map by a leader, which may be a rectilinear or a straight-line segment. In conventional setting,
the labels along a side of the map form a single stack of labels in which labels are stacked consecutively one by one in a column, and the two end sides of a label stack must be prespecified the sides of the map.
However, such a setting may be in conflict with generation of a better boundary labeling, measured by the total leader length or the number of bends of leaders. As a result, this thesis relaxes this setting to propose the boundary labeling with {\em flexible} label positions, in which labels are allowed to be placed at any nonoverlapping location along the sides of the map so that they do not necessarily form only one single stack, and the two end sides of label stacks are made flexible too, so that they may adapts to the desired objective criteria.
In this scenario, we investigate the total leader length minimization problem and the total bend minimization problem under various variants, which are parameterized by the number of sides to which labels are attached, their label size, port types, and leader types. It
turns out that almost all the total leader length minimization problems using nonuniform-size labels are NP-complete except for one case, while the others can be solved in polynomial time.

近年來，圖形繪製為一個經常受探討的議題。尤其是以正交畫法受到注目，其只用到垂直與水平線段來做圖形的畫法，因此容易讓人辨識圖形的樣貌。在平面及非平面的圖形上，都有研究其正交畫法。在此，我們只針對非平面圖做討論。正交畫法中所討論的議題有許多，像是減少每條邊上所需的彎曲數量，減少圖中總共所需的彎曲數，減少邊之間的交叉數量，等等。在過去的研究中，圖形中節點的形狀，往往會根據其維度的變化，而有所改變。此外，節點上每邊之間的距離，也會因為維度過高，而有所減少。因此，為了針對節點能有固定大小，而且每邊之間也有固定的距離。我們提供了一個演算法，將每個節點放在格網上，並且以對角線的方向來做擺放。同時，線段則是畫在格線上，用以清楚區別每條線段。使得圖形能夠兼顧，節點形狀以及每邊之間有清晰的呈現。在此方式下，我們的目標在於，求得每邊最多所需要用到的彎曲數量，以及整個圖形最多所需要使用到的面積。所得到的結論為，在圖形中節點維度不超過6以上，每邊能以最多2次彎曲畫出；點維度不超過8以上，每邊能以最多3次彎曲畫出。面積則是限制在O(n2)的大小。此外，針對圖形是否能夠，以每邊皆無彎曲的方式下畫出，我們證明其為一個NPC的問題。

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